Tone-Metric Analysis: A Novel Approach to the Study of ... · iii phantasticus preludes and...

Tone-Metric Analysis: A Novel Approach to the Study of

Form and Performance in North German Baroque Organ Music

by

Federico Andreoni

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Faculty of Music

University of Toronto

© Copyright by Federico Andreoni 2020

ii

Tone-Metric Analysis: A Novel Approach to the Study of

Form and Performance in North German Baroque Organ Music

by

Federico Andreoni

Doctor of Philosophy

Faculty of Music

University of Toronto

2020

Abstract

This dissertation studies the principles and applications of tone-metric analysis, a novel

analytical approach based on the assumption that beat (or tactus) is a combinatorial set of strong

and weak beats. As such, tactus can be studied using mathematical tools such as Pascal’s and

Sierpinsky’s triangles, which help to quantitatively describe and model, through graphic

constructs such as tone-metric waves and trees, the relationship between rhythmic, melodic, and

harmonic designs as unfolding functions of the tactus’s regularity. The specificity of these

functions, and their lengths and positioning within a piece, have consequences for the

performance and perception of music, the analysis of which is at the core of this dissertation.

Therefore, waves and trees can be used to describe phenomena such as the perception of tension

and relaxation in music, and to initiate the analysis of how agogic elements can be integrated and

rendered in performance.

Because of its ability to describe and graphically model the elusiveness of music’s unfolding in

time, tone-metric analysis is an ideal tool for the analysis, performance, and perception of

repertoires that display a high degree of phrasal and formal variety, and are improvisatory in

character. This dissertation studies primarily North German organ music, focusing on stylus

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phantasticus preludes and fantasias, chorale-based pieces, and fugues by Johann Sebastian Bach,

Nicholaus Bruhns, Dietrich Buxtehude, Nicholaus Hanff, and Johann Pachelbel. For comparison,

the dissertation also explores brief passages from later repertoires (classical, romantic, and

extended tonal) by Wolfgang Amadeus Mozart, Frédéric Chopin, and Alban Berg.

A major novelty of tone-metric analysis is its approach to the combinatorial and geometrical

nature and features of tactus, and the consequent generation of an algorithmic basis for the

analytical discourse. This approach raises questions about the nature of the relationship between

tactus and other musical parameters such as melodic, harmonic, and rhythmic elements, and

about their role in shaping phrasal and formal designs in music governed by a steady beat. The

thesis thus recommends a new look at the fundamental nature of tactus as a conceptually

combinatorial and fractal space for music.

iv

Acknowledgments

First and foremost, I would like to thank my supervisor Don McLean for his invaluable insights

and encouragement throughout my doctoral studies at the University of Toronto. Without him as

a mentor, this journey would have not been as fascinating, productive, and inspiring.

I would also like to thank the members of my doctoral committee, Kevin Komisaruk, Harald

Krebs (External Examiner), Ryan McClelland, and Mark Sallmen, for their insightful

suggestions and careful guidance during the dissertation’s writing process.

A special thank you to my wife Afra Saskia Tucker for her wonderful presence and unceasing

support; to my family and the many students, colleagues, and friends who have contributed to

my musical and intellectual journey; and to all the composers and performers who have enriched

my life through the beauty of their art.

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Table of Contents

Acknowledgments.......................................................................................................................... iv

Table of Contents ............................................................................................................................ v

List of Tables ................................................................................................................................ vii

List of Figures .............................................................................................................................. viii

List of Examples ............................................................................................................................ ix

Chapter 1 Introduction to a new theory of meter and form ......................................................... 1

1.1 Tactus, geometry, and performance ............................................................................................ 3

1.2 Research components .................................................................................................................. 4

1.3 Terminology ................................................................................................................................. 7

1.4 Repertoire .................................................................................................................................. 11

1.5 Corollary to Chapter 1 (a primer on tone-metric graphs) ........................................................ 12

1.6 Outline of the dissertation ......................................................................................................... 41

Chapter 2 Historical and theoretical perspectives ..................................................................... 42

2.1 From tactus to meter and beyond in analytical methodologies across the centuries ................ 43

2.2 Meter, form, and hierarchies ..................................................................................................... 44

2.3 Roman numeral analysis and form ............................................................................................ 46

2.4 Theorizing perception ................................................................................................................ 48

2.5 The origin of efflorescence and tone-metric waves ................................................................... 51

2.6 Scientific developments and studies in beat perception ............................................................ 56

2.7 Historic-analytical contextualization of the repertoire analyzed .............................................. 60

2.8 Perspectives on the relationship between performance practice and music analysis ............... 62

2.9 Conclusions................................................................................................................................ 66

Chapter 3 Conceptualizing the musical space ........................................................................... 67

3.1 Tactus, combinatorics, and fractal spaces ................................................................................ 68

3.2 Using fractals to model tactus ................................................................................................... 82

3.3 Conclusions................................................................................................................................ 84

Chapter 4 Analytical method ..................................................................................................... 87

4.1 Modeling purely binary and mixed binary-ternary types of metric organizations .................... 88

4.2 Pure binary type of metric organization .................................................................................... 88

4.3 Mixed binary-ternary type of metric organization .................................................................... 94

4.4 Pivots and moments of articulation ........................................................................................... 97

4.5 Tone-metric trees ..................................................................................................................... 101

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4.6 Summary and further thoughts ................................................................................................ 103

Chapter 5 Analytical applications............................................................................................ 108

5.1 Durch Adams Fall, BuxWV 183 ............................................................................................... 109

5.2 Chorale-fantasia on Nun freut euch (mm. 45–80) ................................................................... 117

5.3 Magnificat primi toni (mm. 1–11; 76–91; and 139–141) by Buxtehude ................................. 125

5.4 Magnificat primi toni (first verse; mm. 1–17) by J. Praetorius ............................................... 132

5.5 Fugue on the Magnificat primi toni by Pachelbel ................................................................... 136

5.6 The Prelude in e minor by Bruhns. Mm. 1–5 and 6–10. .......................................................... 140

5.7 Conclusions.............................................................................................................................. 145

Chapter 6 Expanding the repertoire ......................................................................................... 146

6.1 Chorale harmonization on O Haupt by Bach .......................................................................... 146

6.2 Piano sonata in C major, K. 279/189d, iii, by Mozart. Mm. 1–10. ......................................... 152

6.3 Prelude in E minor, Op. 28 no. 4, by Chopin. Mm. 1–13. ....................................................... 155

6.4 Sonata, Op. 1, by Berg. Mm. 1–11. .......................................................................................... 162

6.5 Conclusions.............................................................................................................................. 168

Chapter 7 Conclusions ............................................................................................................. 169

7.1 Summary .................................................................................................................................. 169

7.2 An essay ................................................................................................................................... 171

7.3 How does tone-metric analysis fit into the field of music theory? ........................................... 175

7.4 Personal remarks ..................................................................................................................... 177

7.5 What next? ............................................................................................................................... 178

Bibliography ............................................................................................................................... 181

vii

List of Tables

Table 1.1. Synoptic table summarizing the core repertoire analyzed in this dissertation. ............ 12

viii

List of Figures

Figure 3.1. Pascal’s triangle. ......................................................................................................... 71

Figure 3.2. Pascal’s triangle with binary combinations and permutations of T and A. ................ 72

Figure 5.1. The tablature of the Prelude in E minor by Bruhns. Mm. 1–5. ................................ 141

Figure 5.2. The tablature of the Prelude in E minor by Bruhns. Mm. 1–5. ................................ 144

ix

List of Examples

Example 1.1. Score and level-1 tone-metric analysis of the Prelude in C major. Mm. 1–3. ........ 15

Example 1.2. Score and tone-metric levels 1 and 2 of the Prelude in C major. Mm. 1–3. ........... 16

Example 1.3. Score and tone-metric levels 1–3 of the Prelude in C major. Mm. 1–3. ................ 17



Example 1.6. Analysis of the Prelude in C major (part 1 of 4). .................................................... 19

Example 1.7. Schenker’s analysis of the Prelude in C major. Mm. 1–11. ................................... 25

Example 1.8. Score and tone-metric level 1 of the Fugue in C major. Mm. 1–3. ........................ 26

Example 1.9. Score and tone-metric levels 1 and 2 of the fugue in C major. Mm. 1–3. .............. 27

Example 1.10. Score and tone-metric levels 1– 3 of the fugue in C major. Mm. 1–3.................. 27

Example 1.11. Score and tone-metric levels 1–4 of the fugue in C major. Mm. 1–3................... 28



Example 1.14. Analyses of the fugue in C major (part 1of 5). ..................................................... 29

Example 1.15. The characteristic melodic-rhythmic design of the fugue’s subject. M. 1.3......... 35

Example 1.16. Tone-metric and harmonic analyses of BuxWV 142. Mm. 101–113. .................. 38

Example 4.1. Level-1 tone-metric articulation based on y=2x–1 in Ach Gott. Mm. 1–7. ............ 89

Example 4.2. Level-1 tone-metric articulation of the chorale prelude on Ach Gott. Mm. 1–7. ... 89

Example 4.3. Partial quarter-note level tone-metric articulation of Ach Gott. Mm. 1–7. ............. 90

Example 4.4. Quarter-note level tone-metric articulation of Ach Gott. Mm. 1–7. ....................... 91

Example 4.5. Eighth-note level tone-metric articulation of Ach Gott. Mm. 1–7. ......................... 92

Example 4.6. Tone-metric articulation of Ach Gott. Mm. 1–7. .................................................... 92

Example 4.7. Tone-metric articulation of Ach Gott. Mm. 1–7. .................................................... 93

Example 4.8. Tone-metric articulation of the fugue in G major at the 3/8 level. Mm. 1–6. ........ 95

Example 4.9. Full tone-metric analysis of the fugue in G major. Mm. 1–6. ................................ 96

x

Example 4.10. Tone-metric analysis and pivots of Ach Gott. Mm. 1–7. ...................................... 98

Example 4.11. Wave analysis and tree structure of Ach Gott. Mm. 1–7. ................................... 102

Example 5.1. Analysis of the chorale prelude on Durch Adams Fall (part 1 of 4). ................... 109

Example 5.2. Analysis of chorale-fantasia on Nun freut euch. Mm. 45–80 (part 1 of 4). .......... 117

Example 5.3. The Magnificat primi toni intonation. .................................................................. 125

Example 5.4. Tone-metric analysis of the Magnificat primi toni. Mm. 1–11. ........................... 126

Example 5.5. Tone-metric analysis of the Magnificat primi toni. Mm. 76–91. ......................... 127

Example 5.6. Tone-metric analysis of the Magnificat primi toni. Mm. 139–141.1. .................. 130

Example 5.7. Tone-metric analysis of the Magnificat primi toni. Mm. 135–141. ..................... 130

Example 5.8. 2/2 tone-metric analysis of the Magnificat primi toni (verse 1). Mm. 1–17. ....... 133

Example 5.9. 4/4 tone-metric analysis of the Magnificat primi toni (verse 1). Mm. 1–17. ....... 134

Example 5.10. Tone-metric analysis of the fugue on the Magnificat primi toni (part 1of 4). .... 136

Example 5.11. Tone-metric analysis of the Prelude in E minor by Bruhns. Mm. 1–5. .............. 140

Example 5.12. Score and TBT tone-metric analysis of mm. 6–10. ............................................ 143

Example 5.13. TBT and TTB wave of the Prelude in E minor by Bruhns. Mm. 6–10. ............. 144

Example 6.1. Harmonic reduction by Lerdahl and Jackendoff of O Haupt, BWV 244. ............ 148

Example 6.2. Tone-metric analysis of O Haupt. Mm. 1–4. ........................................................ 150

Example 6.3. Tone-metric and form-functional analysis of K. 279/189d, iii. Mm. 1–10. ......... 153

Example 6.4. Tone-metric analysis and tree of the E minor Prelude, Op. 28 n. 4. Mm. 1–13. .. 156

Example 6.5. Excerpt of Schachter’s reduction the E minor Prelude, Op. 28 n. 4. Mm. 1–13. . 157

Example 6.6. Score and tone-metric analysis of the Sonata by Berg, Op. 1. Mm. -1–11 .......... 162

Example 6.7. The motivic annotations of Schmalfedt on the sonata, Op. 1. Mm. 1–4. ............. 165

Example 6.8. The three-level reduction by Schmalfeldt of the Sonata, Op. 1. Mm. 1–11. ........ 166

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Chapter 1

Introduction to a new theory of meter and form

I have always been fascinated by the rich and kaleidoscopic variety of harmonic, melodic, and

textural designs of North German organ music. As a child, I enjoyed learning scores by

composers such as Dietrich Buxtehude, Nicholaus Bruhns, and Heinrich Scheidemann; isolating

individual voices of the music to study their designs; and transcribing sections of scores,

maintaining the bass line as such, and rearranging the upper voices in a variety of ways, often

filling in long values with diminutions and embellishments in a re-creative process that spoke to

my personal interests in improvisation.

This analytical and re-compositional work soon made me aware of the modular and

combinatorial nature of chords, in which notes, with the exception of the bass part, can be

rearranged at will to generate textural variations without changing their basic harmonic

functions. Interestingly, the combinatorial features of chords are entirely contextualized within

the piece’s metric framework, which is established a priori by composers and improvisers, and

which provides a frame of reference for the unfolding of rhythmic, and, ultimately, harmonic and

melodic functions. Musicians who wish to study and practise composition or improvisation1 in

North German Baroque style (for example, to improvise a prelude in the so-called stylus

phantasticus), do need to refer to the metric framework determined by the underlying beat in

order to distribute strong and weak accents in a harmonically and stylistically coherent manner,

1 Improvisation can be considered as an extemporary form of composition and, therefore, as based on the same

metric, as well as harmonic, melodic, and rhythmic principles.

2

to determine the unfolding of harmonic patterns based on those accents (e.g., to establish the

workings of consonance and dissonance relationships), and to determine melodic phrases,

cadential patterns, and, ultimately, the formal organization of pieces.

While the role of the beat in defining local treatment of consonance and dissonance within or

across measures and its ability to shape rhythmic organization have been amply discussed in the

literature, to my knowledge only a relatively small amount of analytical-theoretical research has

systematically dealt with its role in shaping rhythmic units, harmonic patterns, and melodic

designs beyond its immediate effects within or between measures. In this dissertation, I address

this gap by: (1) studying the beat as a form of distribution of strong and weak accents; (2)

exploring how this metric distribution influences the shape of rhythmic patterns, melodic

phrases, and harmonic designs; and (3) investigating how those metric structures influence both

performance practice and listeners’ appreciation of music.

The goal of this dissertation is to study beat and rhythm from a mathematical perspective to

understand if and how the mathematical organization of those elements can inform our analysis

and performance of music. As such, this work should be understood as a Versuch,2 or Essay,

conceived as a trial or attempt. The association I make between this dissertation and the

longstanding and varied literary form of the Essay is two-fold: on the one hand, I aim at freely

sketching and elaborating in a personalized way ideas about otherwise usual subjects (following

the examples of Michel de Montaigne and Francis Bacon);3 on the other hand, I aim at

systematizing my personal inquiry into a logically organized system (following John Locke and

David Hume). In the case of this dissertation, I mapped distant associations between meter and

combinatorial properties of a given set of numbers onto each other, to elaborate a complex

system of principles and an algorithmic approach to music analysis for the interpretation of the

score. Therefore, the originating algorithm can be understood as a tool for the creation of new

knowledge, to be abandoned as new properties of music appear to emerge from the mapping of

that knowledge onto the score; in other words, once the mathematically and non-musically based

analytical tool determined by the algorithm is in place (see Chapter 3), the analysis can be

2 For a text on the Versuch tradition in music, see Carl P. E. Bach. Essay on the true art of playing keyboard

instruments, William J. Mitchell, ed. (First ed., New York: W.W. Norton & Company, 1949). 3 For a discussion about these authors and the following ones in this paragraph, refer to Chapter 7.

3

carried out as a discussion of the musical properties that emerge from the algorithmic

organization of the underlying combinatorial principles (see Chapters 4, 5, and 6).

In the next paragraphs, I introduce the main topics that are investigated in this dissertation. A

“Corollary” section provides a primer on the analytical approach, showing the resultant graphs

for three varied examples.

1.1 Tactus, geometry, and performance

In his Der vollkommene Capellmeister (1739),4 Johann Mattheson (1681–1764), probably the

most influential and well-known theorist of the German Baroque, explains that the early

seventeenth-century German musical beat, then called Takt (meter), was organized in the same

way as Renaissance tactus. According to Mattheson’s view, meter, or tactus, could be duple

(e.g., modern 2/2 time signature) or triple (e.g., modern 3/4 time signature); duple meter was

made of two components of equal length, downbeat (or thesis) and upbeat (or arsis): TA; triple

meter was also made of thesis and arsis, but the former was twice as long as the latter: TTA.

According to this perspective, tactus can be understood as a series of discrete binary impulses

(either T or A) of different lengths; as I will show in detail in Chapter 3, those combinatorial and

permutational properties5 can be studied through numerical-geometrical constructs such as

Pascal’s and Sierpinsky’s triangles. Through the novel analytical methodology here presented

and called tone-metric analysis, this dissertation shows how the study of those combinatorial and

permutational features of tactus6 can inform us about tactus’s role in building rhythmic, melodic,

and harmonic designs, thus influencing the overall unfolding of music beyond its algorithmically

generated metric output. This ability of tactus to influence all aspects of the musical texture (i.e.,

meter, harmonies, and melodies) can be graphically represented through (1) tone-metric waves,

4 Ernest C. Harriss. Johann Mattheson’s Der vollkommene Capellmeister: a revised translation with critical

commentary (Ann Arbor: UMI Research Press, 1981), 365. 5 Combinatorial permutation. 6 Combination: any arrangements of elements. Permutation: an ordered set of elements. From now on, I will use the

word beat as the individual pulsations (e.g. modern time signature units; quarter note pulsation in a 4/4 time

signature), and tactus to indicate the regular repetition of a fixed number of strong and weak accents (T and A) per

measure (the same as Mattheson’s meter).

4

which describe the sense of motion and formal organization of pieces, and (2) tone-metric trees,

which describe the relationship within and among waves and phrase groupings (these terms will

be defined in the Terminology section in Chapter 1).

Tone-metric analysis introduces two main novelties into the panorama of contemporary

analytical theories. First, it is based on an algorithmic notion of music-unfolding, as it argues that

the organization of tactus-based music repertoire can be interpreted as determined by a specific

set of numerical parameters that are derived from a combinatorial reading of music and that

remain constant throughout a piece. Such a reading of music generates unique graphic

representations (tone-metric waves and trees) for each piece; these representations can be used as

a basis for analytical discourse, and as a starting point for the study of the performance and

perception of music. Second, tone-metric analysis is based on combinatorial, and not musical,

principles, and can therefore be applied to pieces independently of their style and genre. The

non-musical basis for tone-metric analysis allows for the generation of analytical considerations

that can be used along with other forms of analysis—such as Schenkerian, roman numeral, and

form-functional approaches.7

In the next sections of this chapter, I introduce the research components of this dissertation and

describe the core concepts on which tone-metric analysis is based; moreover, I explain the key

definitions employed in the analytical method and introduce the repertoire analyzed.

1.2 Research components

The dissertation comprises three research components: (1) the analysis of the combinatorial and

geometrical properties of tactus and rhythm; (2) the creation of tone-metric waves and trees—

analytical tools used to study the relationship between metric organization, melodic designs, and

harmonic patterns; and (3) the analysis of seventeenth-century North German organ music (with

7 The study of the interaction between tone-metric analysis and these independent analytic streams will be briefly

analyzed; however, it is beyond the scope of this dissertation to study the full extent of the complex web of

analytical ramifications stemming from these relationships.

5

a brief exploration of the potential application of the approach to the analysis, performance, and

teaching of music beyond that core repertoire).

The first research component establishes the basic principles behind the creation of tone-metric

graphs. As noted earlier, during the German Baroque tactus first and meter later were described

as binary forms of organization made of thesis (T), or downbeat, followed by arsis (A), or

upbeat. In this dissertation tactus is considered a specific permutation of the two-element

combination of the binary set8 made of T and A (e.g., in 2/2 the specific permutation TA among

the 4 combinations TA, AT, AA, and TT of the binary set created by T and A). In this

dissertation, I study the combinatorial and permutational features of binary sets by using Pascal’s

triangle and its isomorphic fractal geometrical variant, Sierpinsky’s triangle; when mapped onto

Pascal’s and Sierpinsky’s triangles, each meter produces a unique geometrical configuration,

which can be used to generate a geometric sequence specific to that time signature.9

The second research component, which constitutes the core of the theoretical basis for the

analytical work of the dissertation, deals with applications of the aforementioned geometric

sequences to the creation of the analytical algorithm. The mapping of an algorithm-based

distribution of the digits of each sequence onto the score allows for the segmentation of the score

into metric layers, which are then superimposed to form what I call tone-metric waves. Those

waves, which are graphic representations of the superposition of the mathematically generated

layers of metric information, are descriptors of the metric unfolding and segmentation in music.

Because the layering of the sequence-based organization of metric units is generated by the

interplay between the regularity of tactus and the variety of rhythmic patterns of the piece, waves

show a variety of shapes and depths, whose articulation is characterized by what I call pivots, or

moments marking the end of a crest and the beginning of a wave’s descending path. An

alternative way to describe waves is through what I call tone-metric trees, or tree-like graphic

structures connecting the layers of waves and designed to help us interpret the relationship

between the metric structure of a piece (described by the waves), and the piece’s harmonic,

melodic, phrasal, and formal articulation.

8 Set of elements that can be of only two types. 9 See definitions in the Terminology section. Combinatorics and fractals have often been used to describe musical

phenomena, from melodic structures to pink noise, and to compose music; however, to my knowledge, they have

never been employed for the analytical description of the relationship between tactus and sonic events.

6

Waves, pivots, and trees contribute to the analysis of efflorescence, which can be defined as the

variety of shapes and depths of waves and used as a descriptor of how metric parameters (tactus

and rhythm) influence the harmonic and melodic unfolding in music. From an experiential point

of view, efflorescence graphically models the sense of stretching and compression of melodic

and harmonic phrases, describing the perceptual byproduct of rhythmic saturation of metrical

subdivisions. In other words, the term efflorescence describes both the layering of metric units in

waves, and the sense of perceived expansion and contraction generated through the sieve of

musical parameters (i.e., harmonies and melodies).

Analogously to how integral calculus provides a mathematical description for continuous

change, efflorescence describes both the position and momentum10 of musical events as discrete

points in time, and the organization of those events as parts of continuous musical motion. As

such, efflorescence does not explain form as a phenomenon generated by harmonic or melodic

functions alone, nor does it provide the reader with prototypical analytical categories and labels

(e.g., “sentence” or “exposition”). Instead, it generates a cross section of the piece’s metric, and,

therefore, harmonic and melodic flux, which can be used to analyze phrasal articulations.

Note that in tone-metric analysis the term phrasal articulation differs from a Classical definition

of the same; while in the analysis of Classical repertoire, phrases are generated by defined

harmonic and melodic boundaries, tone-metric phrasal articulations do not necessarily have to be

defined by those boundaries. For example, a tone-metric wave can end in the middle of a

melodic design or a harmonic progression, as it is solely generated by the metric structure of the

piece and is not determined by specific melodic and harmonic elements. The comparison and

integration of the information provided by the tone-metric wave, and the harmonic and melodic

analyses, are used in tone-metric analysis to explain why and how listeners often perceive a

sense of projection, or pivoting, rather than ending, of a musical idea into the next phrase.

In tone-metric analysis, melodic and harmonic paths are interpreted in relationship to the

underlying combinatorial and cyclical organization of meter, which leads to their recursive

collapse into single pivotal moments throughout the piece. This collapse can be seen as

metaphorically analogous to the notion of collapse of the wave function in physics (the collapse

of wave functions form at a specific observation from multiple Eigenstates, or characteristic

10 The impetus gained by an object in motion.

7

possible wave states, to a single one). In tone-metric analysis, the collapse creates a variety of

potential readings of formal organization in music, and finds interpretational fulfillment in the

specificity of each performance, which is one among many readings of the score.11

Finally, the third research component of the dissertation relates to tone-metric analysis as a

system based on an algorithmic approach grounded in combinatorial-numerical features of

tactus, rather than an approach based on musical relationships such as melodic and tonal rules; as

a result, tone-metric analysis can model any repertoire based on a regular meter, independently

of its style and genre. For this reason, I briefly show by way of illustration (in Chapter 6) how

tone-metric analysis can be: (1) applied to study Classical, Romantic, and post-tonal repertoires;

and (2) used as a complement to other analytical approaches. Since it is not based in musical

principles, tone-metric analysis does not conflict with other music-analytical theories; its

findings can be used to provide a complex and nuanced view of pieces by combining other

analytical methods with its tactus-based approach.

1.3 Terminology

In this section, I summarize and define key terms that are used throughout the dissertation, and

that represent new or modified music-theoretical concepts.

Beat, tactus, meter, and rhythm. In this dissertation, beat indicates the regular pulsations in a

piece (e.g., the quarter note pulsation in a 4/4 time signature); tactus indicates the regular

repetition of strong and weak accents (T and A) per measure (the same as Mattheson’s meter,

roughly corresponding to indication presented by the time signature of a piece nowadays: e.g.,

4/4); rhythm describes the individual discrete relational time units that quantify the length of

each sonic event (e.g., an eighth note); and meter (not to be confused with Mattheson’s meter)

describes the integration of tactus and rhythm into a general descriptor of time articulation in a

11 The development and explanation of these concepts through the aid of graphic representation will be provided in

Chapter 3.

8

piece (e.g., tone-metric waves are descriptors of meter, as they describe rhythmic unfolding as a

function of tactus).

Tone-metric wave.12 Tactus and rhythm combine to generate meter, which is a descriptor of the

unfolding of rhythmic values (and, indirectly of the melodic and harmonic functions attached to

those values) as functions of tactus. That unfolding can be represented as a tone-metric wave, a

waveform graph representing the moment-to-moment changes in the relationship between tactus

and rhythm as the piece unfolds. The wave thus becomes a descriptor of how shorter and longer

rhythmic values are affected by tactus in generating phrasal and formal articulations, and a form

of metric analysis against which analysts and performers can interpret the unfolding of melodic

and harmonic components of music.

Tone-metric tree. Tone-metric waves are generated by the buildup of later material from the

previous material. As such, they analytically describe music-unfolding as an interconnected

continuum. Based on the layered structure of waves, we can build tone-metric trees, which are

branching structures representing how a note within a piece can be organized and hierarchized in

relation to surrounding ones to create groups of sonic events based on tone-metric principles. As

such, tone-metric trees can also be used as an indicator of the hierarchy of melodies and

harmonies within the metric context, as they show (at an analytical glance, so to speak) how

events occurring in the future are related to previous events, and can provide insights into chord

distribution as functions of a tone-metric reading of the piece.

Efflorescence. A phenomenon that emerges from the reading of tone-metric waves, pivots, and

trees. It can be seen as a way to account for the characteristic surface flourishes in the repertoire.

More precisely, efflorescence is a measurable descriptor of both the amplitude of waves, as a

layered superposition of algorithmically generated tone-metric levels (e.g., counting the number

of levels at any given moment), and as the influence of metric articulation on the unfolding of

the melodic-harmonic texture, describing moments of textural saturation and improvisational

flourish, such as the flowery post-cadential patterns characteristic of North German Baroque

organ repertoire.

12 From now on, alternatively indicated as tone-metric waves or simply waves.

9

Tone-metric analysis. The analytical methodology presented in this dissertation. This approach is

based on the principle that metric organization in music can be modeled by combinatorial and

permutational principles of beat organization, and that such organization can be used as a

platform for the interpretation of music-unfolding. Tone-metric analysis understands musical

scores, improvisations, and performances as phenomena built on meter, or as a series of discrete

units of rhythmic information13 organized according to pre-determined accent-based structures.

The unfolding of these units generates a continuous flow of musical output that can be

graphically described as a series of connected waveforms. Because those waves are not

descriptors of individual harmonic and melodic parameters, but functions of the distribution of

metric elements in time to form the musical texture of pieces, tone-metric analysis can be said to

be a tool for the investigation of compositional processes and musical form from a metric

perspective.

Tone-metric pivot. A tone-metric pivot is the point where the end of a wave crest and the

beginning of the descent towards the subsequent trough occur. Pivots indicate articulations in the

texture of pieces that describe how shorter and longer rhythmic values affect the steady

underlying tactus to generate phrasal and formal articulations. As I will show in Chapters 4, 5,

and 6, tone-metric pivots are useful to describe how standard formal elements (e.g., sentence and

period) can be contextualized within the musical texture as a function of tactus.

To conclude, tone-metric analysis is a multi-step process of filtering and mapping of metric

structures. As Figure 1 summarizes, we map the originating tactus to generate a specific numeric

sequence for each time signature (e.g., 1, 2, 3, 5, 9, 17, and so on; see the Corollary to Chapter 1,

and Chapter 3, for more details regarding how those sequences are generated). The sequences are

the result of a mapping process that transforms the familiar linear metric distribution of tactus

into its fractal analogue by mapping its combinatorial properties (tactus is a fundamentally

binary set of strong or weak beats) onto Pascal’s and Sierpinsky’s triangles to generate specific

numeric sequences. Each tactus produces a specific sequence, whose digits describe the position

at which each note is found within the piece. Once mapped back onto the score, the larger-and-

larger gaps between adjacent digits within each sequence are filled by the metric and rhythmic

13 Mansuripur Masud, Introduction to information theory (Englewood Cliffs: Prentice-Hall, 1987). Information

theory is the mathematical study of the coding of information as sequences of symbols and impulses, and of how

fast that information can be transmitted, for example, through computer circuits.

10

activity of the music. The sequence repeats within these gaps to create self-similar and recursive

(i.e., fractal) metric units, at subsequent levels (e.g., quarter notes, eighth notes, sixteen notes).

By feeding those numeric sequences and the piece’s rhythmic articulation into an algorithm (see

Corollary), and mapping the resulting data onto the score, we generate tone-metric waves. These

are graphic representations of the multi-level fractal analogue of the originating linear metric

structure of the piece. They are constructed according to the fractal perspectives of recursivity

and self-similarity; that is, they are built on the same algorithmic procedure at all levels of

magnification (e.g., whether we consider a one-measure unit or a multi-measure “one-page”

wave). Finally, waves can be read through moments of articulations (or breaks in the levels)

known as “pivots,” and organized in hierarchical form through the expansion of “tree structures”

that result in a modeling of the unfolding flow of tactus through the complexities of rhythmic

surface described as a process of “efflorescence.”

Figure 1. Diagram summarizing the theoretical background of tone-metric analysis.

11

1.4 Repertoire

In this section, I present the repertoire treated in this dissertation, and explain the rationale

behind its selection.14 The repertoire selected for and analyzed in this dissertation is, despite the

large number of works examined in preparation for this study,15 quite small. The final selection,

which is in no way meant to be exhaustive or representative of the enormous variety of organ

compositions produced during the German Baroque, seems nonetheless appropriate to represent

the insights of the analytical method through the study of significant cases. The selection offers a

glimpse into the variety of genres, formal designs, and compositional techniques, and it gives the

reader an idea of how tone-metric analysis approaches various formal situations and performance

issues. The choice of pieces by different composers speaks to the desire to apply the method to a

variety of idiomatic patterns and stylistic situations. To summarize, the repertoire was selected to

showcase how tone-metric analysis can provide analytical insights into both a variety of forms

(e.g., chorale-fantasia, toccata, and Magnificat setting) and textures (e.g., imitative and

homophonic writing, accompanied solo chorale tunes, and improvisatory and rhapsodic

character).

A question that arose during the preparation of this dissertation was whether the tone-metric

analytical methodology could be applied to repertoires beyond seventeenth-century organ music

as well. To answer this question, I have dedicated one chapter (Chapter 6) to the application of

tone-metric analysis to repertoire later than Baroque music for comparative purposes, examining

excerpts from pieces by composers from later periods: the Piano Sonata in C major (K.

279/189d, iii) by Mozart, the Prelude in E minor (Op. 28, no. 4) by Chopin, and the Piano Sonata

Op. 1 by Berg.16 In these analyses, I show that the tone-metric method reveals several insights

14 For all scores and analytical-musicological sources related to the repertoire analyzed, see the bibliography. All

scores and tablatures used in this document are from IMSLP. https://imslp.org/wiki/Main_Page. Last accessed 15

September 2020. 15 A summary and catalogue of the major composers and organ works of the time can be found in Corliss R. Arnold,

Organ literature: a comprehensive survey (Metuchen: Scarecrow Press, 1995). 16 One exception to this organization is constituted by my analysis of an excerpt from the four-part chorale

harmonization of the chorale tune O Haupt voll Blut und Wunden (BWV 244–63), which I have analyzed with the

tone-metric method in Chapter 6. This harmonization allowed me to bridge with the analytical approach to Baroque

music presented in earlier chapters with the broader analytical discourse (applicability of tone-metric analysis in

support of other analytical techniques) related to later repertoires.

12

distinct from approaches more typically used to study this repertoire. Chapter 6 thus illustrates

how tone-metric analysis can be applied beyond Baroque repertoire, and shows how insights into

later repertoires can reveal analytical elements that can be mapped onto existing analytical

approaches to the same repertoire.

The following table summarizes the core repertoire studied in this dissertation:

Author Title Location Reason for inclusion

J. S. Bach

(1685–1750)

Fugue in G major

(BWV 577)

Ex. 4.8–4.9 Study of one-metric wave construction

N. Bruhns

(1665–1697) Prelude in E minor Ex. 5.11–5.13 Study of metric organization and performance

practice D. Buxtehude

(1637–1707) Chorale-fantasia on

Nun freut euch

(BuxWV 210)

Ex. 5.2

Study of phrasal and formal organization, echo

effects, harmonic structures, and performance

practice Chorale-prelude on

Durch Adams Fall

(BuxWV 183)

Ex. 5.1 Study of the relationship between pivots and tone-

metric waves, and the articulation of musical

phrases Magnificat primi toni

(BuxWV 203) Ex. 5.3–5.7 Study of the role of tone-metric analysis in

informing the analysis of sectional subdivisions of

pieces N. Hanff

(1663–1711) Chorale-prelude on

Ach Gott Ex. 4.1–4.7 Study of the underlying fractal structure of tactus

works J. Pachelbel

(1653–1706)

Fugue on the

Magnificat primi toni

Ex. 5.10 Study of texture in fugal writing

J. Praetorius

(1586–1651)

First verse of the

Magnificat primi toni

Ex. 5.8–5.9 Study of metric structure

Table 1.1. Synoptic table summarizing the core repertoire analyzed in this dissertation.

1.5 Corollary to Chapter 1 (a primer on tone-metric graphs)

Before venturing into the intricacies and details of the analytical work of this dissertation, I

would like to give the reader a general idea of how tone-metric graphs are constructed, and

inform our analysis. Tone-metric analysis describes how tactus and rhythm unfold; their

unfolding is not necessarily linked with how harmonic and melodic phrases unfold (and,

therefore, they are not related to a more traditional view of musical form); for example, it is not

uncommon to see melodic or harmonic phases end, and an underlying stream of fast notes

continue into the next musical phrase. The interplay between alignment and misalignment of

13

metric factors (i.e., both tactus and rhythm) with harmonic and melodic ones generates a sense of

stability and closure or of tension and instability, respectively.

Tone-metric analysis, which aims at modelling those perceptual phenomena, describes this

variety of states through the use of graphic waves, which are wavelike surfaces built through

algorithmically generated layers of numeric information (more on this in Chapter 3). Tone-

metric waves describe metric units created by the integration of the steady underlying tactus and

the varied rhythmic patterns found within each individual piece. According to this perspective,

waves (and tone-metric analysis more generally) can be understood as a metric analogue to more

traditional form-analytical methodologies based on harmonic and melodic considerations such as

roman numeral or form-functional analyses. Ultimately, the analytical output of tone-metric

analysis is meant to be compared with melody- and harmony-based analytical approaches, as it is

through this interaction that a fuller view of the piece’s structure surfaces.

In this corollary, which anticipates the analytical work explained in depth in Chapters 3, 4, 5, and

6, I analyze three pieces: Bach’s Prelude and Fugue in C major, BWV 846, from Das

wohltemperierte Klavier (“The Well-Tempered Clavier”), and mm. 101–113 of Buxtehude’s

Praeludium in E minor, BuxWV 142). These three pieces/excerpts exemplify three different

forms of musical organization: the prelude by Bach is a broken-chord four-part harmonization

moving at a steady sixteenth-note pace; the fugue is a short imitative piece; and the section from

the Buxtehude prelude is an example of the improvisational character typical of the so-called

stylus phantasticus. I analyze those three pieces and excerpts through the study of tone-metric

waves, and with the support of roman numeral and formal analyses for comparison.

As we will see in more detail in Chapter 3, the first step in tone-metric analysis consists of

transforming the familiar linear rhythmic distribution of beats into their fractal variant, by

considering the combinatorial features of binary and ternary tactus, and mapping those features

onto Pascal’s triangle, and both modulo-2 and modulo-3 based Sierpinsky’s triangles. This

mapping generates arrays of digits that are unique to each tactus. The reading of those arrays

allows us to determine specific geometric sequences that, mapped back onto the score, can be

used to create a fractal analogue to the familiar linear succession of beats.

More specifically, each metric unit (e.g., the quarter note in 4/4) occupies a specific position

within a piece: for example, in 4/4 quarter notes occupy positions 1, 2, 3, 4, 5, 6, and so on; this

14

string of positions is described by the formula y=x+1. After the filtering process provided by

tone-metric analysis, this familiar linear-artihmetic distribution of 4/4 quarter notes becomes the

sequence 1, 2, 3, 5, 9, 17, and so on, which is described by the formula y=2x-1.

Because of the larger-and-larger gaps between adjacent digits in the tone-metric fractal version

of metric distribution, the rhythmic saturation of pieces produces not a linear but a curved

distribution of metric units (e.g., 4/4 quarter notes) and their subdivisions (e.g., eighth- and

sixteenth-notes) across a piece (the specific rules for creating these curvilinear graphs are

outlined in detail in Chapter 3). These graphs, which are a descriptor of the effects of tactus on

rhythmic values, are called tone-metric waves (see top analysis in Figure 1.1), are unique to

tone-metric analysis and differs significantly from a more traditional “arith-metric” view on

meter (see bottom analysis in Figure 1.1).

Figure 1.1. Tone-metric (top) and metric (bottom) analyses of the Prelude in C major. Mm. 1–3.

The graphic analyses of Bach’s Prelude in C major, from the Das wohltemperierte Klavier (“The

Well-Tempered Clavier”) Book 1 in Figure 1.1 highlight two different approaches to the study of

the first three measures of the piece. The tone-metric analytical graph is characterized by smooth

and wavy ondulations, each of which is a descriptor of a specific metric grouping (see the rise of

the wave in mm. 1–3, and its its descent and rise in m. 3), while a more traditional metric

analysis describes a measure-by-measure regularity determined by the consistency of the

rhythmic pace. The fundamental difference between creating a traditional form of metric analysis

15

implies an arithmetic (“arith-metric”) summation process, while a tone-metric reading of the

same implies a sense of projection that, as we will see in more detail in the next paragraphs and

chapters, is determined by the principles of self-similarity and recursivity: each metric span is

produced as a result of the application of the same algorithm at all metric levels.

Before providing a full tone-metric analysis of a piece, I will explain, using a step-by-step recipe,

how tone-metric waves are built. I will do so using as an example mm. 1–3 of Bach’s Prelude in

C major, from the Das wohltemperierte Klavier (“The Well-Tempered Clavier”) Book 1. The

first step in building tone-metric waves consists of applying a multi-layer segmentation of the

metric structure of the piece, to articulate the basic rhythmic unit of the piece, in our case the

quarter note, as the piece is in 4/4. This process starts by laying down a first segmentation level,

level 1 (please refer to example 1.1), to indicate where quarter notes 1, 2, 3, 5, 9 occur. As will

be shown in detail in Chapter 3, digits 1, 2, 3, 5, and 9 correspond to the position at which each

metric unit (i.e., each quarter note) is found following a reading of the combinatorial and fractal

properties of the 4/4 time signature. Based on that sequence, we lay down digit 1 (which stands

for level 1) at positions: quarter note 1, quarter note 2, quarter note 3, quarter note 5, and quarter

note 9. While the original sequence proceeds ad infinitum, in our example, as we are dealing

with only twelve quarter notes (due to the fact that our example is only three measures long), we

stop with the layering process at quarter note 9.

Example 1.1. Score and level-1 tone-metric analysis of the Prelude in C major. Mm. 1–3.

Once we have established the articulation of level 1, we proceed by laying down level 2 to

describe the quarter-note articulation that level 1 could not describe, following the same layering

principles and sequence (i.e., 1, 2, 3, 5, 9) used for level 1. Unlike the articulation of level 1,

however, the layering “2” does not proceed ad infinitum, but is recursive, as it restarts each time

16

we encounter a new “1.” In the next paragraph, I describe how this process works. I exemplify

the results in example 1.2.

Example 1.2. Score and tone-metric levels 1 and 2 of the Prelude in C major. Mm. 1–3.

First, we consider the level-1 unit of mm. 1.3–2.1, which is characterized by an initial “1” (m.

1.3), a quarter note gap (m. 1.4), and an ending “1” (m. 2.1). We segment this unit in quarter

notes through a level-2 articulation, by using the same layering principles and sequence (i.e., 1,

2, 3, 5, 9) that we used for level 1. In other words, we write “2” over m. 1.3 (which corresponds

to digit 1 in the sequence), m. 1.4 (digit 2 in the sequence), and m. 2.1 (digit 3 in the sequence).17

Similarly, we articulate the unit of mm. 2.1–3.1, which is characterized by an initial “1” (m. 2.1),

a three quarter-note gap (mm. 2.2, 2.3, and 2.4), and an ending “1” (m. 3.1). We segment this

unit in quarter notes by writing “2” over m. 2.1 (which corresponds to digit 1 in the sequence),

m. 2.2 (digit 2 in the sequence), m. 2.3 (digit 3 in the sequence), and 3.1 (digit 5 in the

sequence). Finally, we articulate the unit of mm. 3.1–forward, which is characterized by an

initial “1” (m. 3.1), and a three quarter-note gap (mm. 3.2, 3.3, and 3.4). We segment this unit in

quarter notes by writing “2” over m. 3.1 (which corresponds to digit 1 in the sequence), m. 3.2

(digit 2 in the sequence), and m. 3.3 (digit 3 in the sequence).18

As shown in example 1.3, we build level 3 following the same procedure used to lay down level

2, and considering the quarter notes 7, 8, and 9 as one unit, and 11 and 12 as another.19

17 The sequence starts on the initial “1” and stops on the ending “1” of the unit; as this unit is three quarter notes

long, we do not use digits 5 and 9 from the sequence to describe its segmentation. 18 Knowing the metric regularity of this piece, we can assume that the metric and sequential distribution continues

undisturbed. 19 Again, we assume metric continuity and can therefore predict that quarter note 12 will be part of our sequence.

17

Therefore, we add digit 3 on quarter notes 7, 8, and 9 (corresponding, within the unit, to

sequence members 1, 2, 3), and 11 and 12 (corresponding, within the unit, to sequence members

1 and 2).

Example 1.3. Score and tone-metric levels 1–3 of the Prelude in C major. Mm. 1–3.

Now that we have completed the quarter note subdivisions of the excerpt, we can start to

generate the eighth-note articulation, following the same procedure as before. First, we

determine the boundaries of our units (which are now between two adjacent quarter notes); we

then use the sequence 1, 2, 3 etc. to determine where the data points of the segmentation have to

be inserted (as each eighth-note unit is made of only three adjacent eighth notes, we only need

digits 1, 2, and 3 from the original longer sequence); finally, we insert the next level following

the previously described rules.


18

As example 1.4 highlights,20 the addition of a further level at the eighth-note value necessarily

implies that four digit 2s have to be added to segment quarter notes 1 and 2. This occurs because,

as I will describe further in Chapter 3, levels 2 and above are built relative to the previous ones.

We repeat the operation adding level 3 segmenting quarter notes 3, 4, 5 and 6, and level 4 to

quarter notes 7 and 8. Because level 2 stops at m. 3.1, we have to consider the lowest free level

(in this case level 3 in m. 3.1) as the beginning of our next unit (mm. 3.1–3.2); level 3 is also

used for the unit of mm. 3.2–3.3, and again level 4 for the unit of mm. 3.3–3.4 and 3.4-forward.

The same procedure is then used to fill in the remaining sixteenth-note articulations of the

excerpt, adding the appropriate levels as the music unfolds. The result is summarized in example

1.5:


The final tone-metric wave layout follows, for this and all other pieces, the same set of

procedures. The result of applying the algorithmic procedure for the sequential layering of the

score produces waves made of various levels; those levels, taken together, can be read as

efflorescence, which is a quantifiable indicator of how meter unfolds, or how rhythm and tactus

relate moment-to-moment throughout the piece. Efflorescence can be quantified by counting the

number of layers of which each wave is made at any moment: in example 1.5, for instance,

20 The level patterns are not descriptors of absolute rhythmic values, but of the sub-divisional articulation of tone-

metric units established via the layering based on the originating numerical sequence, and each level is built above

the lower existing level. For these reasons, as the reader can see in example 1.5, eighth-note iterations are

represented by different levels across the excerpt. Notice that, following this organization, necessarily level 2

appears as both articulating the quarter note level in mm. 1.3–3.4, and the eighth-note level in mm. 1.1–1.2.

19

efflorescence is “3” (as it includes 3 levels) in m. 1.1–1.2, “4” in mm. 1.3–2.2, and again in m.

3.1–3.2, and “5” in mm. 2.3–2.4 and 3.3–3.4. In this dissertation, however, I will not use or

describe efflorescence as a form of absolute quantification based on counting the levels, but as a

descriptor of the relationship between tactus and rhythm; in other words, I will focus exclusively

on the formal and musical role and significance of efflorescence.

The complete tone-metric wave analysis of the Prelude in C major, BWV 846, is the following:

Example 1.6. Analysis of the Prelude in C major (part 1 of 4).

20

Example 1.6 (continued, part 2 of 4).

21

Example 1.6 (continued, part 3 of 4).21

21 In this and following examples, digits in the higher levels of tone-metric waves are not always clearly visible.

However, as the analytical significance of tone-metric waves is primarily due to their ability to describe metric

density relationships between levels, I decided not to privilege smooth shading over digital clarity.

22

Example 1.6. Score, and tone-metric and formal analyses of the Prelude in C major.

Example 1.6 includes roman numeral, formal, as well as tone-metric analyses of the piece. My

roman numeral analysis articulates the piece in formal units of irregular length (i.e., mm. 1–4, 5–

8, 9–11, 12–15, 16–19, 20–23, 24–31, and 32–35). Moreover, the harmonic annotations show a

number of conventional preludizing paradigms: the I-ii4/2-V6/5-I opening; the model-sequential

to cadential progressions of mm. 8–11 and 16–19: IV4/2-ii7-V7-I (to V and I respectively); and

the tonic pedal with elided [V7/IV] plagal extension at the end, mm. 32–35. In addition, and

following Schenker (see further below), it is shown that the main syntactical harmonies (scale-

steps) of the Prelude are: I (mm. 1–19), IV (mm. 21–23), V (mm.24–31), and I (mm. 32–35).

23

Roman numeral analysis helps to identify crucial stylistic features (e.g., preludizing paradigms)

and formal articulatory moments from a melodic-harmonic point of view, but cannot tell us

anything about the metric organization of the piece. Tone-metric analysis, on the contrary, can

tell us something about how the underlying metric structure works but cannot tell us anything

about the actual note content of the piece. As a consequence, in the following paragraphs, I show

how: (1) tone-metric waves are built; (2) they can be used to analyze the metric articulation of

the piece; and (3) tone-metric and roman numeral analysis can be integrated into a holistic

analytical approach that can provide us with insights into both harmonic-melodic and metric

aspects of music.

As I will discuss in detail in Chapters 4, 5, and 6, the algorithmically generated waves of tone-

metric analysis can be used to describe, in both quantitative and qualitative fashion, the growth

and cumulative aspects of music-unfolding. The reiterative process of tone-metric graphs results

in what can be seen as a corollary of the perception of music flow, functioning as an intermediate

element combining both the tactus-based mechanical repetitiveness of metric unfolding and the

combinatorial-sequential derived organization that lies “beneath” the sound surface, to produce

an ever-growing sense of formal expansion and accumulation.22

The wave analysis of example 1.6 shows that tone-metric waves proceed by articulating the

piece in four-measure units throughout. This regularity corresponds to the regularity of a

traditional formal analysis, at least for the first 8 measures (initial voice-leading cadential

progression of mm. 1–4, where the wave drops 2 units; and model/sequence in mm. 5–8, where

the wave drops 3 units). However, from m. 9, numerous mismatches between tone-metric and

traditional harmonic and formal analyses occur,23 as the boundaries of waves rarely correspond

to the boundaries of melody- and harmony-based formal units. For example, the wave of mm. 9–

13 does not correspond exactly to the cadential progression ii7-V7-I in G major, which ends in

22 In example 1.6 (and following), I discuss only primary tone-metric wave structures, that is waves that end with a

drop of at least two units (for more details on the different types of waves, see Chapter 4). As the reader is not yet

familiar with tone-metric graphs, before venturing into detailed analytical work I would like to add a few words

concerning how the graphs are built. I will do so by using once again the first measures of the Prelude in C major. 23 In example 1.6, I have provided my own harmonic analysis following the roman numeral analytical tradition.

Throughout the centuries, however, other approaches to harmonic analysis of this piece have been explored, such as

the continuo-like numeral approach in Frederick Iliffe, The forty-eight preludes and fugues of John Sebastian Bach

(Novello, 1938), 2. In this dissertation, I use a roman numeral approach instead of a figured-bass continuo such as

Illife’s because of the connection I wish to make between harmonic progression, formal units, and the comparison

between them and tone-metric analysis.

24

m. 11. Why? From a tone-metric perspective, a pivotal moment occurs in the underlying metric

structure at m. 11: from a traditional harmonic-formal perspective, the bass G (m. 11) of the local

G major cadential progression can be interpreted as prolonged through m. 12 to create a

harmonic pivotal link that blurs the boundaries between the cadence in m. 11 and the beginning

of the model/sequence of mm. 12–15. Measure 12 can therefore be considered as both an

extension of the cadence and the beginning of the model/sequence that follows (via the common

bass note G). Both tone-metric and traditional harmonic-functional analyses indicate a point of

interest in this pivotal moment of mm. 11–12.

Similar situations can be found in mm. 16 and 20, in which, again, the mismatch between the 4-

measure regularity of the tone-metric structure and the variety of the harmonic-formal analysis

can be integrated via common tones (E and C bass notes in mm. 16 and 20, respectively). Notice

how all these pivotal measures are based on dissonances that require resolution. Even the arrival

on the dominant V7 in m. 24 (again, via a dissonance that requires resolution) is, tone-metrically

speaking, arriving one-measure too soon. Here, its pivotal role can be seen as anticipatory of the

melodic design repetition in the inner-voice middle C occurring in mm. 25–28 and 29–32 (notice

how those identical melodic designs are supported by the same harmonic patterns: 6/4, 7/4–3). In

other words, tone-metric analysis helps us to identify the double role of m. 24 as both a dominant

arrival concluding the previous 4-measure progression (mm. 21–24), and the beginning of the

dominant pedal of mm. 24–31. A performance following more closely a tone-metric approach

would consider m. 24 as being the end of a four-measure section, and as a pivot towards the two

melodically nearly identical 4-measure sections that follow, thus emphasizing m. 25 as salient;

vice-versa, a performance following more a harmony-based approach would emphasize the

dominant arrival in m. 24 as a point of formal saliency. Ultimately, both performance approaches

could be combined to generate a moment of particular interest and tension in mm. 24–25.

Finally, the tonic pedal enters in m. 32, once again, tone-metrically speaking, one measure earlier

than expected. Here too, the tonic pedal enters with a flat 7th requiring resolution, as if this

anticipation of the tonic arrival can find its confirmation only through the delay produced by the

actual tonic-based IV6/4-V7-I prolongation that follows in mm. 33–35.

The underlying metric structure creates expectations related to articulatory moments in the

metric texture that can be either matched by harmonic and melodic features, or not. When tone-

metric and traditional melodic and harmonic analyses match, we obtain straightforward

25

analytical insights; on the contrary, when the analytical results of tone-metric and traditional

approaches do not match, we are forced to inquire why a dichotomy between the two exists.

This, I argue, has the potential to provide novel insights into the relationship between metric,

melodic and harmonic unfolding, and, ultimately, into the large-scale formal organization of

music. Neither tactus nor rhythm alone, but only their interaction, and their interaction with

melodic and harmonic patterns, can be used to determine the kinds of formal considerations I

presented in the previous paragraphs.

With my analysis of example 1.6, I have shown that the relationship between tone-metric and

roman numeral analysis can provide insights into the organization of the piece that would be less

evident from a purely harmonic reading/hearing. However, tone-metric analysis can be used in

combination with other analytical approaches. For example, Schenker’s well-known analysis

(see example 1.7) shows that mm. 1–11 can be grouped into a formal unit (based on the voice-

leading principle of descending parallel tenths [Oberdezimen] C/E, B/D, A/C, and G/B) that

harmonically progresses from C major (m.1) to G major (m. 11) within a larger tonic-

prolongational coupling [Koppelung] that connects m. 1 to m. 19.

Example 1.7. Schenker’s analysis of the Prelude in C major. Mm. 1–11.

Unlike tone-metric analysis, however, Schenker’s approach does not articulate these first 11

measures into formal units of mm. 1–4, 5–8, and 9–11, but includes a Dehnung (expansion) of

the opening tonic prolongation two quasi-four-measure units spanning, respectively, mm. 1–7

and 8–11 (see the measure accounting between the staves: Schenker’s graph 1…234, 1234).

Schenker’s compelling analysis of the piece is based on the interplay of contrapuntal and

harmonic elements (i.e., the descending tenth voice-leading patterns and their underlying

harmonic prolongation), which determine its structural unfolding into interpretive units that are

tightly linked to both melodic and harmonic principles. Unlike Schenker’s approach, tone-metric

26

analysis is concerned with the study of how, based on an algorithmic regularity of tactus, the

unfolding of the metric structure interacts with melodic and harmonic patterns to generate larger-

scale formal units. The deviations from the underlying metric structure of harmonic- and

melodic-based formal units constitute an added value of tone-metric analysis, as the approach

allows us to contextualize the role of tactus and rhythm within the larger context of the musical

discourse.

This brief comparison of the differences between tone-metric, roman numeral, and Schenkerian

analyses of the first 11 measures of the Prelude, provides analysts, listeners, and performers with

further insights into different ways of grouping the material of those measures when they

analyze, listen to, or perform the excerpt: for example, the analyses agree that the G major chord

in m. 11 is a locally stable point of arrival (albeit, a dividing dominant within the broader tonic

prolongation of mm. 1–19) , but describe the path to reach this chord in different ways. With this

in mind, we can decide whether an emphasis on metric, or contrapuntal and harmonic, structure

might provide a more convincing and fuller reading of the piece, or whether, as is commonly the

case, a combination of the approaches might be preferable. The comparative analysis carried out

based on example 1.7 shows how discrepancies between harmonic and tone-metric views can

shed light on elements of interest in the formal-organizational domain of pieces.

In our second example, the fugue in C major following this prelude, these discrepancies become

even more evident due to the complexity and variety of the rhythmic patterns that characterize

most fugues. As for the prelude, here too I will describe (this time in a summarized way) how the

tone-metric graph of the fugue is built, using mm. 1–3 as an example.

Example 1.8. Score and tone-metric level 1 of the Fugue in C major. Mm. 1–3.

27

In example 1.8, level 1 is built following the same procedures seen in the analysis of the prelude

(see examples 1.1–1.6 as a reference). The difference compared to the tone-metric analysis of the

prelude is that, in the fugue, the “1” in m. 1.1 is in parentheses; the parentheses indicate that no

note is actually performed (i.e., it is substituted by a pause or a tie that connects previous with

later notes). Since tone-metric analysis is based on both predetermined accentuation patterns and

composed-out notes, we have to assume that, even when a note is not present, the necessary

underlying accentuation patterns for the performer provide enough metric information for us to

understand that a beat is nonetheless present in such situations.

In example 1.9, the tone-metric analysis includes both levels 1 and 2:

Example 1.9. Score and tone-metric levels 1 and 2 of the fugue in C major. Mm. 1–3.

Again, level 2 in m. 2.2 is indicated in parentheses because of the tie in the score that prevents us

from hearing the entrance of a new note; however, the articulation of tactus implies the presence

of a beat in m. 2.2.

Moving on to level 3, we obtain the following:

Example 1.10. Score and tone-metric levels 1– 3 of the fugue in C major. Mm. 1–3.

28

The full articulation at the eighth-note level results in the following graph:

Example 1.11. Score and tone-metric levels 1–4 of the fugue in C major. Mm. 1–3.

Once again, notice that levels 3 and 4 in example 1.11 are in parentheses in m. 1.3: because of

the dotted note in the score, no new note is struck in m. 1.3; however, because of the

predictability of the underlying tactus, we have to assume that some kind of articulation is, at

least theoretically, taking place over the dot in m. 1.3.

For the sixteenth-level articulation:


Finally, we fill in the thirty-second-note patterns as follows:

29


Notice that, in m. 3.1 in example 1.13, the dot does not generate a parenthetic version of the

graph because, in this measure, the lower F# is actually entering under the dot. The complete

tone-metric wave graph of the fugue is provided in example 1.14:

Example 1.14. Analyses of the fugue in C major (part 1of 5).

30


31


32


33

Example 1.14. Score, and tone-metric and harmonic analyses of the fugue in C major.

Example 1.14 does not provide the extensive roman numeral harmonic analysis that was shown

for Example 1.6. This is, in part, because harmonic analysis of fugal textures is notoriously

challenging, since the priority of the contrapuntal thematic activity can sometimes create

apparent chord changes every eighth note, further complicated by motivically driven root

34

movements and freely resolved suspensions. The C major fugue is also the locus classicus of

stretto fugue and it is fascinating to see how much harmonic variety Bach can bring to the

various subject and answer entries. The subject itself potentially tilts towards IV (do-re-mi-fa),

the thirty-second-note details reflected on the tone-metric graph being of critical importance in

clarifying the melodic role of the fourth step. And in the unusual subject (alto), answer (treble),

answer (tenor), subject (bass) entry sequence of the exposition, the tilt to IV is actually realized

in a brief tonicizing motion (initiated at the end of m. 5), and the move recurs in more typical

fashion in the final plagal extension (m. 24). Other selected harmonic features shown on

Example 1.14 are directed towards understanding the main syntactical harmonies (scale-steps),

which are quite clear: I (mm. 1–7, exposition), V (m. 10, contrapuntally filled cadence), vi (m.

14, PAC in A minor), ii (m. 19, PAC, previous build-up of mm. 17–19 being towards ii, D minor

with a tierce de Picardie), V (m. 21.3, IAC to V pedal), and I (mm. 24–27, tonic pedal, with

plagal extension paradigm reminiscent of the end of the prelude).

From a tone-metric perspective, example 1.14 is particularly illustrative because it shows how

rhythmic differences can interact with tactus to generate much more complex tone-metric wave

structures than the ones seen in example 1.6. The base algorithm described earlier in this

corollary (see example 1.6) is here used again to generate all analytical tone-metric levels and

produce a metric “density” (e.g., mm. 14–17) that is described through the superposition of

numerous levels. This density, which is not only rhythmic but, broadly speaking, metric,

represents the integration of the accentuation patterns of tactus and their influence on the

arithmetic summation of note values. Such density is the product of the perturbation of the

steady underlying tactus by the different rhythmic patterns; this perturbation contributes to

postpone and, at times, eliminate the next expected occurrence (i.e., beat) of tactus, thus

rendering most metric articulations throughout the piece fuzzy; this fuzziness is a feature that is

clearly rendered by tone-metric waves (e.g., the ends of waves do not always occur on a

downbeat).

As a consequence, in tone-metric analysis the wave structure does not indicate formal

distinctions in a traditional analytical way (e.g., differentiating between subject and answer,

exposition or stretti),24 or even necessarily indicate standard harmonic articulations, but

24 This thematic-based analytical approach is better exemplified by in Illife, The forty-eight, 3.

35

highlights how the unfolding of metric factors, built through the accumulation of rhythmic

values within the context of the underlying tactus, can affect our perception of melodic and

harmonic features as specific points of interest. In the next paragraphs, I will list a few examples

of how waves can help us identify and explain points of interest in the unfolding of the piece.

The analysis of the tone-metric wave’s pivots shows a relatively simple tone-metric articulation

of the piece. In mm. 1.3, 4.2, 15.1, 17.1, and 24.4 the metric articulation occurs over the

following rhythmic figurations:

Example 1.15. The characteristic melodic-rhythmic design of the fugue’s subject. M. 1.3.

The metric articulation in m. 4.4 matches the register switch in the upper voices descending to

the octave below, and, again, matches the occurrence of the rhythmic design in example 1.15.

Furthermore, in m. 9.1 the metric articulation coincides with the end of the model-sequence of

m. 8, and, in m. 19.1, the leap down a minor 6th between A and C# in the upper part is what

breaks the texture, confirming the presence of an articulatory moment. Things become more

complicated in mm. 21.1 and 25.1, in which there does not seem to be any correspondence with

specific harmonic, rhythmic, and melodic articulations in the note content of the piece, which, on

the contrary, appears to emphasize textural, harmonic, and thematic continuity over separation,

an issue I discuss in the following paragraphs.

A combination of harmonic and tone-metric analysis, however, can help us identify and

understand in more detail relevant aspects of the piece’s formal structure. For example, the

single wave of mm. 9–15 includes different melodic and harmonic points of interest (e.g., the

stretto in m. 10.3–10.4; and the cadence in A minor in m. 14.1–14.2). What the chunking

highlighted by the wave in this section of the piece makes us see more clearly is the growing and

uninterrupted metric tension of mm. 9–15, which is built through a series of imitative points

culminating in the A minor cadence in m. 14.1. Interestingly, and in conformity with traditional

fugal-thematic analysis, while the C major is immediately reestablished from a subject entry

point of view in m. 14.2, the harmonic unfolding of mm. 14–17.1 appears to complicate, through

a series of subsequent momentary tonicizations, the sense of return to C major. The efflorescence

of the tone-metric graph suggests a reading of the passage that emphasizes a structural-harmonic

36

lingering (if not quite a prolongation) of A minor until m. 17.1. In other words, from a tonal-

thematic perspective there is a clear caesura in m. 14.1, which temporarily reestablishes C major;

however, from a tone-metric perspective (which emphasizes the effects of meter on form), mm.

14.1–17.1 can be read as a battleground between A minor and C major. As noted above in our

review of main syntactical harmonies, the A minor arrival point (vi) is on a larger trajectory

towards ii(#) and V, within which larger motion the temporary allusion to a C major return is

more parenthetical than recapitulatory.

Another example of how tone-metric analysis can help us identify points of interest in the piece

is found in mm. 15–17. This wave is, metrically speaking, a single unit, and includes the only

“incomplete” stretto of the piece, with the G major version of the subject (answer) that appears

three times in stretto, but with the second appearance in the soprano voice in mm. 15–16) being

incomplete. In this case, the melodic interplay between voices generates an imitative texture that

is entirely enclosed into a single wave: the wave of this example is primarily of thematic

significance.

The wave of mm. 17–19 marks a long modulation towards D minor; notice how the end of this

wave coincides with a melodic point (the high A in the soprano) and not the actual D major

cadence (which occurs half a measure into the next wave). Again, this is an example of how

tone-metric waves are not necessarily telling us where harmonic boundaries are, but how metric

tension is built to reach those points. From a perceptual point of view, the D major cadence in m.

19.3 can be seen as a necessary ending, or offshoot, of the arrival on the high A that begins the

same measure, not the main point of interest of the passage. In fact, mm. 19–21 act as a

modulatory bridge that brings the piece back to the concluding C/G major tonic areas (whose

beginning is marked by the wave starting in m. 21).25

From a tone-metric perspective, the final cadence in C major in m. 24 is not marked by any wave

articulation. This is because the element of interest in this part of the piece is the sudden switch

from the long flow of sixteenth notes in mm. 20–23.3 to the slower eighth-note flow in m. 23.4:

from a tone-metric perspective, the V-I cadence in m. 24 is, at least rhythmically, an offshoot

25 The drop in m. 25.1 is simply the continuation of the drop occurring in m. 24.4, and therefore belongs to the

previous wave.

37

(and, therefore a simple continuation) of the eighth-note flow that started earlier, and is not

signaling any metric points of interest.26

In the previous analyses, I have shown the variety of musical features that tone-metric analysis

can help highlight. For the third example, I will now analyze a more extreme case of formal

irregularity: mm. 101–113 of Buxtehude’s Praeludium in E minor, BuxWV 142 offer a glimpse

into what is probably the most characteristic element of North German Baroque organ music, the

main subject of this dissertation: the improvisational character of some sections from the so-

called free pieces in stylus phantasticus. The tone-metric graph for this piece follows the same

rules as for the prelude and fugue by Bach; I will therefore provide only the complete analysis.

26 The final measures (mm. 25–27) do not exhibit any two-step drop, and are therefore not primary waves, but

simply the post-thematic ending of the piece, as they appear after the last thematic presentation and do not contain

any new statement of the subject/answer pair.

38

Example 1.16. Tone-metric and harmonic analyses of BuxWV 142. Mm. 101–113.

39

Example 1.16 (continued).

The harmonic analysis provided in example 1.16 shows the kind of analytic “hindsight” typical

of any roman numeral approach. In labelling, we begin with overall tacit awareness of the E

minor home key. The choice of key orientations—in moving from the initial A minor of the

excerpt through the never-quite realized (dominant-arriving) progression in C major, then

through the sleight-of-hand and texturally calming return to E minor followed by the highly

elaborate cadence in B minor/major—cannot adequately capture the listener’s perception of

being buffeted about in a recitativo-like series of progressions, and the performer’s challenge to

find an appropriate way to navigate through, and convincingly present, the material that

characterizes the stylus fantasticus. But what does tone-metric analysis say about the piece that

roman numeral analysis was not able to highlight?

In example 1.16, the first wave (mm. 101–103) includes the opening two measures of the

excerpt, which are characterized by the 16th- and 32nd-note rhythmic design and underlined by

the V6/5–I harmonic motion in A minor. The second tone-metric wave of mm. 103–105 does not

end on a harmonically defining point, but on a melodic and textural point of interest, as it

finishes right after the texturally unusual E–F#–G motion (played up an octave from the

otherwise middle-of-the-range solo melodic line) in the soprano part that functions as a melodic

and metric pickup to m. 105. The third wave (mm. 105–109) contrasts the previous wave by a

progressive slowing down of the rhythmic values in the soprano line and a confirmation of the

40

tonic E minor. Finally, the 4th wave (mm. 109–113) starts by confirming again the tonic E minor

and is characterized by fast-paced rhythms throughout. The pivot in m. 109 also arrives in the

middle of an E minor harmonic progression and is misaligned with the E minor arrival in m. 108;

however, m. 109 can be read as a harmonic pickup (built as V6/5 with a suspension E–D in the

bass) of the confirmed E minor harmony in m. 110.

Ultimately, the combination of both harmonic and tone-metric analytical approaches can be used

to generate a musically meaningful performance of the excerpt, as all points of harmonic arrival

and tone-metric articulatory moments can be highlighted in performance to render the

improvisatory and elusive character of the excerpt.

Example 1.16 shows how waves can help us organize, at least metrically, an improvisatory

excerpt into well-defined segmentations. An important aspect of tone-metric analysis that is

highlighted by this example is that its approach, fundamentally based on numerical-geometrical

relationships and not on purely musical ones, initiates a holistic path to music analysis:

boundaries between waves are not consistently harmonic, melodic, or metric, but can be

analyzed as corresponding to harmonic, melodic, or metric boundaries interchangeably. In other

words, sometimes waves are articulated following melodic endings; at times they mark harmonic

cadences; other times they indicate rhythmic changes; and, finally, they are often indicators of

the recurrence of the underlying tactus. For example, sometimes waves are divided by

differences in rhythmic features (in example 1.16, first and second waves), other times the

subdivision is determined by harmonic boundaries (see articulation between second and third

waves). This is a feature of major importance in tone-metric analysis, because it provides the

ability to highlight boundaries that would be less noticeable according to specific music-

analytical approaches such as melodic, harmonic, or rhythmic analysis alone.

In this Corollary, we have seen how tone-metric analysis can help us identify specific pivotal

moments and overlaps between formal units (example 1.6), model the influence of metric

parameters in identifying moments of tensions and relaxation in the texture of the piece beyond

its melodic, harmonic, and formal boundaries (example 1.14), and help analysts and performers

to identify sectional subdivisions within a non-classically conceived formal structure (example

1.16).

41

To conclude, tone-metric analysis constitutes a complementary form of analysis that can be

applied to any metrically based music and provides insights into the relationship between

melodic and harmonic elements, and the underlying metric structure, to describe how this

relationship can inform our understanding of formal structures and articulations in music. While

it does not deny or contradict more traditional forms of analysis, it provides insights into how,

for example, the unfolding of metric regularity can influence our perception of the loosening and

blurring of formal boundaries (e.g., example 1.6); how metric irregularity can contribute to

create moments of tension and relaxation in both small- and large-scale organization of music

(e.g., example 1.14); and how harmonically unstable progressions that cannot be fully explained

through classical harmonic analysis can be clearly modeled through metric structures (e.g.,

example 1.16).

1.6 Outline of the dissertation

The dissertation comprises seven chapters. Chapter 1 is this introduction, which presents its main

research components, distinctive terminology, and repertoire focus, plus the Corollary of

introductory analyses. Chapter 2 is dedicated to the contextualization of current research relevant

to this dissertation through a literature review. Chapter 3 deals with the creation of a

mathematical-geometrical model for a unified tone-metric method. Chapter 4 discusses the set of

rules necessary to build the analytical algorithm that generates the wave graphs and tree

structures; it also presents some considerations about kinesis (musical motion) and form. Chapter

5 provides in-depth analyses of a number of repertoire examples from the North German

Baroque organ school based on the model established in Chapter 3 and the algorithm developed

in Chapter 4. Further analytical, performance, improvisational, and pedagogical applications of

the algorithm to later repertoire examples are found in Chapter 6. Chapter 7 concludes with

summary of and commentary on the findings, limitations, and suggestions for further research.

Chapter 2

Historical and theoretical perspectives

In this chapter, I provide an overview and critical discussion of historical and analytical-

theoretical secondary literature relevant to my research on the association between tactus and

harmonic, melodic, metric-rhythmic, and formal designs in seventeenth-century North German

organ music. More specifically, I explain how the features of past and current analytical

approaches have shaped my understanding of tactus as an underlying combinatorial framework

for the unfolding of music. This chapter is divided in the following sections: (1) From tactus to

meter and beyond in analytical methodologies across the centuries; (2) The origin of

efflorescence and tone-metric waves; (3) Scientific developments and studies in beat perception;

and (4) Historical-analytical contextualization of the repertoire analyzed.

43

2.1 From tactus to meter and beyond in analytical methodologies across

the centuries

In her Tactus, Mensuration and Rhythm in Renaissance music,1 Ruth DeFord lists six definitions

and descriptions for tactus:2

(1) The physical motion, such as a series of taps or movements of the hand in the air, that measures time

in performance. I shall call this action and the unit of time to which it corresponds the “performance

tactus.” (2) The time unit that serves as a standard of reference for various aspects of rhythm, such as the

rate of contrapuntal motion, dissonance treatment, and syncopation, in a composition. I shall call this unit

the “compositional tactus.” (3) The time unit that functions as the theoretical standard of measure under a

given sign. I shall call this unit the “theoretical tactus.” (4) A time unit of the mensural structure

corresponding to any of the above definitions. I shall call this time unit the “tactus-unit.” (5) The abstract

quantity of time corresponding to any of the above definitions. I shall call this quantity the “value of the

tactus.” (6) The concrete quantity of time corresponding to any of the above definitions. I shall call this

quantity the “duration of the tactus.” It is analogous to a modern metronome mark.

DeFord’s wording highlights that tactus in the Renaissance was primarily understood as a “time

unit” and “quantity.” Interestingly, in her summary DeFord does not describe Renaissance tactus

in terms of accentuation patterns, which became instead central in the definition of meter during

the Baroque. For example, according to Caplin, in his Der Vollkommene Capellmeister3

Mattheson indicates that music is organized according to “a primary division of the measure into

equal meters (our duple and quadruple meters) or unequal meters (triple). The latter are made up

of two parts, the first (thesis) lasting twice as long as the second (arsis),4 just as in the original

tactus theory.”5

The notion of Renaissance tactus as a pattern of quantities of time, and the Baroque idea of meter

as a form of accent distribution of those patterns are further developed and combined in tone-

1 Ruth I. DeFord, Tactus, mensuration and rhythm in Renaissance music (Cambridge: Cambridge University Press,

2015), 51–52. 2 For an introduction to the variety of words (including tactus) used to describe time organization in music used

throughout the Renaissance and early Baroque, see Roger M. Grant, Beating time and measuring music in the early

modern era (New York: Oxford University Press, 2014), 15–42 and 261. 3 Ernest C. Harriss, ed., Der Vollkommene Capellmeister (Ann Arbor, Michigan: UMI Research Press, 1981), 365. 4 For a discussion about the role of arsis and thesis in Mattheson’s theory with respect to how arsis and thesis are

used in this dissertation, see Chapter 1, pages 2–3. 5 William Caplin, “Theories of musical rhythm in the eighteenth and nineteenth centuries,” in The Cambridge

History of Western Music Theory, ed. Thomas Christensen (Cambridge: Cambridge University Press, 2002), 660.

44

metric analysis, whose final analytical product, as we will see in Chapter 3, is a graphic

representation of meter as a binary set of strong and weak accents studied from a combinatorial

perspective. But how did we get from Renaissance tactus and Baroque meter to the

combinatorial approach to time unfolding used by tone-metric analysis? In the following

paragraphs, I will trace a brief history of the main music-theoretical ideas that brought us there. I

will describe these developments by grouping them in three historic-analytical approaches to the

role of meter in formal analysis, discussing, for each of them, their significance in shaping the

creation of tone-metric analysis: (1) Meter, form, and hierarchies; (2) Roman numeral analysis

and form; and (3) Theorizing perception.

2.2 Meter, form, and hierarchies

The story begins in German music-theoretical circles during the nineteenth century. Soon after

the Baroque, a strong association between meter and harmony developed; this association found

its systematization in the first half of the nineteenth century with Moritz Hauptmann (1792–

1868). In his work, Hauptmann argued that meter and harmony are both integral generating

elements of musical form6 (an approach that was later revived by David Lewin (1933–2003) 7

and Richard Cohn,8 who described the affinity between metric and tonal processes in music

through their well-known analyses of Brahms’s Capriccio, Op. 76 No. 8. Hauptmann’s view was

soon contrasted by the ideas of Ernst Toch (1887–1964),9 who was chiefly concerned with the

relationship between melodic and formal designs, and stated that melody, not meter, was,

together with harmony, a leading element in determining formal organizations in music.

In this context, in which melody and harmony had become form-generating elements par

excellence, two analytical tendencies developed: one that emphasized the primacy of melody,

6 Moritz Hauptmann, The nature of harmony and metre (New York: Da Capo Press, 1991). 7 David Lewin, “On harmony and meter in Brahms’s Op. 76, No. 8,” Nineteenth-century Music 4.3 (1981): 261–

265. 8 Richard Cohn, “Complex hemiolas, ski-hill graphs and metric spaces,” Music Analysis 20.3 (2001): 295–326. 9 Ernst Toch, The shaping forces in music: an inquiry into the nature of harmony, melody, counterpoint, form (New

York: Dover Publications, 1977).

45

and one that emphasized the equal importance of melody and harmony, in the generation of

musical form. Major exponents of the former tendency were Rudolph Reti (1885–1957),10 who

argued that only linear-intervallic relationships and melodic designs constitute viable

compositional “forces” driving the harmonic discourse, and Ernst Kurth (1886–1946),11 who

considered harmony as a function of melodic unfolding;12 major exponents of the latter were

Heinrich Schenker (1868–1935), who, in his Free Composition,13 created the basis for a

hierarchical understanding of form based on the relationship between harmonic and melodic

relationships, and Carl Schachter, who applied Schenker’s principles to generate a systematic

hierarchical approach to harmonic analysis.

Unfortunately, none of those approaches considered meter as a primary factor in shaping long-

term harmonic, melodic, and formal relationships. For example, in his theory, Schenker, the most

influential of all theorists cited, relegated meter to a foreground phenomenon (overall,

Schenker’s theory is about the composing out of harmony and delaying of closure through voice-

leading transformations, and privileges a narrative that speaks primarily of large-scale formal

relationships based on harmonic-tonal features, not on meter). And Schachter, a leading

Schenkerian pedagogue and analyst, was one of the first to address metric relationships as form-

generating elements.14

It is with the research of Schenkerian scholar William Rothstein that meter gained a prominent

position within Schenkerian analysis. In his Phrase rhythm in tonal music,15 Rothstein presents

the notion of hierarchies of meter, which is tied to, and derived from, Schenkerian principles of

tonal organization applied to meter and rhythm. A concept relevant to the establishment of

10 Rudolph Reti, The thematic process in music (New York: Macmillan, 1951). 11 Thomas Christensen, ed. The Cambridge history of Western music theory (Cambridge: Cambridge University

Press, 2001), 927. 12 There is some irony in this gap, since metric and hypermetric organization play such crucial compositional as well

as perceptual roles in Bruckner’s work, a key repertoire for Kurth’s analytical applications. 13 Heinrich Schenker, Free composition: volume III of new musical theories and fantasies = Der freie Satz,

(Hillsdale: Pendragon Press, 2001). 14 Harald Krebs, Review of Schachter 1976, 1980, 1987, and Rothstein 1989, Music Theory Spectrum 14.1 (1992):

82–87. 15 William N. Rothstein, Phrase rhythm in tonal music (New York: Schirmer Books, 1989). Harald Krebs 1992

reviews the three Schachter articles (footnote 43) and Rothstein monograph. More complete Schenkerian-informed

analytic treatments of meter, hypermeter, and metric dissonance have been applied to the music of Robert Schumann

and Johannes Brahms: see Harald Krebs, Fantasy pieces: metrical dissonance in the music of Robert Schumann

(Oxford University Press, 1999); and Ryan C. McClelland, Brahms and the scherzo: studies in musical narrative,

(Ashgate, 2010).

46

hierarchies of meter is what Rothstein calls “phrase rhythm,” a term “embracing both phrase

structure and hypermeter.”16 Phrase rhythm includes various compositional techniques such as

overlap, extension, expansion, lead-in, and elision. As a parallel to melodic phrase constructions,

“phrase rhythm” tries to integrate melodic and rhythmic elements in the musical continuum. This

integration assumes that rhythmic phrases, and tonal and melodic patterns, are co-dependent, and

that phrase rhythm is somehow generated by tonal and melodic constraints (an approach

consonant with Schenker’s privileging of harmonic and voice-leading principles).

From the works of Schenker and Schachter tone-metric analysis has gained the understanding of

music as a hierarchical systematization of melodic and harmonic functions; from Rothstein, the

notion that those hierarchies can be transferred to metric organization. Beyond Schenker,

Schachter, and Rothstein, tone-metric analysis argues that it is through the study of metric

organization according to algorithmic principles that new insights into the formal organization of

harmonic and melodic patterns can emerge.

2.3 Roman numeral analysis and form

Nowadays most analytical approaches to tonal music are based on roman numeral analysis,

which is a technique that helps us model and organize in a coherent analytical system the basic

building blocks of the harmonic discourse as well as the form-functional reading of musical

excerpts. More specifically, it assists analysts in the categorization of cadential functions (e.g.,

half cadence, perfect authentic cadence) and formal-functional categories (e.g., sentence or

period, presentation or continuation); and it is useful for the identification and description of

harmonic progressions and modulations. Moreover, roman numeral analysis often uses

generalized tonal relationships (e.g., tonic, pre-dominant, and dominant) to determine and

describe tonal and phrasal unfolding in individual pieces. However, roman numeral analysis can

be applied successfully only when the piece consistently obeys harmonic and phrasal designs

based on common-practice tonal relationships, which is not always the case with repertoires such

16 Rothstein, Phrase rhythm, 12.

47

as North German Baroque organ music, filled with abrupt changes in the direction of the

harmonic discourse, and often led by long chant-like and psalm-tone-based tunes not based on

strict formal-tonal rules.

The usefulness of roman numeral analysis becomes more evident and widely accepted in the

analytical context of later repertoires, such as formal analysis of music from the Classical era. In

fact, roman numeral analysis is used as a starting point for the analysis of form according to

some of the most important contemporary approaches, such as those of Hepokoski and Darcy,17

and Caplin,18 and provides a harmonic framework for the definition of cadential patterns and

progressions that are, according to those theories, determinant features of formal organization

(one thinks of the harmonic aspect of the role of the medial caesura for Hepokoski and Darcy,

and the nuanced understanding of I6 as a pre-cadential signal for Caplin). In particular, in

Chapter 6 I will discuss aspects of Caplin’s form-functional analysis through the study of an

excerpt from the piano sonata in C major (K. 279/189d, iii) by W. A. Mozart; the study will

center on the argument that Caplin’s approach accounts only marginally for the influence of

meter on form, although it does consider the measure as the basic unit of the formal discourse,

and standard phrases as normally made of basic two-measure units, in the Formenlehre tradition

developed by Arnold Schoenberg (1874–1951) and Erwin Ratz (1898–1973).19 As a counterpart

to this approach, I suggest a tone-metric view of the same excerpt, and show how an integrated

view of tactus with harmonic and melodic elements provides additional information on the

excerpt, which can be integrated into Caplin’s analysis and used as a guide for the analysis and

performance of the excerpt.

Both roman numeral analysis and theories of form based on it (e.g., Hepokoski and Darcy, and

Caplin) were useful in the creation of the tone-metric methodology, as they provided me with

excellent means of comparison for the results of tone-metric analysis. They helped me answer

questions such as: how do harmonic-functional and combinatorial aspects of music combine

together in a meaningful and useful music-analytical product? What kind of information can

17 James Hepokoski, and Warren Darcy, Elements of sonata theory: norms, types, and deformations in the late

eighteenth-century sonata (Oxford: Oxford University Press, 2006). 18 William E. Caplin, Classical form: a theory of formal functions for the instrumental music of Haydn, Mozart, and

Beethoven (New York: Oxford University Press, 1998). 19 Norton Dudeque, Music theory and analysis in the writings of Arnold Schoenberg (1874–1951) (Aldershot:

Ashgate, 2005).

48

tone-metric analysis provide that is not included in theories of form based on harmonic

functionality? It is important to note that tone-metric analysis and the analytical approaches

referred to in this dissertation are not mutually exclusive. On the contrary, tone-metric analysis

should be used as an analytical methodology complementary to other approaches to music

analysis, to inform our understanding of formal designs through their combinatorial features and

beyond a purely harmonic-melodic approach to phrasal construction.

2.4 Theorizing perception

This section includes an analysis of two analytical approaches that discuss the role of meter from

a perceptual point of view: the theories of Fred Lerdahl and Ray Jackendoff, and those of

Christopher Hasty.

In their A generative theory of tonal music (from now on A generative theory),20 Lerdahl and

Jackendoff indicate that metric relationships are extremely influential in directing the tonal

discourse in music; however, they also argue that, from a perceptual point of view, the influence

of meter on tonal relationships progressively weakens over time.21 As a consequence, they try to

show that it is possible to analytically model the switch between metric and tonal prioritization,

and to construct an analytical discourse that integrates metric and tonal hierarchies into a unified

musical continuum. As fascinating as this idea is, there are two issues that need to be addressed

with respect to A generative theory.22 First, even assuming that the perceptual effect of metric

relationships weakens over time, its compositional presence and relevance in terms of

organization of the musical material remains steady throughout the piece and should be taken

into account in a music-theoretical analysis; and second, because metric and tonal relationships

20 Fred Lerdahl, and Ray Jackendoff, A generative theory of tonal music (Cambridge, Mass: MIT Press, 1983). 21 “As the listener progresses away by level from the tactus in either direction, the acuity of his metrical perception

gradually fades; correspondingly, greater liberty in metrical structure becomes possible without disrupting his sense

of musical flow. […] At very large levels metrical structure is heard in the context of grouping structure, which is

rarely regular at such levels; without regularity, the sense of meter is greatly weakened.” Lerdahl and Jackendoff, A

generative theory, 21. 22 Refer to the analysis in Chapter 6.

49

are based on different generating principles, their relationship in a unified analytical model

cannot be created by a simple switch between the two.23

Unlike A generative gheory, tone-metric analysis addresses the (apparent) meter-harmony

dualism not from a purely musical perspective (e.g., analyzing harmonic functions), but by

looking at: (1) sonic events as bits of information (with no specificity about their function as

harmonic or melodic elements), and (2) tactus from a combinatorial perspective. This approach

allows for the generation of a common numerical analytical framework between metric and

harmonic features of music; this framework then allows for the analytical discourse to integrate

metric and harmonic aspects into a unified tone-metric continuum, thus overcoming conflicts

ensuing from the combination of parameters (meter and harmony) based on different generating

principles.

Hasty’s metric theory, discussed in his book Meter as rhythm,24 considers meter as a process of

becoming, rendered by what he calls metric “projection.” As a sonic event unfolds (e.g., a chord

is played), a durational “projective potential” becomes associated with its duration by the

listener; this projective potential, which is the ability of an event to generate in the listener the

expectation about the timing of arrival of the following event, is quantifiable as it is either equal

to, or shorter or longer than the length of the previous sonic event. The expectation in the listener

is to find regularity of duration; however, this expectation can be either fulfilled (if the following

event does occur when expected), or not fulfilled (if the following event does not occur when

expected). The chain of expectations and realizations (or lack thereof) creates the listener’s

perceptual quantification of timing of adjacent sonic events (e.g., a chord can feel too long or too

short relativeto these perceptual aspects of time unfolding, depending on whether or not the

following chord fulfills the durational expectation of the previous chord). To summarize, for

Hasty meter is built and perceived step by step, according to a sort of short-term predictive

pattern.

Tone-metric analysis includes Hasty’s notion of metric organization as a generative process

based on the influence of an event on the following one (projective character); however, it also

explains that the basis for this relationship lies not in the durational element of individual

23 Refer to Chapter 6. 24 Christopher F. Hasty, Meter as rhythm (New York: Oxford University Press, 1997).

50

rhythmic values, but in the interaction between those rhythmic elements and the underlying

tactus, which is an algorithmically conceived articulatory structure of time unfolding. In this

sense, tone-metric analysis combines Hasty’s projective character of metric structures with a

cyclic understanding of the same, based on tactus’s algorithmic and recursive nature. Hasty’s

theory and tone-metric analysis are not mutually exclusive but can interact with each other, as

Hasty’s theory looks at music from a perceptual point of view (conceiving meter as a

phenomenon that we listen and adapt to), while tone-metric analysis studies music from a

compositional-structural perspective (conceiving meter as a function of a repeating pattern of

accents that exist even without us listening to them, and unfolding through measures that are

always equal to each other from the accentual perspective).

The combination of tone-metric and Hasty’s theories can inform us how the underlying metric

structure of pieces, which exists even when we do not listen to it, can be perceived. Moreover,

both Hasty’s approach and tone-metric analysis conceive of meter and rhythm as elements whose

interaction creates a unified continuum. However, for Hasty this continuum is articulated

through quantifiable rhythmic relationships between adjacent sonic events generating the piece’s

beat, which is, therefore contextual to rhythm and has to be constantly reassessed; for tone-

metric analysis, on the contrary, this interaction springs from the beat’s recursivity, and is,

therefore, a preconceived and quantifiable phenomenon that emerges from the underlying tactus,

over which metric and rhythmic relationships are organized.

To summarize, Hasty’s integration of processive and projective characters of metric organization

shares with the analytical approach developed in this dissertation the idea that meter projects its

influence on rhythmic unfolding in music,25 and that there is a recursive element in metric

organization through which one sonic event influences the following ones. However, tone-metric

analysis differs from Hasty’s theory in that, while Hasty suggests that meter is generated by a

perceptual reading of the unfolding of rhythmic patterns, tone-metric analysis supports the notion

that meter is generated as the integration between the combinatorial properties of the pre-

conceived tactus (understood as an organized sequence of arsis and thesis) and rhythm. The

musical analyses in Chapter 5 will illustrate this approach.

To conclude, the work of Lerdahl, Jackendoff, and Hasty has had an important influence in the

25 For a definition of tactus as used in this dissertation, see the Terminology section of Chapter 1.

51

creation of tone-metric analysis. In particular, their work highlighting a perceptual approach to

music analysis has shaped my understanding that there exists a flexible and nuanced view of how

metric, melodic, and harmonic details of individual pieces unfold and are organized into formal

units, beyond the notion of a formal organization based on strict modularity (e.g., rigid

subdivision of the musical discourse into pre-defined structures such as sentences or periods).

2.5 The origin of efflorescence and tone-metric waves

Music is both a discrete and a continuous phenomenon: discrete because it is made of specific

sound frequencies and rhythmic values, and continuous because of its ability to generate a sense

of melodic and harmonic unfolding that are perceived by the listener as a flow of musical ideas

and motion. The present research aims at modeling, in a single graphic representation, both

discrete and continuous features of music, in a similar way to which calculus in the seventeenth

century described the relationship between digital (discrete) versus analog (continuous) motion.

As I will show in the next chapters, the underlying combinatorial structure of tactus allows us to

digitize what appears to be the continuous flow of music, generating a pixelated blueprint (tone-

metric wave) of the textural flow for the description of the unfolding of metric, harmonic, and

melodic relationships. This pixelated blueprint and the related tree structures allow us to model

the emergence of elaborated details of the musical surface as “efflorescence” (see definition in

Terminology section of Chapter 1).

Partitioning and analytical organization of efflorescence is challenging because, when

efflorescence is perceived (i.e., when it becomes a stimulus perceived by the listener), the

analytical scaffolding is momentarily suspended; conversely, when the analytical perception is

processed, efflorescence disappears. In other words, if we focus our attention on individual notes

and chords (discrete units), no flow can be perceived; contrarily, if we focus on the metrical

unfolding of music, we lose the details of the individual constituent elements of that flow.26 This

26 A parallel with the Gestalt principles of proximity, closure, and good continuation emerges here. Brett D. King,

and Michael Wertheimer, Max Wertheimer & Gestalt theory (New Brunswick: Transaction Publishers, 2005).

52

problem highlights an important difference between efflorescence and the traditional notion of

diminution in music.

The existence of improvisational and compositional treatises, which provide lists of diminutions

to be applied as fill-ins for specific melodic intervals – for example, those of Spiridione (1615–

1685)27 and Girolamo Diruta (1554–1610)28 – speaks to the modular character of the

compositional and improvisational practices of the Renaissance and early Baroque; conversely,

the notion of efflorescence provides an analytical tool for the description of how flow in

becoming can be parceled, quantified, and described. In a repertoire such as North German organ

music, tone-metric analysis and its emergent efflorescence perceptually translate the modularity

typical of Renaissance and Baroque diminution practices into a sense of dynamic flow, thus

bridging compositional and perceptual aspects of music. In other words, efflorescence is an

emergent property of the metric unfolding of piece, and, therefore, not the same as the

Renaissance and Baroque notion of diminutions, which are pre-conceived melodic patterns.

The ability of efflorescence to model the integration of compositional modularity and metric

flow speaks to the dualistic nature of music composition and improvisation as both modular and

transformative processes. Historically, numerous scholars of various disciplines have discussed

such dualism in both music and the arts more generally. Some of those scholars come from a

more philosophical perspective, some from a more music-theoretical point of view. In the

following paragraphs, I will discuss some of the ideas that have influenced my work towards the

creation of tone-metric analysis; in particular, I will focus on aspects of the work of German

mathematician Gottfried Wilhelm Leibniz (1646–1716) and French philosopher Gilles Deleuze

(1925–1995),29 and aspects of fractal theory.

The analytical rationale provided by tone-metric analysis is centered on a process of

transformation and evolution of the sonic texture analogous to what Leibniz identified as plica ex

27 Edoardo Bellotti, ed., Preface to Nova instructio pro pulsandis organis, spinettis, manuchordiis, etc: pars prima

(1670); pars secunda (1671) [by Spiridion] (Colledara: Andromeda, 2003). 28 Murray Bradshaw, and Edward Soehnlen, Il Transilvano [by Girolamo Diruta, 1593] (Bologna: Forni editore,

1969). 29 For introductory discussions on these topics, see Robinson Keith, “Events of difference: the fold in between

Deleuze’s reading of Leibniz,” Epoche: A Journal for the History of Philosophy 8.1 (2003): 141-164; Adam

Wilkins, Modes, monads and nomads: individuals in Spinoza, Leibniz and Deleuze (State University of New York at

Stony Brook, 2008); and John Fauvel, Raymond Flood, and Robin J. Wilson, Music and mathematics: from

Pythagoras to fractals (Oxford: Oxford University Press, 2003).

53

plica, or the process through which “finitely extended parts of matter” unfold from one

another.30 Connecting back to Leibniz’s work, Deleuze describes, throughout his (in several

senses) baroque study The Fold,31 how this unfolding is tied to the relationship between the

biological concepts of preformism and epigenesis, which he uses to explain the generative

process in the Baroque architectural understanding of space, and the relationship between

structure and ornament. Preformism develops the idea that each organism is generated by

another organism of the same nature, plica ex plica; applied to music, this concept implies that

surface details (e.g., embellishments and diminutions) are ‘folds’ of, or of the same nature as,

large-scale structures from which they originate.32 Epigenesis is defined as the development of

organisms through differentiation. In musical terms, this implies that surface details present

features that are unique to, and not merely replicas of, the structural fabric of pieces, and unfold

in a dialectic discourse that constantly renews itself.33

This understanding of musical phenomena as both preformistic and epigenetic also subtends the

mathematical relationships governing fractals, which are abstract objects that are both self-

similar (preformism) and continuous (epigenetic).34 As I will demonstrate in Chapter 3, fractals

can be used to create a music-analytical space in which it is possible to describe tactus as the

preformistic and recursive aspect of music (since it is the basis of music-unfolding and repeats

itself always identically from one measure to another), and music-unfolding as epigenetic and

continuous (since melodic and harmonic patterns acquire musical meaning through their

becoming). More specifically, as we have seen in Chapter 1, tactus in this dissertation is

understood as an algorithmic process that alternates strong and weak accents in specific

combinations of thesis and arsis;35 these can be studied through combinatorics, by using the

30 Rocco J. Gennaro, and Charles Huenemann, New essays on the rationalists (New York: Oxford University Press,

1999), 161. 31 Gilles Deleuze, The fold: Leibniz and the Baroque (Minneapolis: University of Minnesota Press, 1993). 32 These are enticing ideas from a musical-theoretical perspective since they tie in neatly with Schenker’s concept of

the replicative yet differentiated continuity of voice-leading transformations across structural levels – his epigraph

motto to Free Composition of “idem sed non eaodam modo” (“the same but not in the same way”) –, and with the

notion that surface details are fully determined by, and of the same tone-metric nature as, tactus unfolding. 33 In Schenkerian terms, this recognizes that surface diminutions typically do not relate to deeper level voice-leading

patterns in fully nested or replicative ways, but are rather more florid and intuitive in their musical motivic

associations. 34 In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the

same shape as one or more of the parts); continuity can be defined as a property of certain functions, which can be

called continuous if the output depends on the input so that a small change in the input only leads to a small change

in the output. This effect is commonly known as the butterfly effect. 35 See Mattheson quote in Chapter 1.

http://en.wikipedia.org/wiki/Mathematics

http://en.wikipedia.org/wiki/Similarity_(geometry)

54

fractal-numeric triangle of French mathematician and philosopher Blaise Pascal (1623–1662)

and its geometric isomorphic version, the triangle of Polish mathematician Wacław Franciszek

Sierpiński (1882–1969).36 The mapping of tactus onto the Pascal/Sierpiński triangles produces

specific numerical patterns that, mapped back onto the piece, allow for the subdivision of the

piece into sections that are used to build tone-metric waves, in a process that will be explained in

Chapter 3. The steps just indicated are possible because of the fractal properties of: (1) recursion,

or the determination of a succession of elements (as numbers or functions) by operation on one

or more preceding elements according to a rule or formula involving a finite number of steps;

and (2) self-similarity, or the quality or state of having an appearance that is invariant upon being

scaled larger or smaller.37

The use of a geometric/fractal space to describe tone-metric waves and, ultimately, formal

analysis in music is, to my knowledge, entirely new in the panorama of music theory research.

However, geometry and mathematics have been amply used to map out musical relationships.

For example, wave-like graphs have been used in the past to represent not only sonic phenomena

like actual sound waves, but also the formal flow and structure of music, e.g., in addition to

Kurth, above: Viktor Zuckerkandl (1896–1965), Wallace Berry (1928–1991), and Frederick

Neumann (1899–1967).38 In this dissertation, I use wave-like graphs to visually describe how

meter (tactus plus rhythm) influences the combination of harmonic and melodic features to

generate what we can define as a “sense of motion” or a perception of unfolding in music.

Waves are generated by the mapping of the combinatorial features of beat and rhythm, and not

by any other musical element (e.g., melodic or harmonic considerations). As such, they do not

provide any details about the actual harmonic and melodic content, relationships, and hierarchies

among sonic events; for example, they cannot say anything about how two chords are

functionally related. Rather, harmonic and melodic content is drawn into the analytical

observation through the tone-metric approach and wave representations. In order to graphically

represent such musical relationships within a wave structure, I use graphic trees, which are

36 For an introduction to Pascal’s triangle, see A. W. F. Edwards, Pascal’s arithmetical triangle (London: C. Griffin,

1987). It is also known as the triangle of Niccolò Fontana Tartaglia (1499–1557). 37 Benoit B. Mandelbrot, The fractal geometry of nature (San Francisco: W.H. Freeman, 1982). 38 Justin London, “Rhythm in twentieth-century theory,” in The Cambridge history of Western music theory, ed.

Christensen Thomas (Cambridge: Cambridge University Press, 2001), 695–725.

55

organizational structures widely used to describe hierarchies in, for example, linguistics (Noam

Chomsky),39 and music (A generative theory by Lerdahl and Jackendoff).

Theorists have made considerable use, especially in the past few decades, of a variety of graphic

renditions and geometric models as analytical tools for the description of musical relationships.

For example, neo-Riemannian transformational theories (e.g., by David Lewin40) describe triadic

transformations through the use of planar segmentations through which intervallic relationships

can be built. The geometrical arrangement and understanding of triadic relationships as based on

intervallic associations was influential in my research, as it provided me with the first basic

intuition that the segmentation of melodic intervals could be mapped onto the temporal domain

and be used to describe musical flow as an emergent property of discrete and quantifiable units

(similarly to how triadic relationships could be described in terms of their constituent melodic

intervals beyond traditional tonal functionality).41

In this section, I have presented an overview and critique of some aspects of past and present

analytical approaches to tonal repertoire. To conclude, I would like to point out that tactus and

accentuation patterns are, in the work of the above-cited scholars, only occasionally integrated

into the analytical discourse. None of the theories noted above is concerned with tactus from an

ontological perspective, which, one might think, ought to be a priority. This is not only an

abstract philosophical inquiry, but a practical music-analytical issue. Tactus, as I have defined it

so far, is a sequence of accents, algorithmically conceived and automatically carried out

throughout pieces, as a motoric feature of the music-unfolding; beyond this motoric feature,

there is nothing musically derived in the notion of tactus (e.g. tactus does not spring from

39 Noam Chomsky, “Three models for the description of language,” IRE Transactions on Information Theory 2.3

(1956): 113–124. 40 David Lewin, Generalized musical intervals and transformations (New Haven: Yale University Press, 1987). 41 The understanding of the concept of triad as described by neo-Riemannian theories is based on a conceptual

intervallic mapping. This generalized approach to mapping allows theorists to generate the Tonnetz as a uniform

field that does not account for the length of the sides of triads. This mapping in turn allows neo-Riemannian

theorists to assert that triads can be flipped around any axis (3rds and 5ths), and to create an analytical system based

on the notion that, because any flipping can occur, triads can transform into each other. However, as I hope to

demonstrate in a future paper (in preparation), triads’ sides (particularly in this early tonal repertoire) do have

specific sizes, and always form scalene (rather than equilateral) triangles, which do not allow for indiscriminate

mapping around any axis. The implications of these limitations in triadic transformations are numerous: for

example, we cannot simply integrate major and minor triads through flipping, as major and minor triads belong to

two complementary, but distinct, systems. In the paper, I describe how these limitations of triadic transformations

can be linked to forms of harmonic organizations such as the concept of “negative harmony” elaborated by Ernst

Levy, and Siegmund Levarie in their A theory of harmony (Albany: State University of New York Press, 1985).

56

harmonic, melodic, or even rhythmic configurations or considerations). As I have already

suggested, considering tactus not as a musical but as a combinatorial-mathematical-geometrical

space can help us shed some light on its role in shaping compositional features, perceptual

elements, and performance practices in music, a question that seems to have interested scientific

thinkers and attracted the curiosity of music scholars alike in the past few centuries. In the next

section, I will discuss a few non-musical approaches to time unfolding in music that are relevant

to the definition and analysis of tactus.

2.6 Scientific developments and studies in beat perception

Over many centuries, the use of cross-domain mapping42 involving arithmetic, geometry, and

astronomy has helped musicians and music scholars to define, explain, and categorize sonic and

musical phenomena. The connection between astronomy and music, for example, is a

particularly fascinating one for its rhetorical power. From the ancient theory of the “music of the

spheres”43 to the understanding of music of Johannes Kepler (1571–1630) as a mirror of the

cosmos in his Harmonices Mundi,44 astronomy has fascinated musicians and scholars alike for

centuries.

In his Musurgia universalis,45 Athanasius Kircher (1602–1680) framed musical knowledge in

terms of relationships between the universe, and music and musical instruments. As musicologist

Kerala Snyder notes, Kircher provides one of the most striking and fascinating descriptions of

the organ as a mirror of the macrocosm,46 and describes how cadences are related to the planets’

42 Lawrence M. Zbikowski, Conceptualizing music: cognitive structure, theory, and analysis (Oxford: Oxford

University Press, 2002). 43 Jamie James, and Robert M. Williamson, “The music of the spheres: music, science and the natural order of the

universe,” American Journal of Physics 64.5 (1996): 667. 44 E. J. Aiton, A. M. Duncan, and Judith V. Field, The harmony of the world [by Johannes Kepler] (Philadelphia:

American Philosophical Society, 1997). 45 Athanasius Kircher, Musurgia Universalis. Reprint Der Deutschen Teilübersetzung Von Andreas Hirsch,

Schwäbisch Hall 1662, Melanie Wald ed. (Kassel: Bärenreiter, 2006). 46 Kerala J. Snyder, The organ as a mirror of its time: North European reflections, 1610–2000 (Oxford: Oxford

University Press, 2002), 8–9.

57

orbits:

Kircher developed his own system for adapting the harmony of the planets to human vocal polyphony. In

tune with his geocentric worldview, the earth functions as the bass, the proslambanomenos, the lowest

pitch in the ancient Greek system of music theory. He orders the traditional seven planets above it, with

the soprano and tenor each consisting of groups of three alternating consonant and dissonant planets and

the sun between them in the alto, which “directs the other planets, making them lively and sounding with

its rays.” The dissonant planets Saturn and Mars are mediated in the soprano by the consonant Jupiter;

likewise, dissonant Mercury comes between consonant Venus and Moon in the tenor: “So it is with the

grim planets Saturn and Mars; what evil would they cause in the lower world with their poisonous

cooperation, if the good Jupiter did not stand between them, conciliate, and temper them; what are Mars

and Saturn other than dissonances?” The musical example that he offers […] indeed consists of an

alternation of consonances and suspended dissonances.47

From today’s analytical perspective, Kircher’s view of the relationship between music and the

macrocosm can hardly be considered a reliable source of information (on astronomical grounds

alone!); however, at the time when Musurgia universalis was written, this description of a

cadence as generated by the motion of planets posited a narrative of great rhetorical power that

was generally accepted as a possible explanation of musical phenomena.

An example of how this mindset influenced the composition and understanding of musical

repertoires in the early Baroque in Germany is found in Mattheson’s Der vollkommene

Capellmeister, in which Mattheson declares that Dietrich Buxtehude successfully translated

cosmic phenomena into music. According to Mattheson, Buxtehude conceived of seven, now

lost, keyboard suites as depicting “the nature and qualities of the planets.”48 Mattheson’s

comments are echoed in recent times by the Dutch organist Piet Kee, who entertained the

possibility that Buxtehude’s Passacaglia in D minor, BuxWV 161, could have been conceived as

a description of the moon cycle. According to Kee,”49 the piece’s form, its number of variations,

and the tonal relationships between sections contribute to the large-scale understanding of the

piece as a metaphorical representation of specific cosmic relationships. Mattheson’s reading of

Buxtehude’s seven suites and Kee’s interpretation of Buxtehude’s Passacaglia are but two

examples of how, in the Renaissance and early Baroque, these pieces mirrored cosmic

relationships and represented them in musical terms, proportions, and form through the

47 Snyder, The organ, 82–83. 48 Ibid., II/4.73. 49 Piet Kee, “Astronomy in Buxtehude’s passacaglia,” Organist’s Review (reprinted in 2007) 93.3, (1984/2007): 17–

27.

58

combination of harmonic and melodic designs that, in themselves, are generated (according to

these authors) from nature.

As rhetorically fascinating as these perspectives might be, they remain irrelevant from a

scientific point of view, and bear little or no consequence for any music-analytical observations.

Conversely, arithmetic, geometry, and physics played important roles in the advancement of the

understanding of how sound is generated and combined to create musically meaningful patterns.

Pythagoras, for example, was deemed to be the first to have systematically discovered that music

intervals are governed by numerical and geometrical ratios. Centuries later, during the

Renaissance and the Baroque, Aristotle’s ancient theories of motion were still systematically

used to describe phenomena such as harmonic tensions and resolutions. The influence of science

was not limited to the music-theoretical domain: throughout the nineteenth century, many new

experimental musical instruments were created thanks to the integration of science and art,

combining ancient knowledge and new technologies. For example, Hermann von Helmholtz

(1821-1894) devised the siren, a mechanical instrument able to produce musical intervals

through the spinning of a wheel in which air is injected into holes found at the distances of

Pythagoras’s ratios.

As technology progressed into the digital era, and more powerful computers were developed,

fractal geometry, made popular, among others, through the research of Benoit Mandelbrot (1924-

2010) and beautiful digital visual renderings, started to attract more attention among music

researchers and composers, and found application in both music analysis and composition, and in

the study of sound phenomena such as pink noise.50 Within the context of this broad and ongoing

research about the properties and applications of fractals in music, this dissertation explores the

potential of fractalism in elucidating and modeling metric relationships. More specifically,

starting in Chapter 3 I will show that fractal geometry can help model the relationship between

tactus, meter, rhythm, and harmonic and melodic patterns, and provide insights into the

algorithmic/predictive mechanics of music-unfolding as a function of those fractal structures.

Interestingly, research on beat perception has shown some evidence for a relationship between

50 John Fauvel, Raymond Flood, and Robin J. Wilson, eds, Music and mathematics: from Pythagoras to fractals

(Oxford: Oxford University Press, 2003).

59

beat perception and fractal patterns;51 for the predictive nature of beat processing; and the

interaction between two distinct circuits subtending beat (striato-thalamo-cortical) and rhythm

(cerebello-thalamo-cortical) processing, both of which are also the primary circuits of motor

control.

The neurobiological underpinnings of musical mechanisms underlying beat and rhythm

perception show (also through neuro-imaging studies) that musical processes (both passive and

active; i.e., listening and performing) are the result of the engagement of motor-related areas of

the brain. This engagement suggests that the interplay between cognitive-theoretical constructs

and neurobiological processes is an underlying factor of our understanding and perception of

time unfolding and articulation in music. Out of this view of time perception, I argue, springs our

everyday notion of time articulation (as tactus and rhythm) in music, which is nothing more than

what cognitive linguist and philosopher George Lakoff and cognitive neuroscientist Mark

Johnson call a “primary metaphor,” or a metaphor that is not only, or primarily, a literary device

but the “pair[ing] of subjective experience and judgment with sensorimotor experience.”52

Music cognition and perception, and the study of the neurobiology of music, have provided me

with useful insights into the neurobiological nature of time organization, and have helped me to

shape my understanding of tactus and rhythm as “primary metaphors” describing time-related

neurobiological processes. How are these concepts contextualized within music-theoretical and

analytical approaches with respect to my research? In the next section of this chapter, I will

discuss the music-theoretical background on which I have built tone-metric analysis. I will focus

in particular on the discrete/continuous dualism in music and the notion of fractals in tone-metric

analysis, as well as the rationale for using fractal geometry and an algorithmic process to explain

relationships between tactus, rhythm, melody, and harmony.

51 Michael Hove, et al. “Interactive rhythmic auditory stimulation reinstates natural 1/f timing in gait of Parkinson’s

patients,” PLoS ONE 7.3 (2012): e32600. 52 George Lakoff, and Mark Johnson, Philosophy in the flesh: the embodied mind and its challenge to Western

thought (New York: Basic Books, 1999), 49.

60

2.7 Historic-analytical contextualization of the repertoire analyzed

The preparatory work for this dissertation required investigation into historical-musicological

surveys, monographs on individual composers, analytical studies, sources on organ building and

temperament, compositional and improvisational practices and styles, critical apparatus of music

editions, and recordings and their critical notes. While I refer the reader to the bibliography for a

representative – although not exhaustive – list of sources consulted in preparation for this

dissertation, I will here discuss those sources that were particularly useful in its preparation.

With specific reference to North German music, The stylus phantasticus and free keyboard

music of the North German Baroque by George Webber and Paul Collins53 has been useful as a

general overview of major areas of musicological research on North German organ and keyboard

repertoire; in particular, their discussions about the classification of music styles, the notion of

stylus phantasticus in eighteenth-century writings and its relationship to rhetoric, and the

analysis of sample works were particularly useful to my research. Kerala Snyder’s seminal

Dietrich Buxtehude, organist in Lübeck54 and The organ as a mirror of its time: north European

reflections, 1610–200055 provided me with useful information into, respectively, Buxtehude’s

life and work in relationship to his socio-cultural world, and organ building and the cultural

milieu within which compositional practices of North German Baroque music flourished.

Primary sources about organ and keyboard improvisation in seventeenth-century Northern

Germany are, unfortunately, lacking. My knowledge about improvisational techniques is based

on: (1) primary sources (e.g., treatises by Banchieri, Spiridione, and Diruta among others) of the

Italian Renaissance and early Baroque, which constituted one of the primary stylistic models in

the development of North German organ repertoire; (2) Giorgio Sanguinetti’s The art of

partimento: history, theory, and practice,56 a reference work that integrates both theoretical-

musicological scholarship and performance insights into the world of partimento; and (3) my

53 Geoffrey Webber, and Paul Collins, “The stylus phantasticus and free keyboard music of the North German

Baroque,” Music and Letters 88.3 (2007): 487–489. 54 Kerala J. Snyder, Dieterich Buxtehude, organist in Lubeck. New York (Schirmer Books, 1987). 55 Kerala Snyder, ed. The organ as a mirror of its time: north European reflections, 1610–2000 (Oxford: Oxford

University Press, 2002). 56 Giorgio Sanguinetti, The art of partimento: history, theory, and practice (New York: Oxford University Press,

2012).

61

Master’s degree lessons with organist William Porter, world-renowned improviser and scholar of

both Italian and German Baroque organ music.

On this side of the Atlantic, a relatively recent interest in analytical-historical perspectives on

North German organ improvisation has produced interesting dissertations focused on both

historical and methodological approaches to improvisatory practices in North German Baroque

organ music. More specifically, I studied Karin Nelson’s Improvisation and pedagogy through

Heinrich Scheidemann’s Magnificat settings57 and Michael Callahan’s Techniques of keyboard

improvisation in the German Baroque and their implications for today’s pedagogy.58 These

summarize and integrate theoretical-musicological knowledge about organ improvisation with

practical applications, emphasizing the modularity of North German organ improvisation as

based on the juxtaposition of discrete units of melodic and harmonic patterns (formulae) to

generate the sense of a continuous harmonic and melodic flow. This juxtaposition ultimately

creates an interplay between: (1) the modularity and continuity that tone-metric analysis

quantifies and describes, (2) the modeling and integrative aspects of improvisation and

composition, and (3) the interplay between harmony, melody, and rhythm as generated within

tactus (something I discussed a few pages earlier in this dissertation, when writing about

efflorescence).

While searching for sources about individual composers, I realized that, unfortunately, no in-

depth monographic sources on North German organists, especially of a theoretical nature, seem

to exist at present (with the exception of Heinrich Scheidemann’s keyboard music: transmission,

style and chronology by Pieter Dirksen59); for example, no analytical monographs on Bruhns or

Pachelbel have been published in either musicological or theoretical domains.

The exploratory work carried out in preparation for this dissertation brought me to investigate

critical notes to published scores. The Broude Trust edition60 of music by Buxtehude, for

example, has been particularly useful for the richness of the sources cited, and the interesting

57 Karin Nelson, Improvisation and pedagogy through Heinrich Scheidemann’s Magnificat settings (PhD diss.

Gothenburg: University of Gothenburg, 2011). 58 Michael R. Callahan, Techniques of keyboard improvisation in the German Baroque and their implications for

today’s pedagogy (PhD diss. Eastman School of Music, University of Rochester, 2010). 59 Pieter Dirksen, Heinrich Scheidemann’s keyboard music: transmission, style and chronology (Aldershot:

Ashcroft, 2007). 60 Kerala Snyder, and Christoph Wolff, eds. The collected works [of Dietrich Buxtehude] (New York: Broude Trust,

1987).

62

editorial policies: for example, the pedal staff is often missing in the edition, as the editors

wanted to include the pedal part within the manual staves to reproduce the arrangement of notes

of the tablature that served as a basis for the modern edition (this has important consequences, as

the texture and role of the bass line and pedal part, not necessarily played with the pedals, can

help performers to rethink pieces from the perspective of voice distribution between manuals and

pedal).

Among the large number of recordings consulted during the preparatory work, I would like to

cite Harald Vogel’s CD set for the complete organ works of Buxtehude,61 which made an impact

in the field of early music since its first release, as the first sound recording edition that included

extensive critical notes, and the stop lists and temperament features of the organs used in the

recordings. Additionally notable are the complete works of Bruhns and Hanff62 by William

Porter, considered by early music scholars among the most valued editions of this repertoire

published in North America.

2.8 Perspectives on the relationship between performance practice and

music analysis

Tone-metric analysis is a methodology that can inform the work of performers from both

analysis- and source-related performance practice perspectives, as it provides both a

mathematical-analytical approach to the modelling of musical form and a methodological

platform for an interpretation of musicological sources in light of how the information provided

by those sources can be applied in performance. But how does tone-metric analysis enter the

dialogue between analysis and performance? In the following paragraphs, I will discuss this

relationship in detail.

61 Harald Vogel, Complete organ works [of Dietrich Buxtehude] (Detmold: MDG Gold, 2007). 62 William Porter, Complete organ works of Nicolaus Bruhns and Johann Nicolaus Hanff (Seattle: Loft Recordings,

1998).

63

With respect to the primary repertoire analyzed in this dissertation (i.e., North German organ

music), it is difficult to identify sources that directly discuss performance-related issues or

provide performance-related indications. To my knowledge, contemporary analytical

publications about North German organ repertoire focus heavily on the study of how melodic,

harmonic, and rhythmic patterns influence the phrasal unfolding of music, from John Butt’s

chapter on German and Dutch keyboard music before 1700 in Alexander Silbiger’s Keyboard

Music Before 170063 to Peter Williams’s The organ music of J. S. Bach.64 While those works are

extremely useful from an analytical perspective, as they provide us with insights into both

detailed and large-scale articulations of patterns and phrases, they are not meant to provide

musicians with suggestions on how to perform passages, and provide little or no indication about

how to articulate the patterns and phrases they analyze.

I would argue that, ironically, the written indications that have had some of the most significant

impact on performance practice of this repertoire are those that are not based on score analysis.

For example, the registration indications provided by Vogel in his Buxtehude CD booklets,65

which have been considered the ultimate standard in terms of how to register a piece by

Buxtehude based on a soundscape he would have been familiar with, or the performance

indications by Snyder in her Dieterich Buxtehude, organist in Lübeck,66 which are born from her

studies and understanding of organological features of instruments and primary sources and not

necessarily from a systematic score analysis.

To conclude, I found it difficult to identify significant studies that qualify as performance

practice sources for North German music. Works like those of Butt and Williams informed by

analytical insights into the melodic and harmonic style of North German organ music, and those

of Vogel and Snyder have provided me with information on how to approach the soundscape that

Buxtehude in particular would probably have privileged. However, almost nothing is known

about North German organ performance practice (as, unlike for French, Italian, and Spanish

music, no compositional treatises and no embellishment tables were left to us from Northern

Germany). Moreover organ music of this period does not produce the kind of systematized

63 Alexander Silbiger, Keyboard music before 1700 (Schirmer Books, 1995). 64 Peter Williams, The organ music of J. S. Bach (Cambridge University Press, 2003). 65 Vogel, Complete organ works, 2007. 66 Snyder, Dieterich Buxtehude, 1987.

64

phrasal and formal organization that attracts the attention of musicologists and theorists

interested in performance practice of this repertoire (I believe this is one of the reasons why

organ music of this era is rarely featured in contemporary music performance or analytical

journals); and organists interested in this repertoire are more than happy to transmit their

performance practice knowledge of this repertoire orally, from teacher to student, exactly in the

same way that the North German Baroque organists did before them.

From an analytical-theoretical perspective, tone-metric analysis provides an algorithmically

generated background, over which performers can engage in a dialogue about performance

issues. The ultimate goal of tone-metric analysis is to provide the readers with a series of

interpretative values to be applied, at their discretion, in the performance of the piece. Similarly

to the way Schenker’s graphs make aspects of the voice-leading and motivic aspects of music

emerge but do not substitute for the score itself (although Schenker viewed his graphs as ‘part of

the score’), tone-metric analysis only provides a blueprint for the interpretation of the score from

an alternative perspective, which is based on the interaction between tactus, rhythm, and the

note-content of the piece (rather than melodic/thematic or harmonic/functional analysis).

While the graph of tone-metric analysis is conceived algorithmically, the final analytical product

of tone-metric analysis invokes an interpretive act, not a prescriptive indication: its graphs

provide points of musical engagement that can be used by performers in their interpretive

approach to music. A number of analytical approaches have been influential in shaping and

clarifying the role of tone-metric analysis as a performance-oriented analytical methodology.

Some of those theories expressly provide performance indications, others analytical-theoretical

instructions that can stand in relationship to performance practices.

For example, Brian Alegant and Don McLean’s theory of tokens67 studies how melodic-thematic

elements in music have corresponding structural intervallic homologues that emerge from the

underlying motivic structure of pieces when analyzed via a Schenkerian-reductive type of

analysis. Highlighting those analytically revealed melodic-thematic relationships is, for Alegant

and McLean, a first step to address the issue of how the relationship between thematic-melodic

and structural homologues can, and should, guide the performance of the piece.

67 Brian Alegant, and Don McLean, “On the nature of enlargement,” Journal of Music Theory 45.1 (2001): 31–71.

65

Another methodology that aims at relating analysis and performance is the approach to rhythmic

analysis of Wallace Berry. For example, in his article “Metric and rhythmic articulation in

music,”68 he analyzes the rhythmic structure of the Prelude in E minor by Chopin, providing

insights into how rhythm and meter influence tonal relationships and accents, thus bridging

metric analysis with performance practice. In this dissertation, I also analyze the same piece by

Chopin (see Chapter 6, pages 155–162), but from a tone-metric perspective, discussing how

metric structure determines formal boundaries, and how subtle metric details can inform our

performance of those boundaries.

Unlike the previous two examples, other analytical approaches do not explicitly address

performance-related concerns. For example, William Caplin’s theory of formal functions69

primarily aims at defining phrasal boundaries and typologies (e.g., sentences or periods) based

on harmonic functionality, and Janet Schmalfeldt’s “one more time technique”70 provides

insights into a compositional practice typical of compositions in the classical and post-classical

era: the use of avoided cadences followed by the often-varied repetition of the previous cadential

material as a means to generate unexpected lengthening of the closing section of the piece or

section. While these approaches do not explicitly address performance-related concerns, they can

be used by performers to identify points of interest in the articulation of the piece’s form, and

their findings can provide a solid analytical ground over which tone-metric analysis can enter the

dialogue. For example, as I will show through the analysis of an excerpt from the Piano Sonata

in C major, K. 279/189d, iii, by Mozart, on pages 151–155, while Caplin’s approach provides a

theoretical rationale for analytical compartmentalization of pieces into their constituent phrasal

units, tone-metric analysis’ graphic interpretations can inform us about how those boundaries

and phrases are affected by metric structure, and how each boundary and phrase might be

interpreted in performance.

This dissertation provides examples of this relationship between performance and theory (e.g.,

the analysis of the chorale-preludes on Durch Adams Fall by Buxtehude, on pages 109–117, and

68 Wallace Berry, “Metric and rhythmic articulation in music,” Music Theory Spectrum 7.1 (1985): 7–33.

doi:10.1525/mts.1985.7.1.02a00020. 69 Caplin, William E., Classical form: a theory of formal functions for the instrumental music of Haydn, Mozart, and

Beethoven (New York: Oxford University Press, 1998). 70 Schmalfeldt, Janet, “Cadential processes: the evaded cadence and the ‘one more time’ technique,” Journal of

Musicological Research 12.1-2 (1992): 1–52. doi:10.1080/01411899208574658.

66

the prelude in E minor by Bruhns, on pages 140–145). In all these examples, and in tone-metric

analysis more generally, the objective of the analysis is not to perform the tone-metric graph, but

to use the graph to inform analysts and performers of the existence of otherwise hidden points of

articulation (i.e., tactus-derived pivots) in the formal organization in music that surface through

the algorithmic unfolding of tactus and rhythm.

2.9 Conclusions

In this chapter, I discussed theoretical, musicological, and performance-practice-related aspects

that have shaped my understanding of the cultural and musical contexts in which my research is

situated. More specifically, I provided an overview and critical discussion of both the historical-

musicological and theoretical premises on which tone-metric analysis is based, and an overview

of the scientific developments that shaped my understanding of how the role of tactus can be

integrated within the tone-metric analytical discourse. Moreover, I introduced the basic concepts

of the combinatorial nature of tactus, and investigated the secondary literature for its role in a

variety of theoretical approaches and historic-musicological sources.

In the next four chapters, I will show how to build and analyze tone-metric waves (Chapter 4),

and will apply the method to both the core North-German Baroque organ music of the

dissertation (Chapter 5) and to later repertoires (Chapter 6). But first (Chapter 3), I describe the

theoretical framework of tone-metric analysis, to which we now turn.

Chapter 3

Conceptualizing the musical space

In this chapter, I discuss the theoretical background on which the analytical methodology of

tone-metric analysis develops. I first discuss the notion of tactus as a binary phenomenon

characterized by specific combinations and permutations of strong and weak beats. More

specifically, after delineating the features of tactus as a binary phenomenon I explain how its

binary structure can be described through the mathematical array known as Pascal’s triangle and

its geometric-fractal counterpart, Sierpinsky’s triangle. I also provide a rationale for a fractal

approach to meter. I do not consider tactus from a musical point of view, but from a

mathematical-geometrical one. Some of the examples in this chapter might seem removed from

standard musical practices and contexts; however, their musical meaning will become clearer in

the following chapters.

Music is made of sonic objects (i.e., individual notes and chords) associated with specific

rhythmic values (e.g., eighth and quarter notes). From a theoretical perspective, sonic objects and

rhythmic values exist independently from each other (we can imagine a note without a specific

rhythmic value attached to it, and a duration or rhythmic value without a specific note attached to

it). Sonic objects and rhythmic values are, in fact, ontologically different: the former are

physical phenomena perceived as sound waves, and the latter arithmetical constructs for the

organization of sonic events in time. However, in the performance and perception of music,

sonic objects and rhythmic values cannot be separated: simply put, nobody can play or sing a

melody or a chord progression without some sort of durational value attached to it. I refer to the

necessary integration of sonic objects and rhythmic values as sonic events.

In each piece, sonic events are functions of (i.e., unfold in time based on and according to) the

piece’s specific tactus (i.e., an algorithmically pre-conceived recursive pattern of strong and

weak accents). A sonic event contextualized within that tactus acquires a unique position within

the unfolding of a piece (e.g., a chord on beat 3 of measure 1 embodies a perceptual and

68

compositional role that is not found anywhere else in the piece, as the same chord on beat 3 of

measure 2 will necessarily be different, at least in its accentuation, from the chord on beat 3 of

measure 1). This unique entity combining sonic event and tactus of each unit that I call a tone-

metric event. What are the implications of the interaction between tactus and tone-metric events,

and between sonic objects and rhythmic values? And why is it important to define the

relationship between sonic events and tactus as an integrated tone-metric event? In order to

answer these questions, I approach tactus not as a musical phenomenon, but as a set of specific

combinations1 and permutations2 of accentuation patterns.

3.1 Tactus, combinatorics, and fractal spaces

Everyone has experienced the pleasant feeling of tapping a foot to keep the time to a familiar

song playing on the radio. “Keeping the beat” of a piece is not a modern phenomenon, as

Renaissance and the Baroque musicians, especially choir conductors, used to keep the beat (with

their arms and hands, or wooden sticks banged on the floor) during the performance of pieces. In

particular, during the Renaissance and the early Baroque the word most frequently used to

indicate the beat was tactus. Tactus is the perfect passive participle of the Latin verb tangere (“to

touch”) and suggests the image of someone’s hand or foot touching and lifting away from a

surface, like tapping on a table, to indicate the basic pulse of music. Bodily metaphors for the

description of tactus have been used for centuries. One description from the famed English

musician John Dowland (1563–1626) seems to summarize particularly well the notion of tactus:

“Tact is a successiue motion in singing, directing the equalitie of the measure: Or it is a certaine

motion, made by the hand of the chiefe singer, according to the nature of the marks, which

directs a Song according to Measure.”3 Dowland’s quote suggests that the musical discourse is

led by tactus as a form of motion and succession of events, or, better, the change of position

(e.g., of the hand of the lead singer/ensemble director) guiding the unfolding of the music.

1 A combination is a selection of elements from a collection, such that the order of the selection does not matter. 2 A permutation is the arranging or re-arranging of the members of a set into some order. 3 John Dowland, Gustave Reese, and Steven Ledbetter, A compendium of musical practice: musice active

micrologus [by Ornithoparchus, Andreas] (New York: Dover Publications, 1973), 46.

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Bodily metaphors for the description of tactus continued to abound throughout the Renaissance

and the Baroque; this abundance suggests that tactus has always been understood as dependent

on some sort of “binary-ness” (down and up, touch and lift away, rise and fall), even when it was

ternary. For example, as William Caplin states, Johann Mattheson “presents a primary division

of the measure into equal meters (our duple and quadruple meters) or unequal meters (triple).

The latter are made up of two parts, the first (thesis) lasting twice as long as the second (arsis),

just as in the original tactus theory.”4 In other words, all binary and ternary tactus and meter were

understood and organized in the basic binary form of (1) downbeat and (2) the-rest-of-the-

measure. From this perspective, tactus is a form of binary grouping for the organization of equal

and unequal meters. To my knowledge, this basic binary understanding of both duple and triple

meters as a bodily gesture has never been systematized into an analytical theory. In this chapter,

I present an analysis of the features of tactus as a binary construct made of T (thesis), or

downbeat, and A (arsis), or upbeat (or “the-rest-of-the-measure”).

The approach to the order of T and A presented in the previous paragraph might seem to

contradict Mattheson’s indications (presented above) that, for example, ternary time is made of

TTA, as, according to the information presented, a ternary pattern ought to be TAA. However, as

we will see in detail in the rest of this chapter, in combinatorial terms the ordering of T and A

does not really matter, as permutations of T and A are, for the purpose of this dissertation, the

same: TAA and TTA are both combinations of T and A including one element that is present in

double quantity in comparison with the other (e.g., 1 T and 2 As, or 1 A and 2 Ts). I have chosen

to use the T followed by As because: (1) I want to emphasize the distinction between one strong

beat and the weaker following ones; and (2) I wish to make a distinction between the approach

presented in this dissertation, which is combinatorial in nature, and Mattheson’s music-

theoretical approach.

These beat/combinatorial properties will then be systematized into a coherent and

algorithmically-based analytical system. This binary structure shapes our understanding of how

harmonies, melodies, and rhythmic patterns are combined together by composers and

improvisers, and unfold in time. In the rest of this chapter, I will explain how this binary

structure can shape the unfolding of music.

4 William Caplin, “Theories of musical rhythm in the eighteenth and nineteenth centuries,” in The Cambridge

History of Western Music Theory, ed. Thomas Christensen (Cambridge: Cambridge University Press, 2002), 660.

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An empirical manifestation of the fundamentally binary structure of metric organization is seen

in all forms of meter, as, at the smallest sub-divisional level, all forms of metric organization are

binary; for example, 3/4 is ternary at the quarter note level, but binary at the eighth-note level;

and 12/8 is binary at the dotted quarter note level (there are 4 dotted quarter notes per measure),

ternary at the eighth-note level (there are three eighth notes for each dotted quarter note), and

binary at the sixteenth-note level (there are normally two sixteenth notes for each eighth note).

The fundamentally binary structure of meter has a mathematical and perceptual5 explanation,

which is rooted in the combinatorial feature of beat, or tactus. In order to demonstrate this, we

have to step away from music-related concepts, and consider tactus from a mathematical

perspective.

Binary tactus. Let’s take as an example time signature 2/2. As we have seen in Chapters 1 and 2,

tactus is a specific arrangement of the thesis and arsis, appearing in the specific order TA. From

a combinatorial perspective, the three other possibilities in which T and A can be permutated are:

AT, AA, and TT. By considering the theoretical arrangements of T and A without limiting our

attention to the only arrangement that is musically significant (TA), we can contextualize

musical tactus within the larger spectrum of combinatorial properties of a binary system. As I

will show in the next chapters and analyses, this will allow us to geometrically model musical

tactus, and map that geometrical model onto a score for the analysis of how melodic, harmonic,

and rhythmic parameters unfold in time to generate phrase articulations and large- scale form. In

the next paragraphs, I will demonstrate how this analytical process works.

The combinatorial and permutational features of a binary set6 such as “TA” can be studied

through Pascal’s triangle:

5 For a music-theoretical discussion about the perceptual approach to meter-rhythm see Justin London, “Relevant

research on rhythmic perception and production,” in Hearing in time: psychological aspects of musical meter, 2nd

ed. (Oxford: Oxford University Press, 2012). 6 A binary set is a set containing an even number of elements.

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Figure 3.1. Pascal’s triangle.

Pascal’s triangle can be used to model the combinations and permutation of A and T (arsis and

thesis, down and up). Among the various combinations, one will be the descriptor of tactus (i.e.,

TA) studying how the combination of T and A is related to the organization of duple (e.g., 2/2,

4/4) and triple meters (e.g., 3/4, 9/8).

First, I identify all possible combinations and permutations that can occur in a set containing one

or both elements T and A. From a theoretical perspective,7 in such a set there can be: 1 TT

combination (with no A), 1 AA combination (with no T), and 2 permutations of the combination

including both T and A (this combination can present itself as either TA and AT permutations).

The “2” of the “1–2–1” line on Pascal’s triangle describes these combinations and permutations:

“1” describes the TT combination, “2” the TA and AT permutations of the combination

including 1T and 1A, and “1” the AA combination.

In Figure 3.2, the binary combinations and permutations of T and A are highlighted in blue and

red. What does this description of the combinatorial features of a binary set of elements have to

do with tactus and, more broadly, with music? Among the sets represented by “1–2–1,” “2”

7 Some of the following combinations and permutations exist from a combinatorial perspective but do not have a

direct correspondence in music (e.g., AA and TT, or AAAA and TTTT). This discrepancy between combinatorial

and musical aspects of tactus, which is a core point of this research, and is addressed and discussed later in the

chapter, is generated by the fact that I am analyzing the combinatorial nature of a musical phenomenon (e.g. the

combinations and permutations of A and T of tactus), and not the musical nature of a combinatorial phenomenon

(e.g., the tactus of combinations and permutations of A and T). In order to create a combinatorial model of tactus, all

combinatorial and permutational possibilities must be taken into account first; those combinations and permutations

that have no musical validity are then discarded, as explained later in the chapter.

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contains the pattern TA which is the pattern characteristic of tactus, while 1 and 1 only contain

theoretical AT pattern:

Figure 3.2. Pascal’s triangle with binary combinations and permutations of T and A.

The red circle highlights the positions of the 2 AT and TA permutations (“2” permutations) and

the blue circles the position of the AA and TT permutations (“1” AA and “1” TT permutations)

for a the case of a single measure. If we repeat the process for two or more measures, we will see

that only one circle per tactus (e.g., 2 in the “1–2–1” one-measure pattern) is musically relevant

as a descriptor of a viable metric pattern that starts with T and continues with As.

Before continuing, it is important to explain that only the numbers highlighted by circles on

Pascal’s triangles in Figure 3.2 and following are important for defining the combinatorial

patterns that constitute the basis of tone-metric analysis. All other numbers are left in the figures

as a contextual background noise over which the viable pattern of tactus designs emerge. As we

will see shortly, the selection of musically relevant tactus from the broader possible

combinatorial patterns will generate a specific geometric pattern for tactus from Pascal’s triangle

that, mapped onto the score, will allow us to graphically model the unfolding of the piece.

In Figure 3.3, the next line of Pascal’s triangle shows the sequence “1–4–6–4–1,” which

represents two measures of binary meter. If we consider two measures of binary meter 2/2,

which are made of TA/TA instead of one TA, the number of combinations of T and A becomes:

1 combination containing no T (AAAA); 4 permutations of the combination containing 1 T

(TAAA, ATAA, AATA, and AAAT); 6 permutations of the combination containing 2 T (TTAA,

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TATA, TAAT, AATT, ATAT, and ATTA); 4 permutations of the combination containing 3 T

(TTTA, ATTT, TATT, and TTAT); and 1 combination containing 4 T beats (TTTT). This is

summarized on Pascal’s triangle on the line “1–4–6–4–1.” Tactus, which can only be constituted

by the specific TA/TA pattern, belongs to one of the “6” of “1–4–6–4–1.”

Figure 3.3. Pascal’s triangle with the 2/2 combinations and permutations of T and A.

In Figure 3.3, the 2/2 combinations and permutations of T and A for two measures are

highlighted in blue and red. Again, the red circles indicate the musically significant tactus

patterns, which are based on a T followed by a series of As. If we apply the same process to

three and more measures and highlight the results on Pascal’s triangle, we see that binary tactus

designs a line of circles; i.e., digits 2, 6, 20, 70, 252, and so on, in Figure 3.4, which, taken

together, are a descriptor of the musically relevant tactus of a chunk of the piece, each circle

representing a single measure, in which the 2/2 tactus array generated by the positioning of the

AT permutations is marked in red):

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The string of numbers highlighted (2, 6, 20, 70, 252, and so on) is a descriptor of the articulation

of the combinatorial patterns of 2/2 tactus on Pascal’s triangle. This string is made of multiples

of 2, and is, therefore, a subset of even integer numbers. The grouping of even integers (i.e., 2

and its multiples) and their separation from the odd numbers on Pascal’s triangle generate a well-

known fractal figure called Sierpinsky’s triangle modulo-2 (in blue in figure 3.5). Indeed, a

defining feature of Sierpinsky’s triangle is its generation via this filtering of odd and even

numbers on Pascal’s triangle. As a fractal figure, Sierpinsky’s triangle is both self-similar and

recursive, that is, made of the same (self-similar) repeated patterns (recursive) at all levels of

magnification and reduction.8 These fractal features are crucial in the formation of the tone-

metric analytical methodology, as explained later in this chapter and in Chapter 4.

8 Pascal (1623-1662) and Sierpinsky (1882-1969) were among the most brilliant thinkers of their respective

centuries. The former was a physicist, mathematician, philosopher, and theologian; as a scientist, he is particularly

known for his work on probability theory, calculating machines, projective geometry, and for refuting the

Artistotelian notion of the nature abhor vaccum. The latter is known primarily as a mathematician who significantly

contributed to set theory, number theory, theory of functions, and topology.

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Figure 3.5. Sierpinsky’s triangle modulo-2 superimposed to Pascal’s triangle.

But why is this complex mapping process significant musically? So far we have seen that the

binary organization of tactus in As and Ts turns tactus into a binary set of accents. As such,

accent distribution of tactus can be mapped onto a triangular numerical array (Pascal’s triangle).

By definition, the separation of odd and even integers on Pascal’s triangle generates a fractal

figure called Sierpinsky’s triangle, which is characterized by recursivity and self-similarity at all

levels of magnification. To summarize, this mapping process allows us to map musical tactus

onto a numerical space (i.e., Pascal’s triangle) and, ultimately, onto a fractal space (i.e.,

Sierpinky’s triangle). As we will see in the next paragraphs, a reading of musical tactus within a

mathematical-fractal framework highlights specific numerical properties that can be applied to

tactus and mapped back onto the musical score, providing a fractal reading of musical meter that

can be used, as I argue in this dissertation, to generate musically meaningful analytical insights.

The analogy between binary tactus (e.g., AT, ATTT) and its unique relationship with the binary-

ness of Sierpinsky’s triangle (as generated by separating odd and even integers) finds a parallel

in the relationship between ternary tactus (e.g., ATT) and Sierpinsky’s triangle modulo-3, which

is generated by separating 3-and-its-multiples from the other digits on Pascal’s triangle. This

parallelism generates a conceptual bridge between musical meter and its combinatorial-

mathematical and geometrical forms. In other words, binary tactus is described by Sierpinsky’s

triangle modulo-2 only, and ternary tactus is described by Sierpinsky’s triangle modulo-3 only.

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The parallel between binary tactus and Sierpinsky’s triangle modulo-2, and ternary tactus and

Sierpinsky’s triangle modulo-3 provides both an interesting mathematical insight into the nature

of musical tactus and its accentuation patterns, and a generating principle of a fractal alternative

and dynamically evolving framework for the analysis of musical meter, as I will show in

musical-analytical applications in Chapters 4 through 6.

Sierpinsky’s triangle modulo-2 is a subset of the originating Pascal’s triangle. If we map the

combinatorial pattern of 2/2 studied earlier onto Sierpinsky’s triangle modulo-2, we obtain the

following figure:

Figure 3.6. Superposition of Sierpinsky’s triangle modulo-2 and 2/2 tactus array.

In Figure 3.6, the presence of both blue triangles of Sierpinsky’s triangle modulo-2 and red

circles of binary tactus highlights that all even numbers describing 2/2 are found only in the even

number section of Sierpinsky’s triangle, and therefore, that, the 2/2 array descriptor is a subset of

the binary articulation of Pascal’s triangle (described by Sierpinsky’s triangle modulo-2). A point

of interest is that the match between binary tactus and the central even-number array in

Sierpinsky’s triangle modulo-2 (made of 2 and its multiples) is unique to 2/2.

So far we have analyzed 2/2. But what happens when the tactus is another form of binary

organization such as 4/4? As already noted in Chapters 1 and 2, Mattheson’s explanation that

duple meters (both 2/2 and 4/4) are subject to the same rule that wants both T and A to be of the

same length, raises the question whether 2/2 and 4/4 are actually different from each other from a

tone-metric perspective. If both 2/2 and 4/4 were built on the same accentuation structure TA,

the combinatorial structure of the two time signatures would be same and would correspond to

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the one studied in the previous pages (as 4/4 would become TTAA, where TT are always paired,

and AA are always paired, and would present the exact same combinatorial structure as a 2/2

time signature). I argue here, however, that we can, from a combinatorial perspective, understand

2/2 and 4/4 as being built on two different patterns of accentuations: 2/2 on a TA pattern where T

and A are of the same length; and 4/4 on a TA pattern in which a strong beat T is followed by

three weaker beats AAA.9

Tone-metric analysis is not dealing directly with musical accents, but aims to demonstrate the

mathematical-combinatorial principles behind those accents. As a result, this approach allows me

to show significant differences between 2/2 and 4/4 in terms of the metric organization of music

and its performance.10 Alternatively, it would be possible to remain tied to Mattheson’s

explanation and generate graphic representations of 2/2 and 4/4 that are identical; however, in

my view this would lose some valuable interpretive distinctions between accentuation patterns

and related compositional features (e.g., 2/2 rarely makes use of sixteenth-note runs, while 4/4

often does).

According to the explanation provided in the previous paragraph, a 4/4 measure is described as

“TAAA” as it is made, in its generalized form described as a series of quarter notes, of 1 T (first

quarter note) and 3 A (second, third, and fourth quarter notes). This combination is described by

the “1–4–6–4–1” array, which contains the following permutations: no T (AAAA); 4

permutations of the combination containing 1 T (TAAA, ATAA, AATA, and AAAT); 6

permutations of the combination containing 2 T (TTAA, TATA, TAAT, AATT, ATAT, and

ATTA); 4 permutations of the combination containing 3 T (TTTA, ATTT, TATT, and TTAT);

and 1 combination containing 4 T beats (TTTT). Among them, the 4/4 measure is described by

number 4 in its TAAA permutations. If we apply the same process to more 4/4 measures, we

obtain the following articulation of Pascal’s triangle modulo-2, which highlights both 2/2 (in red

circles) and 4/4 (in green circles) tactus arrays for comparison:

9 Various beats and variants such as 6/8, which are binary at one level and ternary at another, will be analyzed

throughout Chapters 4 and 5. In this category falls also the compound 4/4 beat, which is not analyzed in this

dissertation but is certainly an interesting case of relationship between binary level (quarter notes) and ternary (three

eighth notes for each quarter note). 10 The distinction between an initial strong and following weak beats in 4/4 is also true in the modern SWMW view

of 4/4. Although the latter was not the way in which 4/4 was categorized at the time in which the repertoire under

investigation was composed, it would certainly be interesting to see whether tone-metric analysis can be adapted to

incorporate the M category, which is inconsistent with the binarism of thesis-arsis proposed by Mattheson.

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Figure 3.7. Sierpinsky’s triangle modulo-2 with 2/2 and 4/4 tactus arrays superimposed.

Figure 3.7 shows that the arrays of numbers describing 4/4 (numbers 4, 28, 220, 1820, and so on,

in green) belong entirely to the even number subset of Sierpinsky’s triangle modulo-2, as does

the 2/2 tactus pattern (numbers 2, 6, 20, 70, 252, and so onm in red).11 A comparions between

Figure 3.7 and the upcoming Figure 3.9 shows that binary tactus (in both its 2/2 and 4/4 forms)

matches the topology of Sierpinsky’s triangle modulo-2 and not that of Sierpinsky’s triangle

modulo-3 (a comparison of the numeric distributions of red and green arrays in Figure 3.7 with a

similar distribution in Sierpinsky’s triangle modulo-3 in the upcoming Figure 3.9 would

highlight that only in the modulo-2 topology those numbers are all found in the “empty” portions

of the figure, whereas in the modulo-3 topology they occur alternatively in the “empty” and

“full” portions of the space).

To summarize, binary tactus 2/2 and 4/4 are linked only to modulo-2 (and not modulo-3);

however, each describes a unique pattern within modulo-2 Sierpinsky’s triangle. In other words,

2/2 and 4/4 are subsets of modulo-2 Sierpinsky’s triangle but not subsets of each other. As

mentioned earlier in this chapter, this difference of 2/2 and 4/4 within their binary samenss has

important musical implications, both compositionally and from a performance perspective. For

example, 2/2 and 4/4 are binary forms of tactus, but are not articulated in the same way. More

specifically, 2/2 is made of 2 half notes, one T and the other A; 4/4 is made of 4 quarter notes,

11 Recall that each red and green circle indicates the location of the musically relevant combinations of Ts and As in

2/2 (red) and 4/4 (green).

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the first being T, the second, third, and fourth (subdivided into three separate quarter notes) its A.

As we will see in Chapters 5 and 6, this difference has significant implications in performance

and composition, as the internal rhythmic relationships of 2/2 and 4/4 are fundamentally

different. For example, throughout the Renaissance and the Baroque 2/2 (with very few

exceptions) did not contain sixteenth notes, while 4/4 made ample use of them; this difference

generates a series of compositional and performance issues related to the way harmonic

progressions and phrase structures are differently organized in those two forms of tactus (what

we today, with some historical inaccuracy, call two meters). These issues are analyzed in more

detail in Chapters 5 and 6.

Ternary tactus. So far, we have seen how binary tactus works. But what about ternary tactus?

Similarly to binary tactus, ternary tactus is a specific configuration of accentuation patterns (this

time one T accent followed by a combination of two A accents) among the theoretically possible

permutations of sets of three accents: 1 combination containing no T (AAA); 3 permutations of

the combination containing one T beat (TAA, ATA, and AAT); 3 permutations of the

combination containing two T (TTA, TAT, and ATT); and 1 combination containing 3 T (TTT).

This is summarized on Pascal’s triangle on the array “1–3–3–1.” Tactus, which can only be

constituted by one T followed by (one or more) A, belongs to one of the “3” permutations, which

contain one T. Both “3s” can be considered, depending on whether we start counting the

presence of T beats by starting from the left or right of the array (see Figure 3.8).12

12 As for duple meter, only one permutation of the combination including three elements of T and A is musically

viable: TAA, as only one beat, the first of the measure, is a strong T (unless composers or performers decide to

voluntarily add accents; however, this is beyond the scope of this dissertation).

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If we consider two ternary tactus measures (TAA/TAA)13 instead of one (TAA), the number and

type of combinations of T and A beats are shown on the “1–6–15–20–15–6–1” array, where the

second and second last “6” are the indicators of the number of elements in the set (6 elements:

TAATAA). Because the number of T (Strong, Down-) beats in two measures is 2, the left-most

“15” is the number that contains the specific TAA/TAA permutation of a two-measure 3/4 tactus

(“1” indicates the only combination without T; “6” indicates the 6 permutations of the

combination containing one T; and “15” indicates the 15 permutations of the combination of the

set containing 2 T: “15” is therefore the locus of two-measure ternary tactus). By applying the

same process to 3 and more measures, the ternary tactus designs for us the following pattern:

Similarly to what we did earlier, we can group all 3-and-its multiples on Pascal’s triangle and

map onto it a pattern for ternary tactus (in red in Figure 3.9):

13 In combinations such as TA, TAA, and, as we will see later, TAAA, we see that the tone-metric approach is about

considering the binary aspect of any duple, triple, or quadruple meters, which are, as scholars such as Mattheson

pointed out, fundamentally reducible to binary phenomena.

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Figure 3.9. Sierpinsky’s triangle modulo-3 with ternary tactus pattern highlighted in red.

The figure in example 3.9 highlighted in blue is Sierpinsky’s triangle modulo-3, which is

generated by separating 3-and-its-multiples from the rest of the numbers on Pascal’s triangle.

More specifically, this is a form of Sierpinsky’s triangle in which the originating triangle

generates a recursive iteration of the fractal by repeating three times; the result of this iteration is

then itself repeated to generate the next iteration, and so forth.

The example in Figure 3.9 indicates that the ternary combinatorial array is a subset of

Sierpinsky’s triangle modulo-3, as all its elements belong to the binary “white spaces” of the

triangle; and that both the ternary combinatorial array and Sierpinsky’s triangle modulo-3 are

subsets of Pascal’s triangle. How can we now use this information to enlighten our

understanding of musical organization? In the next section, I describe how the link between

tactus configurations and Sierpinsky’s triangles produces isomorphic equivalencies, that can be

used to model the influence of tactus and metric subdivisions in music.

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3.2 Using fractals to model tactus

In Sierpinsky’s triangle it is possible to draw a straight line starting at the origin “a” (“a” being

any of the vertices of the triangle), and passing through any points on the surface or edges of the

figure; as shown in the following example, each point on the line has one and only one

correspondent point on the same line for each iteration of the fractal.

Figure 3.10. Sierpinsky’s triangle modulo-2 (cf. Example 3.10).

Example 3.10 shows the triangle’s iterations (highlighted within a black frame), and their

relationships (indicated with red arrows): a=1, b=2, c=3, d=5, e=9, and so on. The figure in

example 3.15 shows that points “b,” “c,” and “d” articulate the straight line connecting “a” and

“e.” Segments ab, ac, and ad are all sides of different iterations of the ae triangle side. They mark

the articulation of each iteration of the triangle, and are related by the proportions

ae:ad=ad:ac=ac:ab; this relationship indicates that each side of the triangle is half the size of the

triangle found at the immediately larger iteration, or, in other words, that in each larger iteration

the size of the triangle of the previous iteration doubles in length. This length between segments

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ab, ac, ad, and ae is described by the equation y=2x-1.14 This means that, for example, for x=2,

y=3, the rate of expansion described by y=2x-1 results in the sequence 1, 2, 3, 5, 9, 17, 33, 65,

129, 257, and so on; for a=1 and b=2, c=3, d=4, e=9, etc.

Figure 3.11. Sierpinsky’s triangle modulo-3 (cf. Example 3.15).

Its iterations are highlighted within a black frame, and their proportional relationships indicated

with red arrows: a=1, b=2, c=4, d=10, and so on. Similarly, points on the bisecting line in

Sierpinsky’s modulo-3 triangle increase in distance from the origin at a specific incremental rate.

The equation that describes this value is y=3x-2. This means that, for example, for x=4, y=10, the

rate of expansion in Sierpinsky’s triangle modulo-3 is described through the equation y=3x-2, and

results in the sequence 1, 2, 4, 10, 28, 82, 244, and so on, for a=1, b=2, c=4, d=10, etc.

Before concluding this chapter, I would like to discuss a feature common to all these sequences.

Each sequence expands at a unique rate; this expansion is determined by the widening of the gap

between numbers. This widening is related to the ever-expanding structure of Sierpinsky’s fractal

triangle. As I will show in the next chapters, the sequences can be used to map the fractal patterns

of Sierpinsky’s geometry onto musical scores. This mapping allows for the generation of a

14 For x>1 and x=any segments originating in a; ab=bc>0; and y=any segment directly preceding x in the order ab-

ac.

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geometric understanding of tactus and a descriptive graphic model of the unfolding of time and

the articulation of musical form. In musical terms, the widening of the gap between elements of

the sequences provides a mathematical description of the perceptual loosening that necessarily

occurs when listening to a piece of music: humans are exceptionally good at associating events to

generate meaningful patterns, but cannot perceptually perceive more than a relatively small

amount of music at once, due to short-term memory constraints. Because of such limitation, long-

term musical organization can be analytically apperceived but not perceptually inferred. For

example, form-analytical statements in music analysis (e.g., the comparisons between the material

of exposition and recapitulation in a classic-era sonata) are the result of association-based

deductive processes, and not direct perceptual inferences. As such, they rely on the repetition of

specific melodic designs and harmonic concatenations as a basis for the creation of associative

links over time. Through those sonic events, our sense of articulation of the musical space is

perceived as renewed every moment and carried forward, allowing us to generate meaningful

patterns and, ultimately, formal classifications. As will be seen in chapter 4, tone-metric analysis

results in the construction of waves and trees that model this perception of musical momentum and

continuity.

3.3 Conclusions

In this chapter, I showed that tactus can be analyzed through the sieve of combinations and

permutations of thesis T and arsis A as a continuously expanding space that is fractal-based and

generated by the combinatorial and permutational patterns of A and T. More specifically, I

showed that these combinations and permutations of A and T can be described through Pascal’s

triangle, and mapped onto a fractal figure called Sierpinsky’s triangle. This mapping generates a

set of geometrical relationships that allows us to algorithmically define and graphically represent

the continuously expanding space of tactus and its rate of growth.

This expanding space is generated by the recursive layering of specific numeric sequences for

each tactus onto the score. These sequences are generated through a combinatorial reading of the

metric structure of pieces obtained through a mapping onto Pascal’s and Sierpinsky’s triangles,

as shown in the previous pages. As a result, in contrast to the familiar linear arithmetic unfolding

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of metric organization, we obtain a geometric, fractal, and layered version of the same, which is

characterized by self-similarity and recursivity (i.e., a process through which the numeric

sequence is applied to all metric levels, generating self-similar and recursive numeric patterns

across metric layers, and at all metric spans).

To summarize, the derivative arrangement from abstraction to musical application takes the

following steps: understanding tactus and its fundamentally binary accentual nature (as a binary

set of As and Ts); the modeling of this numerical binarity through Pascal’s triangle and its fractal

analogue Sierpinsky’s triangle; and the generation, from the reading of those geometrical figures,

of specific numeric sequences related to each tactus (e.g., 1, 2, 3, 5, 9, 17, and so on for binary

tactus, and 1, 2, 4, 10, 28, and so on for ternary tactus).

As I will show in the next chapters, these steps allow for the transformation of linear-arithmetic

tactus into a fractal construct, and allow for the projection of the metric flow in a non-linear but

layered fashion. More specifically, the newly determined numeric sequences are mapped back

onto the musical score according to a specific algorithmic procedure (shown in Chapter 4) across

all metric levels (e.g., half notes, quarter notes, eighth notes), thus generating a layered metric

structure based on the segmentation of each metric level as determined by the presence of

specific sonic events of specific rhythmic values (e.g., a half note metric unit divided in two

quarter notes will produce a different tone-metric analysis than a half note metric unit divided in

four eighth notes, or any other values).

The interaction of the tactus with the unique sonic events of the piece along with the application

of the same numeric sequence in recursive form to the “gaps” in the first-layer of tone-metric

mapping to its other levels result in the formation of tone-metric waves as the layers are built up

(and down), “pivots” (which occur between levels representing their rise and fall), an alternative

representation of the evolving waves as tone-metric “trees,” and the overall representation of the

interaction between the metrical ground and the ebb and flow of layered surface activity above it

as a kind of “efflorescence” of musical content.

As we have seen briefly in the Corollary section in Chapter 1, and will see in more detail in

Chapters 4 and 5, the application of the underlying Sierpinsky triangle as a sieve represented by

the sequential-geometrical articulation generates a reading of the sonic features of pieces for

which rhythmic, harmonic, and melodic relationships emerge, or surface, as a continuous flow of

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musical information from the underlying geometrical organization of tactus. As just noted above,

I call this surfacing “efflorescence.” Efflorescence, whose theoretical and musical significance

will be analyzed in more detail in Chapter 4 (and shown in the analytical application of Chapter

5), unfolds as a continuous flow of discrete tone-metric events; this double nature of

efflorescence as both discrete and continuous can be modeled through two tone-metric graphic

representations: tone-metric waves and trees.

Chapter 4 will discuss the tone-metric methodology; more specifically, the chapter will introduce

the theoretical foundations of the algorithm on which tone-metric analysis is built, and describe

how both wave and tree graphs are built, in application of the algorithm to the analysis of

musical examples from the repertoire.

Chapter 4

Analytical method

In Chapter 3, I presented the combinatorial foundations of tone-metric analysis, an analytical

approach for the modeling of the integration of tactus and melodic, harmonic, and rhythmic

patterns. More specifically, I examined combinatorial and permutational features of the T

(thesis) and A (arsis) of tactus by using Pascal’s and Sierpinsky’s triangles; showed that each

combination and corresponding permutations design a specific geometrical array on them; and

concluded that binary tactus models a subset of Sierpinsky’s triangle modulo-2, and ternary

tactus a subset of Sierpinsky’s triangle modulo-3. By mapping the geometrical proportions of

Sierpinsky’s triangle onto the score, we can model the unfolding of tactus and its metric

articulation in time, against which we can describe how melodic and harmonic features of the

music develop.

In this chapter, I describe in detail how the mapping of the algorithmic geometrical proportions

of Sierpinsky’s triangle onto the score can model the musical gestures and improvisatory

character of music, and shape phrasal and formal articulations of pieces, which are described

through tone-metric waves and trees.

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4.1 Modeling purely binary and mixed binary-ternary types of metric

organizations

Seventeenth-century North German organ repertoire, like most repertoires, employs two types of

metric organization, which can be classified as “pure duple”1 (e.g., 2/2 or 4/4, which are

composed of binary subdivisions at all rhythmic levels), and mixed binary-ternary (e.g., 3/42 and

12/8, which, as seen in Chapter 3, employ a mix of binary and ternary subdivisions).3

4.2 Pure binary type of metric organization

An example of the pure binary type of metric organization is found in the chorale prelude on Ach

Gott, vom Himmel sieh darein by Nicholaus Hanff, which is a piece in 4/4. In Chapter 3, I

studied how 4/4 tactus, which is characterized by one T and three As, or TAAA, can be modeled

through the numerical and geometrical properties of Pascal’s and Sierpinsky’s triangle modulo-

2. More specifically, Sierpinsky’s triangle unfolds at the rate described by the equation y=2x–1,

which produces the sequence 1, 2, 3, 5, 9, 17.4 Members of the sequence define the position of

sonic events in the score at the quarter-note level; by mapping the sequence onto the score

(through a process already outlined in the Corollary to Chapter 1 and again in the following

paragraphs), we can describe what I call a level 1 tone-metric articulation of the excerpt. For

example, in the piece in example 4.1, “1” indicates the first quarter-note of the first measure; “2”

the second quarter note of the first measure; “3” the third quarter-note of the first measure; “5”

1 Richard Cohn, “The dramatization of hypermetric conflicts in the scherzo of Beethoven’s ninth symphony,” 19th-

Century Music 15.3 (1992): 194. Cohn indicates that “the length of a ‘pure’ metric complex is a power of some

prime integer. ‘Pure duple’ includes complexes of 4, 8, 16, etc. 2 6/8 can also be listed among the mixed binary-ternary time signatures. As a related note, all metric organization is

binary at its smallest sub-divisional level (e.g., 9/8 is made of dotted quarter notes, which are in turn constituted by

three eighth notes each; these are subdivided in two sixteenth notes each). 3 Cohn’s term ‘pure triple’ (Cohn, The dramatization, 194) cannot be used in this case, as my analysis is concerned

not only with the main beat and metric, but also with its rhythmic subdivisions of a time signature, and all metric

organization is binary at its smallest sub-divisional level, even when triple at the level of beat and meter (e.g., 9/8 is

made of dotted quarter notes, which are in turn constituted by three eighth notes each; these are subdivided in two

sixteen notes each). 4 For an explanation of how the sequential organization of tactus is obtained, refer to Chapter 3.

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the first quarter note of the second measure (which corresponds to the fifth quarter-note of the

excerpt); and “17” the first quarter-note of the fourth measure (which corresponds to the

seventeenth quarter-note of the piece):

Example 4.1. Level-1 tone-metric articulation based on y=2x–1 in Ach Gott. Mm. 1–7.

This sequential articulation is used as a basis for a complete wave articulation of the piece, which

is constructed as follows. First, we establish a distribution for the basic time signature of the

piece and map it onto a grid. Ach Gott by Hanff is in 4/4; therefore, we will use a duple meter

distribution, which is obtained according to the geometric sequence for duple meter (1, 2, 3, 5, 9,

17) described in Chapter 3. This distribution is articulated according to the unit of measure of the

time signature; in the case of Ach Gott, 4/4, the unit of measure is the quarter note (e.g., 1 means

the first quarter note of the piece, 2 the second quarter note, 3 the third quarter note, 5 the first

quarter note of the second measure, 9 the first quarter note of the third measure, and so on). As a

convention, this level is numbered with “1”:

Example 4.2. Level-1 tone-metric articulation of the chorale prelude on Ach Gott. Mm. 1–7.

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The analysis in example 4.2 contains the sequential level, or first level 1 above the score. Level 1

is not dependent on the actual note content of the piece, as it describes the underlying fractal

structure of tactus and not the rhythmic unfolding/articulation of the music.5

Once the sequential level is mapped onto the grid, we start mapping sonic events occurring at the

level of the unit of measure in the piece (in our example, the quarter note). Notes appearing

between quarter notes are not considered at this level. From each “1” of level 1, we highlight the

position of each quarter note whenever it appears in the score, following the distribution

provided by the duple meter sequence. For example, considering for the span of level 1 starting

in m. 3, and ending on the downbeat of m. 5 (we consider duple-meter spans as including their

limits, indicated by “1s”), we highlight quarter notes 1, 2, and 3 in m. 4; quarter note 1 in m. 5;

and quarter note 1 in m. 6; those are the quarter notes that belong to the duple-meter sequence

“1, 2, 3, 5, 9, 17.”

Example 4.3. Partial quarter-note level tone-metric articulation of Ach Gott. Mm. 1–7.

The missing quarter notes (e.g., the fourth quarter note in m. 2 in example 4.3) will be mapped

onto the grid using another level, i.e., level 2. Level 2 follows the same procedure used for level

5 The organization of the musical material onto a grid such as the one described above posits some questions. First,

does level 1 tone-metric articulation indicate that it is possible to mathematically determine the unfolding of pieces,

and does something musically relevant always happen wherever a new span is mapped onto the score? And second,

what happens between the limits of the tone-metric spans? The answer to the first question is no; as we will see in

the next paragraphs and chapters, the final analytical product emerges not from individual layers of tone-metric

articulation, but from the superposition of multiple layers and the reading of their relationship. With respect to the

second question about the material contained within each tone-metric span, the answer can be summarized with the

indication “more of the same sequential material.”

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1: starting from the last “1” before the gap in level 1 (e.g., between the third quarter note of m. 1

and the first quarter note of m. 2), we apply the same sequence (1, 2, 3, 5, 9, 17) to the quarter-

note level (see example 4.5). Our system of reference is now the space between adjacent quarter-

notes, and not the whole piece. If, after this step, any quarter notes are still missing from the

mapping, we create level 3 and higher, and use the same procedure until all the quarter notes are

mapped onto the grid. Remember that the mapping always proceeds according to the sequential

distribution for duple meter. Here is the final quarter note mapping:

Example 4.4. Quarter-note level tone-metric articulation of Ach Gott. Mm. 1–7.

Once the quarter- note mapping is complete (see example 4.4), we proceed to map the piece at

the eighth-note level, applying the same layering technique used for previous levels. This time,

our spans are the quarter note level spans. Again, sonic events found in between eighth notes

(e.g., any sixteenth-note events) are not considered at this level:

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Example 4.5. Eighth-note level tone-metric articulation of Ach Gott. Mm. 1–7.

Now that the articulation at the eighth-note level is complete (see example 4.5), we apply the

same procedure to generate the sixteenth-note level.

Example 4.6. Tone-metric articulation of Ach Gott. Mm. 1–7.

As the reader can see by analyzing the grids of the previous examples (4.1–4.6), we do not

simply generate levels by mapping rhythmic spans onto lower levels, because this would be an

arithmetic procedure describing rhythmic, and not tone-metric, relationships. The difference

between rhythmic and tone-metric layering processes can be described as follows: the former is

generated through an arithmetic organization based on the summation or multiplication of note

values (e.g. 1+1+2 eighth notes) and it is absolute in nature; the latter is generated according to

geometric sequences and positioning of rhythmic values on the tone-metric grid (e.g., each

number of the sequence generates the next through the same sequence generator, such as 2x–1,

or “1, 2, 3, 5, 9, 17” fed back into the sequence itself) and it is relational in nature.

The difference between absolute arithmetic and relational tone-metric approaches to rhythm

speaks to the modeling of the perceptual function of rhythmic groupings in terms of the formal

structure of pieces. For example, longer stretches of sixteenth notes embellishing a cadence

(something often found in North German organ pieces) can be modeled by both arithmetic and

tone-metric approaches. However, an arithmetic approach simply looks at rhythmic relationships

as a linear sequence of events, whereas a tone-metric approach allows for the study of rhythmic

relationship as a function of the underlying tactus, thus connecting rhythm with the pulsations

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that precede and follow a specific rhythmic interval, and providing more interpretive insights

into the unfolding of rhythmic patterns than would be captured in a simple rhythmic analysis.

In building a tone-metric grid as the ones seen above, three types of tone-metric layering can

occur: (1) the two outer limits of each tone-metric span can be found at the same level (see level

numbers in black in m. 2.3–2.4 in example 4.7); (2) the leftmost limit can be found at one or

more levels above the right-most limit (see level numbers in black in m. 3.1–3.2 in example 4.6,

where 2 on the left is lower than 4 on the right); and (3) the right-most limit can be found at a

level or more below the leftmost limit (an instance not found in example 4.7). For (1), we

proceed by adding another level to both sonic events, as shown in example 4.5; for (2), the new

level is added starting one level above the right-most span limit; and for (3), the extra level is

added starting one level above the level of the let-most span limit. This procedure preserves the

horizontality of the levels and does not generate gaps between levels.

Example 4.7. Tone-metric articulation of Ach Gott. Mm. 1–7.

Example 4.7 models the complete tone-metric articulation of the first 7 measures of Hanff’s Ach

Gott. The red in mm. 2.3–2.4 shows the result of the application of the layering procedure for

type (1) tone-metric layering; the red in mm. 3.1–3.2 shows the result of the application of the

layering procedure for type (2) tone-metric layering. The numbers in parentheses in the graph

indicate tone-metric articulations where no new sonic events occur (e.g., m. 5). The final

systematic arrangement of all sonic events onto a grid, as shown in example 4.7, is obtained by

the mapping of the numeric sequences presented in Chapter 3. This arrangement preserves the

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temporal relationships between events and generates the tone-metric spans. Their layering

describes the rate of growth of the tactus and the musical texture that it generates.

4.3 Mixed binary-ternary type of metric organization

As we have seen, tone-metric analysis takes the notion of tactus as portrayed in historic sources

through the Renaissance and the early Baroque as a binary-based form of metric organization (an

approach already presented by Mattheson; see Chapter 2) into the realm of combinatorics. This

approach shows that any tactus is ultimately built on a binary structure.6 This point becomes

particularly important when we are dealing with meters such as 12/8, which are a mix of duple

and triple forms of metric organization, and which I call in this dissertation “mixed binary-

ternary” meters.

Because they integrate both binary and ternary elements, mixed binary-ternary types of metric

organization can be described through a mix of y=2x–1 and y=3x–2 equations and related

sequences (respectively 1, 2, 3, 5, 17 and 1, 2, 4, 10, 28). To show how this type of organization

works, I analyze the first 6 measures of Johann Sebastian Bach’s fugue in G major (BWV 577),7

a piece in 12/8, which is binary at the dotted quarter-note level (four dotted quarter-note beats)

and ternary at the eighth-note level (three eighth notes for each dotted quarter note).

The first step is the description of the unfolding of the sequential level, which is binary at the

dotted quarter-note level (1, 2, 3, 5, 9, 17). The description of this unfolding is obtained

according to the same procedure previously applied to binary layering.

6 Through a different path, Hasty too arrived at the conclusion that triple meter is, at its core, an “unequal duple”

(obtained through deferral). 7 Friedrich Griepenkerl, and F. A. Roitzsch, Johann Sebastian Bach’s Kompositionen fur die Orgel; kritisch-

korrekte Ausgabe von Friedrich Conrad Griepenkerl und Ferdinand Roitzsch. [Vol. IX] (New York: C.F. Peters,

1950).

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Example 4.8. Tone-metric articulation of the fugue in G major at the 3/8 level. Mm. 1–6.

Once the dotted quarter-note articulation is fully laid out (see example 4.8), we add levels based

on the ternary type of tone-metric articulation (to describe the eighth-note articulation of the

dotted quarter notes) using the equation y=3x–2 and the related sequence 1, 2, 4, 10, 28 as our

basis. All the rules for the application of this procedure remain the same as for the binary tone-

metric articulation:

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Example 4.9. Full tone-metric analysis of the fugue in G major. Mm. 1–6.

What emerges is a series of small adjacent groupings (e.g., mm. 1.1, 1.2–1.3, and 1.4 in example

4.9) generating larger-scale tone-metric waves. As I show in the upcoming paragraphs and

chapters, the mapping of the algorithmically-generated layering of numerical patterns onto a

visual representation as waves describes how tactus and rhythm (i.e., meter) can be used to:

describe the flow of melodies and harmonies; a sense of forward motion of the musical texture;

and, ultimately, produce insights into the unfolding of the large-scale organization of the musical

discourse.

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4.4 Pivots and moments of articulation

The continuous flow of waves is articulated through tone-metric pivots. These are moments of

articulation between the end of a wave’s crest, which function as a preparation to those pivotal

moments, and the immediately following descent.8 Pivots can be simple or compound. Simple

pivots, which are in the vast majority, indicate moments of articulation between waves in which

a one-level descent after a wave’s crest is followed by an ascent; compound pivots, which are

rarely found,9 indicate moments of articulation where a descent after a wave’s crest continues to

descend at least another level before ascending again. As I show through the analysis in Chapters

5 and 6, simple pivots indicate a motion towards a new tone-metric formal unit, or an articulation

of the musical discourse understood from the perspective of the integration of harmonic or

melodic (e.g., a new phrase) elements; compound pivots generate a sense of continuation from

the previous formal wave.

A qualification about the term pivot is needed here. In tone-metric analysis, the term pivot

indicates an articulation, or fold, in the tone-metric structure, which is numerical-quantitative in

nature, and does not necessarily correspond to a specific musical component or concept. Any

musical considerations related to pivotal articulations (e.g., pivots producing a separation

between musical phrases) are interpretations of the pivotal articulation of the graph. In other

words, pivots produce an articulation in the tactus-derived metric texture of the piece. This

articulation can be either disjunctive (simple pivots) or conjunctive (compound pivots); the

musical significance of each specific articulation remains an interpretive and analytical act.

The wave analysis of the first 7 measures of the Hanff’s Ach Gott shows a tone-metric reading of

the elaboration of the first phrase of the chorale-tune:

8 Here I use the word descent to indicate the downward slope of a wave. This can be either a sudden drop or a

smooth curve. 9 Because compound pivots are rarely found, in the paragraphs below I will show examples of simple pivots almost

exclusively, leaving the explanations related to compound pivots to the moment in which we will encounter them.

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Example 4.10. Tone-metric analysis and pivots of Ach Gott. Mm. 1–7.

In this example, I show how simple pivots can be interpreted (more on compound pivots later in

this chapter). Simple pivots, which highlight the span between the moments right before and

right after the wave drop occurs, are indicated in example 4.10 with grey rectangles underneath

the wave graph itself. As mentioned earlier in this chapter, simple pivots indicate that some form

of articulation (melodic, harmonic, rhythmic, or tactus-based) within the music material across

waves takes place.10 The simple pivots in mm. 2.4–3.1, 3.3–3.4, and 5.3–5.4 are moments during

which a sense of “closure and new beginning” can be projected through the performance of the

piece. For example, over the pivots in mm. 3.3 and 5.3 a leap occurs in one of the voices

(melodic separation); over the pivot in m. 4.1 the arrival on V is denied through a passing A in

the lower part, which initiates the deceptive motion of mm. 4–5 (harmonic function); finally,

over the pivot in mm 2.4–3.1 no melodic, harmonic, or rhythmic change can seen, and, therefore,

the only reason for this articulatory moment remains the simple fact that all melodic, harmonic,

and rhythmic elements coalesce in this point to generate a closure followed by a new beginning

(with the octave leap in the upper part m. 3.1).

These compositional-analytical features, which are graphically rendered in tone-metric analysis

through pivots and the shape of waves, can be highlighted in performance through gestural and

10 To note: (1) pivots are not individual notes, but regions including two adjacent sonic events, and (2) waves are

gestural groupings and not form-functional phrases (e.g., sentences and periods).

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phrasal organization. For example, different articulations can be used to differentiate between the

rhythmically steadier first wave of mm. 1.1–2.4, and the more lively melodic designs and

rhythmic patterns characterizing the second one of mm. 2.4–4.1; similarly, an agogic accent

might be employed over the pivot in m. 5.3 to highlight the change in the harmonic direction and

the double harmonic function of the A minor triad as both tonicized chord and iv of E.

As a descriptive tool, and not per se an interpretational and prescriptive form of analysis, tone-

metric analysis describes how music unfolds (according to the principles illustrated in Chapter 3)

but does not provide a final and definitive interpretive framework. Ultimately, the performance

of a piece remains a complex sonic phenomenon that is shaped by personal choices based on

stylistic considerations, as well as organological features of instruments, acoustic qualities of the

performance space, and personal taste. However, as a form of analytical modeling, tone-metric

analysis can help us: (1) interpret the improvisatory character of music through the articulation

of pivots and waves, and (2) describe in a systematic way the notion of musical gesture. The

improvisatory character of music is modeled by tone-metric waves as the unevenness of sonic

event spans (e.g., in example 4.10, this unevenness can be seen by looking at the wavelength,

calculated also as the space between the grey bars (pivots) below the score): although tactus

persists unchanged throughout the piece, its interaction with the rhythmic patterns produced by

the composers’ harmonic and melodic decisions stretches and compresses, and lengthens or

shortens, the tactus-generated waves, which become frozen cross-sections, or photograph-like

images, of the improvisatory flow of music. From this perspective, each wave can be seen as a

systematic and quantifiable descriptor of the metric aspects of musical gestures, an otherwise

analytically vague concept.

These spans are determined not by a classical understanding of phrase-based groupings, but by

the layering of tone-metric sequential articulations determined within the context of

combinatorial and permutational orderings of tactus and sonic events. This examination of how

waves and pivots can describe and model the unfolding of musical gestures is part of a larger

dialectic discourse about the relationship between the surfacing of musical details over the

underlying algorithmically generated tone-metric structure.

In Chapter 1, I defined this surfacing of rhythmic, melodic, and harmonic patterns from the

underlying fractal structure of tactus as efflorescence. In the same chapter, I also defined

efflorescence as the degree of deviation of a waveform from a waveform generated by a

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hypothetical piece in which sonic events occur only on the level of the normative units of

measure (e.g., one sonic event per quarter note in a 4/4 piece). This degree of deviation, which

can be graphically modeled and quantified (more on this in the analytical Chapters 5 and 6),

allows us to measure the amount of embellishment in a piece, and provides a visual cue of how

much attention is given to details around a cadence. Efflorescence can manifest itself and be

interpreted as embellishments and diminutions. However, embellishments and diminutions are

the byproduct of the application of compositional techniques and processes based on

compositional choices, and are contextual to the music-syntactical surroundings; conversely,

efflorescence is a descriptive-analytical construct based on the fractal sequential patterns

generated by recursion and self-similarity, and is functionally relevant only to the positioning of

sonic events, independent from their semantic and syntactical-musical role. In other words,

embellishments and diminution, and efflorescence share the same surface, but not the same deep

functional structure.11

Precisely because mathematically generated, efflorescence, arising in the form of waves and

pivots, cannot tell us anything about the details of the underlying note content of a piece

(although it can help model the sense of forward motion in, and provide insights into the gestural

character of, the music). For example, no harmonic-functional or melodic-intervallic

relationships can be inferred from waves; to be able to make any musically meaningful

statements, we have to first map the sonic events of the originating score onto the corresponding

wave. As we will see in the next section, this mapping allows us to determine a variety of

hierarchical relationships between events that can prove useful for the analysis and performance

of music.

11 In the present dissertation, I use the terms ‘surface structure’ as the audible unfolding of music as a note-by-note

texture, and ‘deep structure’ as the chunking of sonic events into the melodic, harmonic, and rhythmic functional

groups that make up the larger and larger units of form in music. For discussions about the difference between

surface and deep structures in music see Heinrich Schenker, and Ernst Oster, Free composition: volume III of New

musical theories and fantasies = Der freie Satz (Hillsdale: Pendragon Press, 2001); and in linguistics see Noam

Chomsky, Syntactic structures (Berlin: Mouton de Gruyter, 2002).

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4.5 Tone-metric trees

Once waves and pivots are determined, we can map them onto the sonic events of a composition

or passage to determine how these relate to each other from a tone-metric perspective to generate

melodic and harmonic content, and surface efflorescence. These relationships are represented by

tone-metric trees, which I will now describe.

Each event is positioned at the lowest level on the graphic columns to which it refers. Events are

tone-metrically connected through segments, or branches, which spatially describe (moving left

to right) the temporal unfolding (past-present-future) of tone-metric concatenations. We build

branches from left to right according to the following rules: (1) branches can connect two events

found at the same level, or a lower-level event with a higher-level event but not vice-versa; (2)

more than one branch can depart from a sonic event but only one branch can arrive at a sonic

event; and (3) if two branches can end on the same event, the branch closer to that event is

privileged. In the following example, I build a tree structure into the corresponding wave graph

of Hanff’s Ach Gott according to the rules highlighted above, by connecting the lowest tone-

metric level available for each event:

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Example 4.11. Wave analysis and tree structure of Ach Gott. Mm. 1–7.

Example 4.11 indicates the existence of multiple nested tree subgroups within the large-scale

formal tone-metric analysis of the piece (e.g., mm. 2.1–2.4; 3.1–4.4; and 5.1–6.4). These

subgroups are relevant from both analytical and performance perspectives, as they provide

information that can be used to model gestures from a theoretical and graphic perspective, and

also indicate articulations of the musical texture that can be taken into account when analyzing

and performing the excerpt. Notice that readings of the excerpt through pivots and trees are not

necessarily identical or even similar (i.e., the linkage or separation provided by pivots does not

necessarily correspond to a similar linkage or separation within tree structures). For example, the

tree analysis of mm. 3.1–4.4 and 5.1–6.4 can be analyzed as articulating two-measure phrases, as

each branch grouping corresponds to both harmonic progression and melodic design of each of

those two-measure phrases.

The comparison of waves, pivots, and trees speaks to the complexity of interpretive layers of

musical information. From an analytical perspective, waves and trees help us identify and model

the surfacing of efflorescence; they are not organizational categories so much as “experiential

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groupings;” they describe how the note content of pieces unfolds as an integrated continuum of

meter, harmony, and melody, to generate a semantic of music. Waves and trees constitute the

platform through which the deeper combinatorial organization of music, through the fractal

geometry of tactus and the appearance of sonic events, is bridged together to generate the

surfacing of musical gestures. In the tone-metric context, modeled through waves and trees, the

role of analysts and performers is the identification of which elements are the most significant

from the interpretational perspective, and how these elements might be integrated within a

coherent and convincing rendition of the piece.

While trees visually clarify hierarchical relationships within tone-metric waves, thus rendering

the analytical relationships more instantly perceivable, they do not actually add information to

tone-metric waves. Trees become particularly useful when their organization is linked to other

parameters of music, such as harmonies: for example, as we will see in Chapter 6, the hierarchies

of the tree branching can be associated with specific harmonies occurring at those points, and

provide a tone-metric-based reading of harmonic concatenations and groupings to be compared

with traditional tonal relationships.

Beyond the basic algorithmically generated tone-metric graphic representation, the reading of the

tone-metric information by analysts and performers remains central for the understanding of the

musical discourse, as the interpretational link between the algorithmic and the categorical, as

well as the compositional and experiential aspects of music.

4.6 Summary and further thoughts

In this chapter, I discussed the applicability of the theoretical and combinatorial principles of the

organization of tactus presented in Chapter 3. More specifically, I described the step-by-step

process through which tone-metric waves and trees are built, discussed some of the aspects and

applications of tone-metric analysis, and highlighted some of the strengths of the analytical

method: (1) the algorithmic basis and related repeatability of the analytical results; (2) the

quantification and graphic modeling of the analysis; and (3) the ability of the method to connect

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and model surface details with a deeper structural understanding of tone-metric levels through

the notion of efflorescence.

The analytical process described to obtain tone-metric waves resembles very closely the creation

of a time-velocity Cartesian graph for the description of motion of an object. By graphing

velocity variations on one axis and time variations on the other, we obtain a line that is a

descriptor of the variations in acceleration of the object. In the case of tone-metric analysis, time

is represented by tactus as the articulation of time, and velocity by rhythm as the rate at which

sonic events appear. The wave that results describes tone-metric “acceleration” and corresponds

to what, in tone-metric terms, is efflorescence. As much as acceleration is the rate of change of

velocity with respect to time, efflorescence is the rate of change of the appearance of sonic

events with respect to the underlying metric unfolding, the relative textural saturation of the

musical space determined by the tactus.

The ability to model and quantify the perceptual effects of rhythmic patterns has been the subject

of analytical research. For example, in the 1970s Eugene Narmour categorized three types of

durational relationships: additive (two rhythmic values of the same length are found next to each

other), cumulative (a longer value is found after a shorter one), and counter-cumulative (a shorter

value is found after a longer one). These three types of rhythmic combinations were linked, in

Narmour’s view, to the ability of rhythm to introduce a sense of closure in melodic patterns (e.g.,

a sense of melodic closure can be achieved through durational accumulation).12 While tone-

metric analysis does not claim any direct correspondence between the duration of rhythmic

patterns and the specific sense of directionality in a piece, it does argue that the relational nature

of rhythmic combinations has an effect on the overall shape of tone-metric waves and, therefore,

on the sense of articulation of pieces and articulation of form.

The algorithmic basis of tone-metric analysis provides us with a multi-faceted platform for the

interpretation of music, which generates a variety of options for performers. The analytical and

performance path modeled by tone-metric analysis is neither meant to provide a definitive

solution nor to generate a purely algorithm-based form of music analysis. However, the variety

12 Eugene Narmour, Beyond Schenkerism: the need for alternatives in music analysis (Chicago: University of

Chicago Press, 1977).

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of readings that springs from the study of tone-metric waves and trees does “collapse”13 into one

decision in the moment of performance. As a result of this actuation, each sonic event becomes

unique within the piece; this uniqueness, however, is due not to the specific melodic, harmonic,

or rhythmic functions associated with it (as the same sonic event can return multiple times

throughout a piece with different functions: for example, a C major triad can be a tonic chord in

C major, and a dominant chord in F major), but because of its position. This position is: (1) the

only element belonging to each sonic event that is not repeatable and interchangeable with other

sonic events; and (2) the determinant of the event’s musically relevant function (e.g., the same

event in two different positions has by necessity a different musical function).

The position of individual and discrete musical events is identifiable through the mathematical-

geometrical articulation of music, while their combination into melodic designs and harmonic

progressions acquires semantic significance only when contextualized within the musical

continuum described by waves. In other words, if the individual position of events is identified

and frozen in time, the feel for the continuous motion and musical semantic is lost; if motion is

felt, individual positions cannot be singled out and isolated. The graphic solution proposed in

this chapter partially overcomes this issue, as it allows us to model both the positioning of

individual events and their flow over time. More specifically, the graph shows that meter unfolds

according to specific waveforms, which represent what I like to describe with the image of

“frozen motion.” This paradoxical image, which cannot be described as either motion or stasis,

finds a way to be represented in waves connecting the underlying mathematical articulation of

the tactus space with the surface activity of the music.

As noted at the end of Chapter 3, the relationship between the discrete and the continuous is

modeled by efflorescence, or the emergence of surface details from the sequential-geometrical

underlying architecture of tactus as understood from a combinatorial perspective. This

combination of the discrete and the continuous, which is an integral part of this research and was

at the core of much mathematical research throughout the seventeenth and eighteenth centuries,

was central to twentieth-century research on fractals as well. For example, Pascal was influential,

through his study on derivatives, for the establishment of calculus, which is the mathematical

13 I use the word “collapse” to indicate the ability to identify both position and momentum of a sonic event. In other

words, when a performer plays a note or a chord, both the position of that chord and the musical trajectory

subtended by it become determined.

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discipline that studies instantaneous changes and the properties of curves, and, therefore, the

relationship between instant (discrete) and gradual (continuous) motion. The analytical model

presented in this dissertation strives to bridge discrete and continuous, as well as theoretical

analysis and performance, through the description of efflorescence, to provide analysts,

performers, and, ultimately, listeners with an experiential analytical guide for the understanding

of music outside of traditional syntax- and category-based readings.

We often emphasize a categorical understanding of music over an experiential one14 by setting a

theoretical system of reference (e.g., roman numeral analysis), abstracting from it generalized

categories (e.g., harmonic functions) associated with individual musical events (e.g., chords)

based on the consistency of the presence of those events within the generating principles of the

chosen theoretical approach, and creating forms of syntactically meaningful teleology (e.g.,

functional tonality).

Categorization of events helps us organize and make sense of musical thoughts (occurring as

temporal sequences of unfolding events), which would otherwise be a series of individually

perceived and disconnected phenomena. While categories are generalized descriptive constructs

applicable to any events that conform to the definition provided by the category itself, musical

events are sonic phenomena that constitute the objects of those categories and are not the

categories themselves. The aim of tone-metric analysis is to overcome the dichotomy between

analytical constructs and experiential aspects of music-unfolding for the description of the flow

of music according to the integration of its constituent melodic, harmonic, and metric elements.

In this context, it is valuable to reflect on the somehow paradoxical nature of tone-metric

analysis, whose goal is (and initial research impetus was) the study of the improvisatory nature

of music, through a fully algorithmic graphic modeling of the discrete and continuous properties

of efflorescence.

The algorithmic basis of the methodology guarantees an initially descriptive and not interpretive

basis for analysis rooted in a simple visual descriptive interface (waves and trees). This

simplicity fosters a directness of observations, and describes a sense of musical flow and

14 A category is a set of abstract schemata for the description of facts or objects and of any relationships between

them, and for the creation of complex narratives unfolding in time; a category is therefore characterized by

interpretative relativity.

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articulations, mapping sonic onto visual aspects. These features, I argue, become significant with

respect to the study of instrumental music, and, in particular, for the study of earlier repertoires

such as Renaissance and Baroque music. Renaissance and Baroque scholars, composers, and

musicians, did not understand music in terms of complex theoretical approaches and teleological

paths of invention (apart from their cosmological excursions), but fostered a practical approach

to music making, which was grounded in note-by-note concatenations built through the use of

standardized melodic and harmonic patterns. For example, authors of German fundamenta and

early tablatures were concerned with generating correct treatments of dissonances and chord

concatenations between adjacent chords, and not with the description or modeling of a long-

range teleological understanding of musical semantics and syntax.15 Similarly, a notational

system such as North German tablature was meant to be of practical use: it indicated which keys

to press at a specific moment in time (e.g., the indication “ee” meant “now press the first key to

the right of the two sharps in the third octave”), and was not intended to describe any of the

harmonic- and melodic-functional relationships between each note and its surrounding ones.16

Even Johann Sebastian Bach understood music-theoretical instructions as only one aspect of

music making, the other being the experiential and perceptual: “two fifths and two octaves may

not follow each other; that is not only a vitium [“fault” in theory] but it also sounds bad.”17

“Phenomenological” and “experiential” are two adjectives that well summarize and describe the

nature of tone-metric investigations, which are deeply rooted in an algorithmic understanding of

sonic events understood as specific articulatory moments in the score, independently from any

music-functional interpretation associated with them. In Chapter 5, I show that this algorithmic

understanding and modeling can generate a semantic reading of music for the understanding of

the repertoire under analysis.

15 By the time the repertoire of this dissertation was composed, fundamenta and early tablatures were more than a

century old; however, their examples remained a point of reference for North German organ composers, who

continued to develop their art by exploring the possibilities of contrapuntal writing based a given cantus firmus. For

an overview of the compositional features of the earliest fundamenta and tablatures, see Corliss R. Arnold, Organ

literature: a comprehensive survey, 3rd ed. (Metuchen, N.J.: Scarecrow Press, 1995), 1–20; and Peter Williams, The

European organ, 1450–1850 (Londond: Batsford, 1966). 16 Another element of interest with respect to tablature is that, by writing pitches as letter names in adiastematic

fashion on a parchment instead of as notes on a diagram, musicians would be able to write more music, saving

resources such as ink and quills. 17 Cited in Joy R. Charles, Music in the life of Albert Schweitzer (New York: Harper & Brothers, 1951), 106.

Chapter 5

Analytical applications

In this chapter, I apply tone-metric analysis to study excerpts and entire pieces from seventeenth-

century North German organ music. More specifically, (1) I study the poly-functional role of

tone-metric pivots as indicators of textural, phrasal, and formal articulations; (2) investigate the

notion of efflorescence, or the surfacing of rhythmic, melodic, and harmonic patterns over the

deeper tone-metric articulation generated by the underlying sequential organization of duple and

triple meter; and (3) explore the music’s sense of forward motion, which is elsewhere typically

understood from an intuitive perspective but which, in tone-metric terms, can be analytically

quantified and visually described through wave and tree graphs.

The repertoire under investigation in this chapter includes organ works by Dietrich Buxtehude

(1637–1707), which occupy, for volume, and variety of forms and stylistic idioms, a position of

pre-eminence in the German organ landscape before Bach. I first analyze two pieces by

Buxtehude based on chorale-tunes: (1) the chorale prelude on Durch Adams Fall, BuxWV 183,

focusing on both the role of cadences from a tone-metric perspective, and the relationship

between efflorescence and formal organization; and (2), the section of mm. 45-80 of the chorale-

fantasia on Nun freut euch, BuxWV 210, to study the insights of tone-metric analysis into the use

of effects such as echoes and repetitions.

I then analyze three settings based on the intonation of the Magnificat primi toni. In the analysis

of Buxtehude’s Magnificat primi toni, BuxWV 203, which is a large-scale fantasia freely

elaborating on segments of the originating Magnificat intonation, I show how the tone-metric

approach can inform our understanding of sectional subdivisions. In the analysis of the first verse

of the Magnificat primi toni by the organist Jacob Praetorius (1586–1651), which, unlike

Buxtehude’s Magnificat is subdivided into verses (each of which is an elaboration on the

originating psalm-tone intonation), I focus on the study of the role of pivots in shaping both local

and general form, and the relationship between the analysis and performance of the piece. In the

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analysis of the Magnificat primi toni by Johann Pachelbel (1653–1706), a composer active in

Southern Germany, I investigate both whether and how tone-metric analysis can provide us with

insights into imitative-fugal writing and into different styles and compositional schools.

The contrasts among the Magnificat settings of Buxtehude, Praetorius, and Pachelbel are

particularly striking from the perspective of the compositional techniques employed and their

performance functions. Buxtehude’s work is a large psalm-tone-based fantasia that seems more

like a prelude to, than an accompaniment in alternatim of, a sung Magnificat. Praetorius’s verse-

based realizations are basically elaborated four-part harmonizations on a straightforward cantus

firmus, presented in long notes (one note per measure), meant to be performed in alternatim with

the choir during the Evening Services. Finally, Pachelbel’s fugue, a piece based on a distant

reminiscence of the first Magnificat tone, is taken from his collection of fugues on the eight

Magnificat tones; these were intended for use during various liturgies.

5.1 Durch Adams Fall, BuxWV 183

Example 5.1 includes score and tone-metric analysis of the chorale prelude on Durch Adams Fall

by Buxtehude.

Example 5.1. Analysis of the chorale prelude on Durch Adams Fall (part 1 of 4).

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111


112

Example 5.1. Tone-metric analysis of the chorale prelude on Durch Adams Fall.

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In example 5.1, the darkening of the tone-metric levels in the graph shows the increase and

decrease of efflorescence relative to the surrounding context—its musical significance is the

explanatory task of analysts and the interpretive task of performers. The example highlights that

this first phrase is articulated in three waves by the simple pivots in mm. 3.1, 4.1, and 5.1. These

pivots, which mark three distinct musical designs (mm. 1–3.1 are an alternation of tonic and

dominant chords; mm. 3.1–4.1 constitute a short chromatic passage made of concatenations of

dissonance and resolutions; and mm. 4.1–5.1 identify the phrase’s cadential pattern).

The tone-metric graph cannot tell us anything about the specific harmonic and melodic content

of the piece; however, as I have just shown, it can indicate that some form of metric articulation

determined by the interaction of the underlying combinatorial structure of tactus and the surface

rhythms built on it occurs. Establishing a relationship between this interaction and the harmonic,

melodic, and, ultimately, formal features of the piece remains the task of analysts and

performers, who can decide whether and how the analysis and/or performance of the excerpt

should emphasize elements such as the change in the harmonic pace, and the harmonic-

functional and phrasal articulation of the excerpt based on those metric parameters highlighted

by tone-metric analysis. For example, I would argue that an interesting performance solution of

this first line of music would be one that highlights the difference, through the use of articulation

and agogic, between the slow harmonic pace of the first two measures, the melodic chromatic

ascent in m. 3, and the V-I cadential motion in m. 5.1.

Score interpretation based on the study of tone-metric analytical waves and pivots is not always

easy, mainly because waves and pivots are generated by non-musical principles of articulation

(i.e., combinatorial features of tactus), and are, therefore, not directly and systematically

dependent on a specific musical function (e.g., melodic, harmonic, or metric). More specifically,

a pivot indicates that a change in harmonic, melodic, and rhythm textures, or the integration of

some or all those parameters, is taking place at a specific point in the score, but does not

(understandably enough) directly indicate whether the change is a harmonic, melodic, or

rhythmic one. It is the task of analysts and performers to determine the nature of this change

through a comparison with the original score. For example, the wave in mm. 7.1 to 7.3, which is

particularly short, signals two significant formal elements: (1) the ending of a harmonic and

melodic sequence with the arrival on V of D in m. 7.1; and (2) the bridging function between the

post-cadential elements (mm. 5.1 to 7.1) and the beginning of the next chorale phrase in m. 7.3.

While the latter (bridging function) does not cause many interpretational issues (the end of this

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wave articulates the beginning of the next solo), the former’s role as the ending of the sequence

in m. 7.1 raises some interpretational questions: is it acceptable to consider a pivot as a

significant moment in the articulation when this articulation interrupts the flow of a model-

sequence? The answer to this question is yes, as pivots and waves can indeed interrupt a model-

sequence, since, as highlighted earlier, the articulation that pivots are indicating might not be

referring to specific changes in melodic designs, but can refer to harmonic or rhythmic

articulations. In the specific case of the wave of mm. 7.1 to 7.3, the wave encapsulates the

dominant harmonic function that concludes the sequence, and acts as a hypermetric pickup to the

next solo entrance of the second chorale phrase in m. 7.3. The role of this short wave is therefore

pivotal, and neither strictly conclusive nor initiating (it remains the task of the analyst and

performer to decide whether the emphasis should be on the conclusive or initiating function in

that half measure). Again, analysts and performers have to interpret these indications in terms of

the larger-scale needs of their analysis and performance. For example, a performer can wish to

emphasize the conclusive character of m. 7, or think of its first half as a pick-up that prepares a

stronger entry of the second melodic phrase of the chorale.

In m. 24.4, a compound pivot appears. As mentioned in Chapter 4, compound pivots are rare and

appear to act as linkage between previous and following material more than a form of separation.

In this case, my personal interpretation of the short compound pivot is that it links the iv–V

motion that will lead to the cadence in F major in m. 29. This interpretation is due to the melodic

continuity given by the stepwise motion of all the voices and the B-flat in the soprano, tied to

connect iv and V. Immediately after, however, the simple pivot in m. 25.1 appears to indicate a

momentary departure from the connectivity proposed by the preceding compound pivot (a

departure obtained through the use of the short eighth-note flourish). Notice how this eighth-note

interpolation, however, is itself immediately undermined by the following simple pivot in m.

25.3, which opens to a return to the previous melodic designs of mm. 23–24. In other words, the

compound pivot suggests a continuation, which is momentarily denied by a short flourish, which

is in turn immediately denied by the next simple pivot.

Tone-metric pivots and waves can be seen as a way (perhaps, in metaphorical terms, a reflective

mirror against which) to summarize and graphically model the perception of music-unfolding.

This role of tone-metric analysis is rendered through the quantification of changes in the musical

texture and is understood from an informational, rather than a strictly musical, point of view.

Tone-metric analysis indicates that some kind of change in the metric information is taking place

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that could signal some kind of melodic and harmonic changes as well; however, it is not

concerned with the understanding and explanation of, for example, phrase structures and form

from a music-semantic point of view.

By integrating all aspects of music as textural information,1 tone-metric analysis overcomes a

reading of music that systematically emphasizes one element over another (e.g., harmonic

functions over melodic segmentation), and, I argue, generates the basis for a flexibility of

analytical discourse that helps performers and improvisers, as much as listeners, to appreciate the

multifunctional role of individual sonic events in shaping musical unfolding (e.g., each note

embodies, at the same time, harmonic, melodic, and rhythmic functions). In Chapter 6, I will

analyze in more detail how this integrative approach of tone-metric analysis can help overcome

some of the limitations of analytical approaches systematically based on a specific alternative

focus, such as those, for example, that, in centering on the harmonic organization of the music

material tend to underplay the role of metric organization as a compositional principle.

As an integrative approach, tone-metric analysis does not always produce results that match

musical intuitions as seen through the sieve of specific analytical features (e.g., harmonic

functions). This potential mismatch between intuition and analysis can result because tone-

metric analysis describes the distribution of musical material and texture as the integration and

mutual interference of different aspects of music: melodies, harmonies, and rhythms. Our

previous analysis of the wave in mm. 7.1–7.3 might help elucidate this concept. By postponing

the cadential V of the wave in mm. 7.1–7.3 beyond the natural downbeat of m. 7, the composer

imparts a sense of weakness to the closing function of that dominant harmony. At one level, V-I

in m. 7 does act as a harmonic cadence; however, because of its “misplaced” metric position on a

weak beat, its ability to impart a sense of closure is weakened; this weakness turns the downbeat

of the next m. 8 into a stronger point of reference for the listener, and the beginning on A of the

second line of the chorale in m. 7.3 into a hypermetric pickup to the next downbeat in m. 8. To

summarize, the V-I harmonic function becomes a link to the hypermetric pickup (mm. 7.3 and

7.4) of the second phrase of the chorale (which, in the original chorale tune indeed starts with a

pickup); and a local harmonic weakening of the V-I in m. 7 makes the same cadence acquire a

hypermetric role as a structural pickup to the next chorale phrase.

1 Here I use the term information as what is generated by a specific sequence of events and conveyed from a source

to a receiver (with respect to music, from composer to listeners).

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But why is this “alternative” view proposed by tone-metric analysis significant, and for whom?

The spectrum of approaches to pre-Baroque and Baroque organ music goes from historical-

musicological, which describe music based on compositional and theoretical indications

provided by original sources, to analytical, which discuss these repertoires through the sieve of a

number of post-Baroque analytical tools for the study of harmonic, phrasal, and cadential

features.2 Tone-metric analysis creates a third methodological axis, as it provides an approach

that is not based on the application of other music-analytical models to the study of pre-Baroque

and Baroque music (tone-metric analysis uses mathematical principles, not musical ones, as the

backbone of the analytical discourse), and is not limited to the stylistic and compositional

constraints of styles and genres. In fact, tone-metric graphs are only modeling how music, as a

form of information and not as a semantic modality, unfolds as a function of tactus; and how

rhythmic components (whose arrangement is related to a specific tactus) can be used to

manipulate and distort the steady pace of the underlying tactus structure, leading to a related

distortion of the melodic and harmonic boundaries of musical phrases . The rhythm-generated

disturbance of the mechanical/algorithmic unfolding of tactus is what generates the metric flow

of a specific piece, and contributes to the creation of expectations about the perception of phrasal

and large-scale forms, and the need for certain phrasings, closures, and pivotal projections. The

semantic-musical interpretation of those gestural, phrasal, and formal articulatory moments lies

entirely in the hands of analysts and performers. Based on these principles and features, tone-

metric analysis provides a novel descriptive approach to phrase structure, harmonic syntax, and

cadential patterns, which I will describe in the next paragraph.

North German Baroque organ music often presents unevenness of phrase length. This is

particularly (but not only) evident in pieces based on pre-existing melodic material (e.g., chorale

preludes and chorale-fantasias), which display phrase structures that are length-dependent on the

note content of the thematic material from which they are derived, and do not obey what is

nowadays considered a standard (classically-derived) two-measure-based phrase organization.

Moreover, while most North German organ repertoire can certainly be considered tonal in its

general harmonic unfolding, it is impossible to systematize its harmonic progressions according

to standard tonal models (e.g., via roman numeral analysis that results in coherent directedness;

see Chapter 2 for a discussion of this topic).

2 See Chapter 2.

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This non-conformity of North German organ repertoire to standardized harmonic and phrasal

constructs is particularly well modeled by tone-metric efflorescence, which is a quantification

and graphic representation of the textural unfolding of a piece represented through waves and

their pivotal articulations. Efflorescence describes the intensity (through wave motion) and

organization (through tree structures) of the continuous sense of motion in music and textural-

phrasal articulation, which is analytically modeled through the discrete pixilation of tone-metric

texture. Pivot and wave analyses show that it is possible to model how multiple factors (e.g.,

melodic, harmonic, and metric designs) can contribute to signal gestural, phrasal, or formal

articulations in music. Performers can use this poly-functional aspect of the analytical

methodology to privilege, according to stylistic and personal preferences, one aspect over the

other at any given moment. In the next section, I study more in detail how tone-metric analysis

and efflorescence can inform performance, by exploring the gestural and formal features in mm.

45–80 of the chorale-fantasia on Nun freut euch by Buxtehude.

5.2 Chorale-fantasia on Nun freut euch (mm. 45–80)

Measures 45–80 of this chorale-fantasia are a particularly illustrative example of how the tone-

metric approach can shed light, specifically through the study of pivots, wave shapes, and

efflorescence, on the relationship between metric organization, and melodic and harmonic

designs, as form generators. I provide the score and wave analysis of mm. 45–80 in example 5.2.

Example 5.2. Analysis of chorale-fantasia on Nun freut euch. Mm. 45–80 (part 1 of 4).

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119


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Example 5.2. Tone-metric analysis of the chorale-fantasia on Nun freut euch. Mm. 45–80.

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Measures 45–68, shown in example 5.2, constitute what is possibly the longest repetitive pattern

of a single melodic-harmonic design ever written during the Baroque. This ostinato starts in m.

45 with a regular four-measure model that is repeated 4 times (4 measures in G major, 4

measures in D major, 4 measures in A major, and 4 measures in D major). In m. 61, this four-

measure harmonic pattern is suddenly interrupted, and a new harmonic pattern begins (1 measure

in A major, and 5 measures in G major). The interruption of the four-measure harmonic pattern,

as well as the sudden shift down a tone from A major to G major occurring in m. 61, generates a

sense of phrasal and harmonic instability, which creates the expectation for a new two-measure

pattern (perhaps a pattern made of an idea in G major followed by a repetition in F major in mm.

64–65). Such an expectation is, however, denied; instead, Buxtehude continues to repeat the

same melodic and rhythmic design in G major, until m. 68, when, finally, the pattern is entirely

dissolved into a completely new idea.

Both the sudden breaking down of a long regular harmonic pattern, and the subsequent

generation of a harmonic expectation whose fulfillment is denied, is highlighted by its wave

structure, as I show in the next paragraphs. The regularity and repetition of wave groupings in

mm. 45–68 mirrors the regularity and repetition of the melodic-rhythmic pattern; this regularity

in the wave is articulated through combinations of compound/simple pivots occurring in mm. 49,

53, 57, and 61. Of these, the first delimits the first 4 measures in G major, the second the four

measures in D major, the third the four measures in A, and the fourth indicates the moment in

which the melodic-rhythmic pattern stops being supported by the four-measure harmonic pattern.

The compound/simple pivots in mm. 49, 53, 57, and 61 suggest, at the same time, connection

and separation between the sections delimited by those measures, which are distinct in terms of

key areas but connected in terms of melodic and rhythmic patterns. Why, then, within each

section are the same melodic and rhythmic patterns separated by simple pivots (e.g., between

mm. 45–47) and not connected to each other? The answer lies in the different use of imitative

texture within and between sections: within sections, the imitative texture is based on the use of

the so-called echo effect, which is here obtained as the repetition of the same material within the

same range in both hands; between sections, however, this effect is nearly nullified precisely

because of the different tonal areas in which the different sections are composed.

The use of repetition is a particularly interesting feature of this musical work. For example, the

exact repetition of the new idea presented in mm. 68–69 (in brackets in example 5.2) and

repeated in mm. 69–70 is highlighted by a compound pivot rticulating them. Wave analysis

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informs us that the compound pivot in m. 68 connects the two halves of that measure, generating

a sense of continuation within that phrase (see underlying bracket); this is, however, denied by

the simple pivot in m. 69, which produces a separation between the two halves of the measure

and projects the second half towards the next wave starting in m. 69.2. According to tone-metric

analysis, the new idea of mm. 68.2–69.1 is a single unit, while the idea starting in mm. 69.2 is

expanded and projected into the next measures.

This view finds compelling correspondence in the indication of manual changes (“O” for

Oberwerk, or great organ, and “R” for Rückpositiv, or positive organ)3 occurring in m. 69.2, on

the pick-up to the beginning of the wave starting in m. 69.3. “R” is introduced in the left hand;

“O”, which is first introduced in the right hand, is then substituted by “R” in m. 70, generating a

sonic continuity that parallels the formal understanding of mm. 69 as being connected to the

following measures. In mm. 71–72, the score indicates another change of manual (through the

indication “O”). This time, however, the change is not supporting a formal separation like the

one we saw in the previous paragraph, since there is no pivot in the wave between mm. 71–72;

on the contrary, a sense of continuation is in place, and m. 72 should be interpreted as an

embellishing echo effect and repetition of the thematic material of m. 71. In other words, while

the manual change in m. 69 coincided with a formal delimitation, the change in m. 72 coincides

with what can be interpreted as a (quasi-parenthetic) musical commentary to the thematic

material presented in m. 71.

The variety of overlaps and misalignments of the melodic-thematic material and formal

organization of mm. 68–72 (which is emphasized by the use of manual changes) generates a

sense of instability that is resolved only starting in m. 73. Here, a tone-metric pivot occurs, over

which the 2–6–7–1 melodic-thematic characteristic figure of mm. 71–72 starts to dissolve to

leave space to the final cadential 7–1 (e.g., mm. 75, 76). The subsequent pivot in m. 77.1 marks

a shortened variant of the same idea. The pivot in m. 79.1 articulates the repetition of the design

already presented in mm. 75–76 and m. 78.

3 North German organs are organized according to the so-called Werkprinzip, or the principle for which the stops of

each manual are grouped together and enclosed in a division separate from the division containing the stops of other

manuals. This finds visual correspondence in an organization of the instrument into pipe divisions that are clearly

distinguishable from each other. For example, the Rückpositiv (literally back positive organ), which normally

corresponds to the pipes of the first manual, is enclosed in a separate case located at the back of the organist; it is

normally used to create echo effects (as is the case in the example I am mentioning on this page), or as solo manual

(e.g., to play the ornamented tune of a chorale while stops on other manuals and pedals accompany it).

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Wave analysis helps us model the formal subdivisions of mm. 45–80, and their relationship with

the thematic-melodic material. In the example discussed in the previous paragraphs, I have

explored how rhythmic, melodic, and harmonic materials in these measures are connected

through both simple and compound pivots to generate formal designs (the analysis also found

support in the manual changes indicated in the score). Although pivots can be defined as either

separating (simple pivots) or unifying (compound pivots) moments, each of them manifests a

high degree of functional specificity related to the surrounding note-content. Moreover, pivots

are significant for the overall shape of waves, and, ultimately, for the form of pieces, as they

interact with the note-content to describe meaningful melodic-harmonic-rhythmic patterns and

large-scale formal subdivisions. This relational approach remains at the core of the tone-metric

analytical discourse, which approaches the analysis of form as a relational and interactive

construct of melodic, harmonic, and rhythmic elements as a function of tactus.

The examination presented in the previous paragraphs raises the question of whether wave

analysis provides us with generalized analytical paradigms (prescriptive approach), or aims at

describing the specificity of the organization of the musical material (descriptive approach). A

useful analysis in this respect is the one proposed by David Temperley in his article “The

question of purpose in music theory: description, suggestion, and explanation.”4 In it, Temperley

gives examples of both descriptive approaches (e.g., Meyer, and Lerdahl and Jackendoff) and

prescriptive approaches (e.g., Rahn, Guck, Kivy, and Schachter), and summarizes both

descriptive and prescriptive approaches as follows:

The methodology of descriptive music theory is primarily introspective. This may seem problematic, in view of

the fact that the mental structures and processes involved are generally held to be unconscious. But it seems

reasonable to suggest that such structures might be made conscious through sustained introspection, or, perhaps,

inferred from other representations that are more readily accessible. A useful parallel may be drawn here with

theoretical linguistics. The reasoning in linguistics is that, while we do not have direct intuitions about (for

example) the syntactic structures of sentences, we do have intuitions about whether sentences are syntactically

well-formed (and perhaps about other things, such as whether two sentences are identical in meaning). By simply

seeking to construct grammars that model these judgments-linguists reason–we will uncover much else about the

syntactic structure of the language we are studying (and languages in general). Similarly, seeking to model our

introspective judgments about (for example) the metrical structures of pieces, or expectations of melodic

continuation, may lead us to posit other mental processes and structures that are not in themselves consciously

available. Of course, unconscious mental representations of music may be-and have been–explored in other ways

besides introspection, notably through psychological experiment and computer simulation. Indeed, these other

methods have an essential role to play in testing the hypotheses of descriptive music theory. Together, these

various approaches can be seen as constituting the musical branch of cognitive science–what has lately come to

4 David Temperley, “The question of purpose in music theory: description, suggestion, and explanation,” Current

Musicology 66 (1999): 66–85.

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be known as “music cognition.”

We might call [Schachter’s] approach to theory the “suggestive” approach. (“Prescriptive” is also a

possibility, but this carries a pejorative connotation that is not at all intended here.) By this view, the

objective of doing theory and analysis is to find and present new ways of hearing pieces, not to describe

the way people hear pieces already. As with the descriptive approach, we might posit a distinction here

between theory and analysis. An analysis recommends a hearing of a particular piece; a theory, on the

other hand, offers general principles of musical structure which might be applied to many pieces.

Whereas a descriptive theory intends to describe some aspect of musical perception or cognition, a

suggestive theory seeks to enhance it in some way. Note the conflict between these two purposes: as

suggested by Schachter’s first quote in particular, suggestive analysis intends not to describe structures

and relationships that are already being heard. (The conflict between the two is also clear in Meyer’s

quote, although here, the opposite view is taken: the goal of analysis is explanation, not education.)

Following these definitions, we can see that, at one level, tone-metric analysis is prescriptive

(e.g., it provides us with a novel form of generalized metric/mathematical principles on which

meter-based music is built), at another level it is descriptive (e.g., it provides us with a way to

organize musical material in formal units). This double nature of tone-metric analysis, however,

does not preclude (and, actually invites) the dialogue between organizational principles and

specificity of the analytical discourse for each individual piece. In other words, music analysis

remains a matter of stylistic and personal interpretations, and cannot be understood as a simple

and direct translation of the graph into sonic events. Performers are often faced with the dilemma

of choosing one among many interpretive nuances, and different interpretive paths can be taken

in subsequent performances; it is assumed that the choice needs to be the best one for conveying

the significance of a specific passage (this significance typically attempts to balance both the

local and larger-scale roles of the passage). But to what extent is the information of tone-metric

analysis and tree structures performable? How is tone-metric analysis helping to shape

performance in a different way than other analytical approaches?

Some of these questions relate to the organological features and architectural spaces in which

organs are found. For example, the traditional location of North German organs in churches with

very different acoustics, from very dry to extremely generous, shapes our understanding of how

pieces should be performed in different contexts. While the interpretation of mm. 68–69

analyzed earlier might require only a small articulation separating the two instances of the idea in

mm. 68–69 in a dry acoustic, in other settings characterized by a much more generous resonance

a larger articulation might be needed, maybe coupled with a rallentando to signal that the end of

the formal section is taking place in m. 68. The interpretation of the information on the score is

also crucial in generating meaningful performance choices. For example, the change in manuals

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required in m. 69 is, by itself, a deterrent for phrasal continuity, as the move from one manual to

another does take a certain (although very small) amount of time, thus defining the need for an

articulatory moment.

The articulatory features just mentioned are present throughout Buxtehude’s compositional

output. A noteworthy example is his Magnificat primi toni, BuxWV 203, whose gestural designs

and formal structures I analyze in the next section.

5.3 Magnificat primi toni (mm. 1–11; 76–91; and 139–141) by

Buxtehude

This composition is a large-scale fantasia divided into sections, each based on a segment of the

originating intonation, which is reproduced here:

Example 5.3. The Magnificat primi toni intonation.

By elaborating on this intonation, Buxtehude’s Magnificat explores both the full potential of the

North German organ, through (1) the ample use of manual and pedal ranges, and fast pedal

passages; (2) thematic dialogue between manuals and pedal; (3) richness of rhythmic and

melodic designs; (4) varied compositional techniques (e.g., strictly imitative textures vs.

rhapsodic gestures) and pairing of contrasting meters for adjacent sections. As a result, with this

piece Buxtehude generates a unique sonic experience for listeners, producing an unparalleled

Magnificat elaboration that is, to my knowledge, the longest and, I would argue, most complex

and innovative Magnificat for organ ever written.

In the following analysis, I focus on three sections of the piece: mm. 1–11, 76–91, and 139–141.

These are significant examples of how tone-metric analysis can offer insights into the formal

structure of music. Here are both score and wave analysis of mm. 1–11:

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Example 5.4. Tone-metric analysis of the Magnificat primi toni. Mm. 1–11.

Because of the rhythmic regularity of the excerpt (which is articulated in a constant sixteenth-

note pulse), this first section of the piece is, from a tone-metric perspective, organized in two-

measure units separated by simple pivots. In both mm. 2 and 5, the pivots split the thematic idea

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into two parts, of which the latter, (i.e., mm. 3.1 and 5.1 respectively) acts as a repetition and a

confirmation of the former (i.e., mm. 2.4 and 4.4 respectively). In m. 7, the pivot separates the

rhetorical flourish on D in m. 6 from its development into a new idea in m. 7. Similarly, the pivot

in m. 9 separates the initial chromatic and syncopated design from its chromatic ascent. Finally,

the pivot in m. 11 highlights the moment in which the tonic D minor arrives followed by the

half-close on the dominant A. These first 11 measures of the piece are a magnificent display of

musical rhetoric that includes thematic presentation-confirmation patterns, long improvisatory

flourishes and elaborations on the tonic harmony, abrupt and unexpected changes in the

harmonic directions, and use of syncopated and chromatic lines. Tone-metric analysis helps us

organize and group this material in a way that highlights these rhetorical aspects beyond a more

traditional thematic and formal analysis.

Another example of how tone-metric analysis can inform our thematic understanding of pieces is

evidenced by mm. 76–91 of the same fantasia (example 5.5).


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Similarly to example 5.4, the metric regularity of the score in example 5.5 generates pivots at a

very regular pace; in this particular example, the application of the algorithm generates an

unusually high number of pivots, which is a feature typical of ternary subdivisions (see examples

4.12 and 4.13 as well). It is important to remember that pivots can inform our understanding of

how melodies and harmonies unfold, provide insights into formal organization of piece, or

simply describe changes in the rhythmic texture of music, but are not necessarily meant to be

emphasized in performance, as they are the byproduct of an algorithmic view on the metric

articulation of music, which is not necessarily relevant from a thematic or harmonic point of

view. In this example, pivots can be interpreted as indicating that each set of three between-pivot

groupings could be slightly articulated to generate a step-by-step feel to the unfolding of this

section, thus highlighting its harmonic character over its melodic linearity. This view is

supported, I believe, by the dance-like time signature, which indicated a subdivision into four

three eighth-note groupings per measure (not two six eighth-note groupings per measure), and by

the ostinato-like pattern that unfolds without rest until the end of the excerpt.

The example highlights the interpretational nature of tone-metric analysis, whose analytical

results derive from both the moment-to-moment integration of the originating algorithmic

information into the metric, melodic, and harmonic features of the piece, and the interpretation of

the same through comparison with aspects such as melodic, thematic, and harmonic analyses. In

the next paragraphs, I explore in more detail how context changes our interpretation of the tone-

metric graph by studying the piece’s cadential progression of mm. 139–141.1 (reproduced in

example 5.6), first considered as an isolated unit and then in the context of the larger sections of

the piece.

This is the wave analysis of mm. 139–141.1:

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Example 5.6. Tone-metric analysis of the Magnificat primi toni. Mm. 139–141.1.

The analysis in example 5.6 shows a single wave rising throughout the passage. From a

performance perspective, this implies a general forward motion throughout. But what happens if

these measures are contextualized within a larger chunk of material such as the passage of mm.

135–141?


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The wave analysis in example 5.7 shows that the graph describing mm. 139–141 remains almost

identical to that in the previous example, except that now m. 139 starts on a pivot, which

provides us with the understanding of mm. 139–141 as a unit somehow separate from the

previous material. In fact, this reading is in line with a common understanding of mm. 139–141

as a cadential unit in itself, with very little connection with the previous material. This reading is

also supported by the clear differences in textural and rhythmic designs that characterize mm.

135–139, and 139–141. This example highlights that tone-metric analysis is a contextual form of

analysis: depending on the amount of material analyzed, the algorithm will produce different

amounts of information (in examples 5.6 and 5.7 the only difference is the lack or presence of a

pivot). It is therefore necessary to carefully consider the context in which our analysis takes

place.

One question related to context within tone-metric analysis relates to the role of lower levels in

tone-metric graphs. When the context becomes so large (i.e. the analysis is carried out on entire

pieces), what happens to the lower levels when their presence in the graph is not matched by a

clear perception, as the sequence progresses towards infinity? For example, is level 1 always

perceivable? One simple answer is: the influence of their lower levels is perceived less and less

as time passes. A more complex and, I believe, interesting answer is: their influence is projected

towards the higher levels, and is felt indirectly through the articulatory parceling of tone-metric

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analysis. In a way, lower levels must be there to generate the basic framework of rhythmic

unfolding (which, from a tone-metric perspective, turns out to be different for each different

tactus). Even though their direct perceptual influence disappears with time because the distance

between articulatory moments within a single level stretches towards infinity, they provide a

unique and always perceived form of rhythmic arrangement. To conclude, they assure that the

specificity of rhythmic arrangements and larger-scale rhythmic form of music are preserved

coherently and consistently throughout a piece: in other words, we can say that the lower levels

transform from felt states of perception to mathematical articulations, or from forms of

perception to underlying mathematically-generated descriptors of metric articulation.

In the analysis of Buxtehude’s Magnificat primi toni I studied the role of pivots in shaping large-

scale formal organization; the notions of tone-metric analysis and efflorescence as analytical

tools for performance practice and interpretation of music; and the importance of the musical

context (and extent of the passage under consideration) in tone-metric analysis. In the next

section, I analyze how tone-metric analysis can be affected by the change of metric context.

More specifically, I analyze how tone-metric analysis can model the difference between 4/4 and

2/2 in the same piece, and indicate the analytical impact of such changes with their consequences

for performance. In the specific example, I analyze the first 16 measures of the Magnificat primi

toni by Praetorius.

5.4 Magnificat primi toni (first verse; mm. 1–17) by J. Praetorius

Both score and wave analysis of the excerpt are reproduced in examples 5.8 and 5.9: the former

example is an analysis of the piece conceived as a 2/2, the latter as if it were in 4/4.

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Example 5.8. 2/2 tone-metric analysis of the Magnificat primi toni (verse 1). Mm. 1–17.

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Example 5.9. 4/4 tone-metric analysis of the Magnificat primi toni (verse 1). Mm. 1–17.

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Historically, time signatures 2/2 and 4/4 differ in that 2/2 is traditionally associated with the lack

of sixteenth notes,5 while 4/4 normally makes ample use of sixteenth notes.6 This difference has

consequences for the performance of pieces, as it impacts the way we understand the speed at

which we can and should perform it, as well as the relationship between its internal harmonic

functions. This difference between 2/2 and 4/4 tactus is highlighted by tone-analysis and the

study of efflorescence, which is, as we saw in the introductory paragraph of this chapter, the

surfacing of rhythmic, melodic, and harmonic patterns over the deeper tone-metric articulation,

generated by the underlying sequential organization of duple and triple meter. A comparison of

the analyses of examples 5.8 and 5.9 highlights that, especially when smaller-value notes (e.g.,

eighth notes) are present, the shape of the waves of 2/2 and 4/4 differs significantly, specifically

from m. 7. The efflorescence (i.e., the shape of waves) of these two versions varies as smaller

values (i.e., eighth notes) are added to embellish the harmonic concatenations of the excerpt. A

reading of this difference from a performance perspective is that the steady progression of tactus

in the 2/2 version is less affected by smaller subdivisions than it is the 4/4 version. This can be

interpreted as follows: the role of eighth notes in 2/2 is less metrically relevant than it is in 4/4,

as in 2/2 they act more as embellishing surface details; as a result, a performance of the 2/2

version could imply a steadier pace (at the half-note level) than a performance of the 4/4 reading

of the same, with fewer local articulatory moments.

In this section, I discussed how tone-metric analysis could be used to model the difference

between two readings of the same piece. More specifically, I analyzed how the same piece

considered in 4/4 and 2/2 produces different tone-metric readings, with consequences for the

understanding of internal metric and harmonic relationships. These different readings can

explain and graphically model what is unique to each tactus (e.g., 2/2 and 4/4), and can be used

by performers as a basis for interpretive choices.

So far, I have shown how tone-metric analysis can be applied to the study of elaborations on

Magnificat intonations composed in Northern Germany. In the next section, I explore how the

methodology can be applied to (1) imitative-fugal writing, and (2) stylistically different

5 This is evident from the analysis of the repertoire and was confirmed in Arnold Dolmetsch, The interpretation of

the music of the XVII and XVIII centuries revealed by contemporary evidence (London: Novello, 1946). 6 An interesting example of this is found in Bach’s two versions (4/4 and 2/2) of the Leipzig chorale prelude for

organ on Nun komm der Heiden Heiland, BWV 661.

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repertoires. I do so by analyzing one of Pachelbel’s fugues on the Magnificat primi toni.

Pachelbel, a major figure of what is known today as the Southern German organ school,

composed a setting of fugues for each Magnificat tone. The compact and clear harmonic and

thematic layout of his fugues makes the pieces ideal platforms for the tone-metric study of form.

5.5 Fugue on the Magnificat primi toni by Pachelbel

Example 5.10. Tone-metric analysis of the fugue on the Magnificat primi toni (part 1of 4).

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Example 5.10. Tone-metric analysis of the fugue on the Magnificat primi toni.

Pachelbel’s fugue on the Magnificat primi toni is, unlike the examples by Buxtehude and

Praetorius analyzed earlier, not based on the thematic material of the Magnificat intonation, but

on the generalized D minor key that, through time, developed as an evolution of the originating

Magnificat intonation.7 In fact, the fugue’s subject, layout, and form, are based less on the

originating Magnificat melodic design, and more on (1) the harmonic relationship between A

major and D minor (nowadays understood as dominant and tonic in D minor), and (2) the use of

B-flat, reciting tone into the D minor key.

Fugue has alternatively been described as a compositional technique and a musical form. This

distinction collapses from a tone-metric perspective, because tone-metric analysis is concerned

with the study of both composition and form as byproducts of the same metric distribution of

sonic events, independently from their textural features. As such, it can be used, as I show in this

paragraph, to study how texture (i.e., melodic and harmonic aspects) influences the perception of

contrapuntal-thematic elements such as subject entries. For example, the pivot in m. 5.1 in

example 5.10 signals a moment of heightened harmonic tension over the first two non-passing

dissonances of the piece (the 2–3 in m. 5.1, and the striking 7–8 in m. 5.2), immediately followed

by the syncopated 2–1 and 4–3 resolutions in the second half of m. 5. The sudden arrival of those

two dissonances works as a harmonic catalyst, generating a heightened sense of drive and need

for resolution, which are used by Pachelbel to prepare the third entry of the subject in m. 5.3.

Another example of how tone-metric analysis can be used to highlight the interplay between

7 This fugue is part of a collection of fugues on the eight Magnificat tones.

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texture and thematic relationships is found in m. 8.3, where three compound pivots mark the

connection between two adjacent waves, thus connecting the underlying circle-of-fifths pattern

of mm. 8–9 that unfolds across those waves and leads to the fourth subject entry in m. 10.1.

From a formal perspective, the study of “highest crest”-to-“highest crest” articulation in tone-

metric graphs8 offers some noteworthy insights. The first of these crest-to-crest spans is found in

mm. 1–5; these include the first imitative point of the subject-answer presentation. The second,

in mm. 5–8.3, includes the third entry of the last confirmation of A minor before the momentary

excursion to F major. The third section, mm. 8.3–16.3/17, presents two main features of interest:

(1) m. 17 marks the point both where the next level 1 takes place and where the highest number

of levels of the piece (8 levels) is first found; and (2) m.17 also marks the beginning of the only

two-part section of the piece (this if we exclude the two-part imitation that opens the fugue) and

the return of the subject in D minor. The fourth section, starting in mm. 16.3–17, features the

presence of a compound pivot in m. 24.2–24.4, which extends the wave to the next highest crest,

found in m. 28.2–28.4; this extension can be interpreted as an improvisatory articulation, or

reflective afterthought, of the last entry of the subject in m. 24.3. The sixth section, mm. 29–31,

marks the pedal point that ends the piece.

This excursus into Pachelbel’s fugue on the Magnificat shows that tone-metric analysis can be

used to investigate fugal writing and to reveal how fugal elements (e.g., subject entries) are

contextualized in the tone-metric continuum. Musical flow described through the discrete pixel-

like structure of waves models the relationships between the underlying geometry of tactus, and

the complexity of harmonic and melodic functions that constitute its graphic representation.

Tone-metric analysis cannot tell us anything about the subject’s features alone, or the harmonic

plan of a piece, but it can help us investigate how the different entries of the subject are situated

within, and interact with, the surrounding musical context, providing us with a flexible and

interactive understanding of form, and, I argue, a fuller understanding of this repertoire. An

advantage of the context- and texture-based foundation of tone-metric analysis is useful for the

study of sectional pieces as well, in which textural and thematic features change constantly from

8 The word crest refers to an element of tone-metric graphs, not (i.e., the crest of a wave) to musical phrasing

elements such as melodic ambitus. Therefore, crests are not indicators of musical formal units, but of purely metric

articulation of pieces.

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section to section.9 In the next section, I show how tone-metric analysis can be used to analyze

the opening 10 measures of Bruhns’s prelude in E minor. More specifically, I explore how tone-

metric analysis can be used to model the features of the two opening sections of mm. 1–5 and 6–

10 (examples 5.11– 5.13).

5.6 The Prelude in e minor by Bruhns. Mm. 1–5 and 6–10.

Example 5.11. Tone-metric analysis of the Prelude in E minor by Bruhns. Mm. 1–5.

9 When referring to Renaissance and Baroque repertoires, the term section is commonly used to indicate a musical

unit characterized by a specific texture or compositional technique, often melodically and harmonically independent

from the material surrounding it.

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First section (mm. 1–5). Mm. 1–5 are described by a recursive and expanding tree structure:

recursive because it is based on the same iterated pattern (m. 1), and expanding because this

pattern generates groupings that become larger with time (mm. 1.1, 1.2, 1.3, 2.1 and 3.1). The

organization of those groupings suggests a chunking of mm. 1–2, and 3–5. Moreover, the only

pivot in the section is found in m. 3.1, at the point of separation between the first mm. 1–2 tree

grouping and mm. 3–5 tree grouping. The tone-metric subdivision of this excerpt appears to

contrast with the entry of the long triple trill in m. 3.3, on a weak beat. The entry of that

characteristic trill is often performed as a point of arrival, or confirmation, of the E minor tonic,

after the long arabesque-like beginning of the piece. However, following a stricter metric

understanding of these measures based on tone-metric analysis, we might want to consider this

trill entry not as a conclusion but as a continuation of the arabesque-like design, perhaps meant

to transform the chromatic intensity of the arabesque into a textural growth of intensity achieved

through the additional of multiple voices and vibrational effect.

There is an interesting documentary and performance practice consideration here. Many of these

pieces were in fact written and transmitted in tablature. Similarly to that way in which modern

notation and particularly musical autographs can convey additional information about musical

structure and performance, tablature conveyed its own type of graphic information, which can

shed light into the organization and realization of a work. In the case of the E minor Prelude by

Bruhns, the voice distribution on a tablature version of the same measures is instructive. In the

tablature reproduced in Figure 5.1, we see that the author started to insert the four lower rests

(which correspond to the four lower voices) right over the entrance of what is, in today’s modern

score, measure 3. This supports the idea that these measures were conceived as a “solo” (mm. 1–

2) followed by a multi-part “crescendo” trill (mm. 3–5), as suggested by tone-metric analysis.

Figure 5.1. The tablature of the Prelude in E minor by Bruhns. Mm. 1–5.

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In fact, the tone-metric reading of this section suggests a sense of continuity and crescendo of

intensity of mm. 3–5, and also that the strong point of arrival is m. 5, both because of the return

of level 1 and because of the thickness of the 6 tone-metric layers of which this beat is

composed. This reading appears to correspond to an interpretation of the triple trill not as an

abrupt change, break in the texture, or point of arrival, but as a continuation of the previous half-

measure with the intention of increasing the textural density from a solo line to a three-part

texture and a pedal point, which, in turns, prepares the stage for the final E minor chord (m. 5).

Second section (mm. 6–10). The first section of this prelude is based on a clear binary meter (its

4/4 measures are made of one T followed by 3 A). However, mm. 6–10 are based on a more

complex time signature. In the tablature mentioned above, the time signature is 18/16 (see Figure

5.2), which, in modern editions, has been changed to become a more familiar 9/8 (see the score

in example 5.2). This editorial choice appears to relate to interpretation of metric groupings, as I

will discuss in the next paragraphs.

The internal groupings (i.e., the sixteen-note runs) that make up this sections are, in the original

tablature, not chunked in any particular way (e.g., 3+3 or 2+2+2 sixteenth notes); however, some

modern editions have taken the liberty to interpret those runs’ groupings, suggesting editorial

solutions for them. For example, in his 1972 edition Beckmann10 grouped then in a 3+3

sixteenth-note fashion; however, the more recent Breitkopf & Härtel edition by Vogel of the

same prelude left the groupings as six-note patterns, mostly on the basis of the fact that there is at

least one manuscript source, a tablature version of the piece, in which this time signature does

not appear.11 In the tablature, there are (1) no sub-divisional grouping indications (e.g., in the

tablature we cannot establish whether each six-note sixteenth-note group was divided into three

binary or two ternary sub-groupings); and (2) no measure subdivisions. The lack of specific

metric groupings in this source can shed some light on the complexity of metric organization in

the piece. This organization can be modeled through tone-metric analysis, and can change the

way we understand, analyze and perform its phrases and harmonic structure. For example, the

lack of a clear indication about a specific grouping of sixteenth-note patterns leaves both of the

possible metric organizations of this section open, as 18/16 can be considered as an alternation of

binary and ternary layering of meters: (1) ternary-binary-ternary (TBT), as in the edition

10 Klaus, Beckmann ed., Samtliche Orgelwerke [by Nicholaus Bruhns] (Wiesbaden: Breitkopf & Hartel, 1972). 11 Harald, Vogel ed., Sämtliche Orgelwerke [by Nicholaus Bruhns] (Wiesbaden: Breitkopf & Härtel, 2008).

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provided; and (2) ternary-ternary-binary (TTB).12 From a performance practice perspective, an

option (3) is also possible: the use of a mix of the mentioned above combinations of binary and

ternary groupings, at both dotted quarter-note and dotted eighth-note levels; this option would be

based on personal performance choices determined by the desire to obtain specific harmonic and

melodic groupings (and perhaps a rather freely improvisatory character).13

In the wave graph provided below in example 5.12, I included the score and tone-metric analysis

of mm. 6–10 with the TBT tone-metric analysis of mm. 6–10, and in example 5.13 I included the

two tone-metric options analyses described.

Example 5.12. Score and TBT tone-metric analysis of mm. 6–10.

12 Groupings based on binary subdivisions at the quarter-note level are not possible: there are eighteen sixteenth

notes per measure, which renders a subdivision in quarter notes impossible. 13 A similar issue in terms of subdivisions and groupings has been highlighted by a number of scholars (e,g., Vogel

and Dirksen) with respect to the first section of the Prelude in G minor, BuxWV 149, by D. Buxtehude. In the

Breitkopf & Härtel edition of the complete organ works of Buxtehude, groupings are forced into a notational

appearance that privileges sextuplets composed of two triplets each, while no indication is found about this kind of

grouping in Buxtehude’s manuscript of the piece (again, neither measures nor sub-divisions are used in tablature

notation).

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Example 5.13. TBT and TTB wave of the Prelude in E minor by Bruhns. Mm. 6–10.

In the example, the two combinations of metric organization are, from top to bottom: TBT and

TTB. Option number one, TBT, at the top, is organized as follows. Level 1 indicates the

sequential articulation of the measure (ternary). As this articulation occurs at the dotted note

level, we include higher levels until the articulation of the dotted quarter note level is covered (or

until we articulate every 6 squares in the graph). The subdivision of the dotted quarter note is

binary (2 dotted eighth notes per dotted quarter note); levels are added until this subdivision is

completely articulated (until the 3 square subdivision is saturated). Each dotted eighth note has a

ternary subdivision; we keep adding levels according to this operation until all squares on the

graph are covered.

Once again, a tablature version of the piece (see Figure 5.2) is particularly instructive, as it

shows that the groupings of sixteenth notes are neither 3+3 nor 2+2+2, but more broadly

organized in groups of 6. This allows a free interpretation of the passage as 3+3, 2+2+2, or even

in mixed groupings, at the performer’s interpretive discretion.

Figure 5.2. The tablature of the Prelude in E minor by Bruhns. Mm. 1–5.

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It is a common performance practice in early music to use a certain degree of flexibility in

combining different accentuation and rhythmic patterns within a section such as the one that is

being currently analyzed. For example, changes in accentuation from binary to ternary groupings

can be employed to emphasize specific harmonic or melodic goals, such as the arrival on E

minor in m. 9.1. In this case, the performance could start the section by using the TBT (ternary-

binary-ternary) combination, switch to TTB (ternary-ternary-binary) on the thematic sixteenth-

note idea on the second beat of m. 8 (this change would also be consistent with the subdivisions

of the left hand in m 8.2 as a quarter note followed by two sixteenth-notes), and, from there,

move through the entrance of the eighth and quarter note in the right hand in m. 9.1 to the last

note of the section as an eighth note (not a dotted eighth note) in m. 10.3. This integrative

TBT/TTB solution would contribute to create a closer relationship between analysis and

performance, which is a goal of tone-metric analysis. More combinations are possible, as a

variety of details can be highlighted within this section. Again, tone-metric analysis would tailor

its range of analytical outcomes to interact with individual performance decisions.

5.7 Conclusions

In this chapter, I analyzed how tone-metric analysis can inform our understanding of a variety of

textural designs in music. In Chapter 6, I will expand the repertoire under analysis to show how

tone-metric analysis can be (1) applied to the study of later repertoires, and (2) be paired with

other analytical methodologies to generate a more sophisticated final analysis of music, by

offering insights into the metric structure of pieces, which would otherwise remain a background

element for most analytical theories.

Chapter 6

Expanding the repertoire

In this chapter, I study how the tone-metric method can be applied to the study of repertoires

other than seventeenth-century North German organ music, and provide insights to complement

other analytical approaches typically used to investigate those repertoires. Specifically, I

compare tone-metric analytical results to four other analytical approaches: (1) the reading of

mm. 1–4 of the four-part harmonization of the chorale O Haupt voll Blut und Wunden, BWV

244, mov. 63, by Johann Sebastian Bach proposed by Fred Lerdahl and Ray Jackendoff;1 (2)

William Caplin’s form-functional analysis of mm. 1–10 of the third movement of the piano

sonata in C major, K. 279/189d, by Wolfgang Amadeus Mozart;2 (3) the insights of Carl

Schachter on the opening thirteen measures of the Prelude in E minor, Op. 28, no. 4 by Frédéric

Chopin;3 and (4) the commentaries of Theodor W. Adorno (1903–1969)4 and Janet Schmalfeldt5

on mm. 1–12 of the Piano Sonata, Op. 1, by Alban Berg.

6.1 Chorale harmonization on O Haupt by Bach

In their A generative theory, Lerdahl and Jackendoff study Bach’s four-part harmonization of the

chorale O Haupt. Their method can be summarized in three stages: first, the organization of the

1 Lerdahl and Jackendoff, A generative theory, 142–145. 2 Caplin, Classical form, 55–57. 3 Carl Schachter, “The triad as place and action” Music Theory Spectrum 17.2 (1995): 149–169. 4 Theodor W. Adorno, Alban Berg: master of the smallest link, transl. by Christopher Hailey and Juliane Brand

(Cambridge: Cambridge University Press, 1991), 36. 5 Janet Schmalfeldt, “Berg’s path to atonality: the piano sonata, Op. 1,” in Alban Berg: historical and analytical

perspectives, ed. David Gable, and Robert P. Morgan (Oxford: Clarendon Press, 1991), 79–109.

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4/4 meter of the harmonization in two two-quarter notes per measure, according to a metric

articulation that sees each measure as based on a “strong/weak” and a “half-strong/weak” beat

pairings (metric pickups are added subsequently); and second, the determination of a tonal

hierarchy between the two chords of each pairing based on the standard notion that the tonic

function is the most tonally relevant, followed by, respectively, dominant and pre-dominant

functions (they also consider root position chords as more hierarchically relevant than the same

type of chords in inversion). The application of this tonal hierarchy to meter generates a hybrid

metric-tonal articulation that can be modeled through a tree structure (shown in example 6.1); the

tree’s shorter branches, which represent less relevant harmonic functions in each pairing, are

attached and subordinate to longer branches, which represent more relevant harmonic functions.

And third, the removal of branches, one by one, based on the tonal hierarchies previously

described, which produces further levels of reductions.

The following is the graphic summary of this process as found in A generative theory:

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Example 6.1. Harmonic reduction by Lerdahl and Jackendoff of O Haupt, BWV 244.6

A generative theory generates an algorithm-based reduction for the description of the tonal

hierarchies of the piece. As seen in the previous paragraph, this algorithm is, at the beginning of

the process, based on the metric organization. However, in subsequent reduction levels (f, e, d, c,

and b) (see example 6.1) this organization loses its metric aspect, and tonal hierarchies remains

the only organizing factor in the reduction process. According to Lerdahl and Jackendoff, this

switch from metric to tonal is possible because, from a perceptual point of view, the role of

6 Lerdahl and Jackendoff, A generative theory, 144.

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meter fades away when larger note groupings appear: “At very large levels metrical structure is

heard in the context of grouping structure, which is rarely regular at such levels; without

regularity, the sense of meter is greatly weakened. Hence the listener’s ability to hear global

metric distinctions tapers and finally dies out.”7 Although we can all easily see how the sense of

meter fades away when larger groupings are perceived over time, we also have to consider that

the “sense of meter” is not the same as meter. The former is a post-compositional phenomenon

generated by the perception of sonic events; the latter is a pre-compositional and algorithmically

organized arrangement of accents and rhythmic patterns which are not necessarily dependent on

the sonic content of pieces (there can be the notion of meter without actual musical content). I

argue here that analytical issues arise when a piece’s analysis and reduction are conceived as

based on “sense of meter,” as the compositional treatment of dissonance-resolution, phrase

structures, melodic shapes, and, ultimately, harmonic syntax, are dependent on a pre-conceived

algorithm-based framework (i.e., meter), and not on how we perceive that framework in time

(i.e., “sense of meter”). In the next paragraphs, I discuss this problem in detail, through the

analysis of accentuation patterns and chord concatenations in the first two measures of Bach’s

harmonization of O Haupt.

I would like to start this analysis by reminding the reader that chorale texts and tunes are paired

together according to the identity of their accentuation patterns. In the chorale O Haupt, the

opening “O” is unaccented and the subsequent “Haupt” accented; in his four-part harmonization,

Bach assigned “O” to a pickup and “Haupt” to a downbeat. The G major chord in m. 1.1 that

harmonizes the word “Haupt” plays, from both musical and textual perspectives, a

compositionally and perceptually relevant role in the unfolding of this first chorale phrase, as

both the first metrically strong beat that appears in the tune and as the chord over the first

accented word of the text. The reading by GTTM does not show, as it is not meant to, how

integrated and perceptually relevant both the textual and accentual functions of that G major

harmony are, and the chord’s musical function as the generator of the characteristic plagal

motion that characterizes the beginning of the first four measures of the phrase. Instead, their

reading highlights the tonal hierarchy between the G major triad in m. 1.1 and the underlying D

major tonality of the first two measures, showing that the tonic function of the phrase is D major

(which modulates to B minor in m. 2.4).

7 Lerdahl and Jackendoff, A generative theory, 21.

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I propose here a tone-metric reading of the first 4 measures of the piece (see example 6.2) to

show how tone-metric analysis can inform our understanding of the tight relationship between

music and text, and complement the reading of A generative theory to generate a more nuanced

analysis of the passage.

Example 6.2. Tone-metric analysis of O Haupt. Mm. 1–4.

The analysis provided in example 6.2 (which maps tree and waves onto each other), and its

comparison with the analysis of Lerdahl and Jackendoff in example 6.1, highlights that, from a

harmonic perspective, the G major chord (IV) on the downbeat of m. 1 plays a secondary role

compared to the D major tonic (I); the same G major chord, however, plays a primary role as the

initiator of the level-1 sequential articulation in tone-metric analysis. This emphasis on the tone-

metric structuring role of G (IV) in m. 1.1 finds a correspondence in the textual accentuation on

the word Haupt, which occurs on the same G major chord. Moreover, contrary to A generative

theory, tone-metric analysis sees V in m. 2.2 as less harmonically relevant than the preceding

ii6/5 and following I; this is due not to tonal constraints but to the specific positioning of V

within the surrounding metric framework.

These hierarchical metric readings closely adhere to the distribution of the textual accentuation

patterns of the chorale: O | Haupt voll Blut und | Wunden (the underlined letters correspond to

the stronger beats). In the text of the chorale, the word “Wunden” is underlined by the ii6/5–V–I

progression in D major of m. 2, in which the ii6/5–V chords corresponds to the accented syllable

“Wun-” and the tonic I to the unaccented syllable “-den.” This correspondence between tone-

metric and textual analyses creates, I argue here, a closer link between music and text than

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Lerdahl and Jackendoff’s reading, thus providing a more convincing link between music and

text.

To conclude our brief discussion of this excerpt, contrary to A generative theory a tone-metric

reading highlights how the first phrase unfolds through the emphasis of pre-dominant harmonies

over dominant and tonic ones, and the downplay of the role of the opening upbeat triad. This is

due to the regularity of the metric articulation of the excerpt, which impacts both the perception

of the piece and its compositional process. Tone-metric analysis does not negate traditional

harmonic hierarchical relationships, but helps us describe how, within the specificity of each

individual musical contexts, that hierarchy can be interpreted to optimize performance choices.

By complementing the metric approach of tone-metric analysis with the harmonic approach of A

generative theory, analysts and performers can ultimately generate a personalized reading of the

piece that prioritizes harmonic, melodic, metric, or textual elements as a function of the final

perceptual results that they want to describe. For example, if the chorale were to be performed in

a liturgical setting, a stricter adherence to tone-metric analysis might certainly be more suitable

than a purely harmonic one, as congregational singing of a chorale harmonization composed

over a specific text implies a strict adherence of the performance to the textual accents (element

highlighted by the tone-metric analysis); however, it remains necessary to have a clear

understanding of the harmonic hierarchy on which the piece is constructed, and this is provided

by Lerdahl and Jackendoff.

In the next example, I discuss another way in which tone-metric analysis can inform other

analytical approaches by analyzing a form-functional reading of the piano sonata in C major by

Mozart (K. 279/189d, iii), mm. 1–10, and providing the complementary tone-metric analysis of

the excerpt.

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6.2 Piano sonata in C major, K. 279/189d, iii, by Mozart. Mm. 1–10.

As mentioned in Chapter 4, tone-metric analysis is a descriptive and not prescriptive form of

analysis, as it is not meant to describe the specific melodic and harmonic functions but their

perceptual flow. More specifically, tone-metric analysis describes the unfolding of melodic and

harmonic events as functions of tactus. For this reason, it often provides a view of pieces that

differs from that of other analytical approaches that do not account for tactus as the underlying

organizing element of the music. Another aspect of interest in tone-metric analysis is the fact that

the methodology understands formal articulations not as based on pre-compositionally

determined schemas (e.g., basic ideas or sentences), but as form in becoming and flux of

interconnected musical gestures and textures. These two elements characterize tone-metric

analysis as a particularly useful tool to highlight the ambiguity generated by interpolations,

which disrupt the regularity of prototypical formal functions such as basic ideas and sentences,

by generating textural and motivic overlaps and, ultimately, formal analytical uncertainties.

These uncertainties are particularly interesting challenges in performance, as they are subject to a

great deal of interpretation. In the next section, I focus on the opening ten measures of Mozart’s

piano sonata in C major, K. 279/189d, iii, to provide a performance-related reading of Caplin’s

form-functional approach, and show how tone-metric analysis can help us generate a bridge

between a theoretical-analytical approach and the performance of music.

From a form-functional perspective, the passage in example 6.3 is made of an antecedent and

consequent phase, each constructed from a basic idea (b.i.) and contrasting idea (c.i.), with an

interjection of material (interpolation) in mm. 7–8. The form-functional approach is based on the

underlying harmonic progression of phrases, which is in turn organized according to cadential

patterns (HC in m. 4 and PAC in m. 10). According to this approach, neither melody nor meter

are taken into account as generating principles of the analysis (although parallels between

melodic and harmonic designs are accepted as indicators of formal identities). In contrast, I

suggest that the mapping of meter and sonic events onto each other obtained through tone-metric

analysis can produce useful insights into the formal analysis of the excerpt. In example 6.3, I

provide a table with both wave and tree analyses of the excerpt, and a summary of Caplin’s

form-functional analysis for comparison:

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Example 6.3. Tone-metric and form-functional analysis of K. 279/189d, iii. Mm. 1–10.

The tree in example 6.4 highlights the presence of tone-metric groupings in mm. 1, 2–3, 4–7, and

8–10. Notice that the trees are constructed as functions of, or as dependent on, the articulation of

tone-metric waves, and not on the thematic, harmonic, or rhythmic material. A comparison of

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tree groupings and Caplin’s analysis (found below the tree analysis), highlights that, while both

form-functional b.i. (basic ideas) in mm. 1–2 and 5–6 entirely belong to their related tone-metric

groupings, the c.i. (contrasting idea) of mm. 3–4 and the interpolation of mm. 7–8 are tone-

metrically divided in two halves by simple pivots in mm. 4.1 and 8.1. These tone-metric

groupings suggest that the HC in m. 4 should not be articulated as an arrival harmonic point, but

as a melodic bridge between mm. 4–5. In support of this reading comes the sixteenth-note

melodic design running through m. 4, which stresses a continuity between m. 3 and the b.i. in m.

5. However, m. 4 and the attached dominant function generate a separate wave, articulated by

two pivots in m. 4.1 and 5.1, rendering this measure, from a tone-metric perspective, both a

bridge between mm. 4–5 and a structural pickup to m. 5.

The interpolation of mm. 7–8 is also divided in two halves by a simple tone-metric pivot. This

demarcation occurs over, and divides the two iterations of, the exact repetition of the design

found in m. 7. This tone-metric divide generates two three-measure sections: mm. 5–7, ending

on the first instance of the repetition, and mm. 8–10, which includes the second repetition and

the final PAC. From a performance point of view, this tone-metric articulation of the repetition

could be interpreted as an echo, or a statement (m. 7) followed by a commentary and

confirmation (mm. 8–10); this performance approach seems even more appropriate when such

repetition, as an interpolation, modifies the standard formal organization of the phrase, as is the

case in this excerpt.

The ability to model and highlight differences compared to more traditional approaches is, I

argue, one of the most interesting and useful aspects of tone-metric analysis. By suggesting a

coherent and often significantly different approach to analytical issues, tone-metric analysis

contributes to generating questions and providing insights into possible alternative solutions for

both analysts and performers. For example, the tone-metric analysis presented in the previous

paragraphs highlights the presence of a continuous sixteenth-note flow in the right hand, and

indicates a textural continuity between b.i. and c.i.. Moreover, the graphic tree in the example

indicates that, at a deeper level, m. 3 is connected to the previous measures, but that, at a more

surface level, it is projecting its influence (via the continuous sixteen-note run) onto the next c.i.,

generating a bridge between the two elements. It remains the task of the analyst and the

performer to decide which solutions should be highlighted in performance, and to what extent

any of those solutions should be emphasized. For example, the difference in analytical outcomes

highlighted in the previous paragraphs regarding mm. 7–8 generates questions about the

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relationship between a standard (and fairly rigid) formal segmentation of this passage (based on

harmonic function and phrasal distortion), and a more fluid and interpretive view of the same,

which is tone-metrically derived. In light of this difference, it is worth noting that even earlier

theoretical works on this repertoire (e.g., from the general The aim and the inner nature of

compositions and, above all, the way in which they arise by Heinrich Christoph Koch (1749–

1816)8 to the specific analyses of Mozart’s sonata by Allen Forte (1926–2014) in Introduction to

Schenkerian analysis,9 which antedates Caplin’s theory, were anchored in a phrase-formal

reading of the organization of the music, which is not the case for tone-metric analysis, as the

latter is intended, so to speak, as the analysis of the performance of form.

To summarize, the regularity of the standard two-measure form-functional articulation of this

period (including its interpolative deformations) is tone-metrically manipulated and denied

through gestural interjections such as the repetition in m. 8, and deconstructed through the

reorganization of form-functional units such as the harmonic articulation in m. 4. This diversity

within the regularity of a classical formal structure is an expression of tone-metric efflorescence,

or the flourishing of surface details organized in relationships to deeper tone-metric organization.

6.3 Prelude in E minor, Op. 28 no. 4, by Chopin. Mm. 1–13.

I now provide an example of the application of tone-metric analysis to the iconic chromatic

passage of the first thirteen measures of the Prelude in E minor, Op. 28, no. 4 by Chopin. This is

the score of these measures:

8 Nancy Kovaleff Baker, and Thomas Christensen, Aesthetics and the art of musical composition in the German

Enlightenment: selected writings of Johann Georg Sulzer and Heinrich Christoph Koch (Cambridge: Cambridge

University Press, 1995). 9 Allen Forte, and Steven E. Gilbert, Introduction to Schenkerian analysis (New York: Norton, 1982).

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Example 6.4. Tone-metric analysis and tree of the E minor Prelude, Op. 28 n. 4. Mm. 1–13.

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In 1995, Schachter provided what is now a well-known analysis of the excerpt:

Example 6.5. Excerpt of Schachter’s reduction the E minor Prelude, Op. 28 n. 4. Mm. 1–13.

Schachter’s well-known reductive analysis of these first thirteen measures focuses on the study

of how the structural B–A–G descent in mm. 1–13 is articulated through a registral shift of A in

m. 9. The reading of these 13 measures from a phrasal point of view articulates the passage in

mm. 1–4, mm. 5–8, and mm. 9–12, ending on a one-measure arabesque in m. 12, connecting to

the new beginning in m. 13. As seen in the score, each of these groups emphasizes a

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neighboring/embellishing gesture (B-C-B for mm. 1–4; A–B–A for mm. 5–8; and F#–A–F# for

mm. 9–12). Tone-metric analysis graphically models this excerpt’s form and sense of flow

through (1) simple tone-metric pivots, which occur every two measures (with the exception of

the first two pivots), and (2) branch groupings, which develop through longer and longer nested

branching groupings (pickup to m. 1; m. 1; mm. 2–3; mm. 4–7; and mm. 8–13).10

A comparison between Schachter’s and tone-metric analysis in example 6.4 highlights that the

first note of the two branch groupings starting in mm. 4 and 8 respectively is the last note of the

melodic designs articulated through the neighboring gestures B-C-B of mm. 1–4 and A–B–A of

mm. 5–8 in Schachter’s study. In other words, there appears to be a discrepancy between the

phrasal boundaries proposed by Schachter and those highlighted by tone-metric analysis. This

apparent discrepancy, however, can be solved if we look more closely at the groupings of tree

structures. The tree in mm. 4–8 is composed of two nested branching groups: mm. 4–5 and mm.

6–8, and the branching group in mm. 4–5 is further subdivided into branching groups in m. 4 and

m. 5. These subdivisions exist because of the regular rhythmic pace of the excerpt: as a result of

this regularity, trees are nested, identical to each other, at the rate of one branching group per

measure (compare this situation with trees created by a variety of rhythmic values, as is the case

with the tree structure of the Piano sonata in C major, K. 279/189d, iii, by Mozart analyzed

earlier). What these regularly nested trees are articulating is a subdivision of the piece in regular

one-measure groupings of different hierarchical significance (i.e., higher-level nesting indicates

fewer tone-metrically relevant branches). If we compare this regular one-measure layering with

the phrasal boundaries layering proposed by Schachter, we see that all the phrasal boundaries of

Schachter’s analysis do actually match some of the pivots of the tone-metric graphic analysis:

when considered at a different structural level, tone-metric analysis provides the same phrasal

articulation as Schachter’s, with additional pivots (not found in Schachter’s analysis) indicating

further points of metric articulation and interest.

The fact that these tone-metric articulations are found at different levels on the tree indicates

their relatively lower hierarchical impact in terms of metric, not of melodic or harmonic

structure. For this reason, any considerations about tree hierarchies can be fully expressed only

10 For the tree analysis, in example 6.4, all chord changes and any notes of the solo lines are taken into account as

sonic events, while repeated harmonies are not.

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within those hierarchies and cannot be exported and mapped onto melodic and harmonic designs

(e.g., we cannot state that, because a sonic event is found at a less tone-metrically relevant level,

it is also less melodically or harmonically relevant). The combination of both analyses,

interpreted by analysts and performers, allows us to establish a complex picture of the piece, in

which metric and sonic properties can be compared while remaining distinct from each other in

terms of their generating principles.

The information provided by the phrasal-melodic and harmonic organization on the one hand,

and the results of tone-metric analysis on the other can inform our understanding and

performance of the excerpt. For example, the complexity of the melodic-harmonic relationships

in mm. 4 and 8, which are brought to light in tone-metric analysis by the rhythmic regularity of

the excerpt, can be emphasized through a performance that generates a sense of melodic-phrasal

ending, and the expectation of the subsequent phrase. This effect can be achieved through the use

of trattenuto and agogic accents where a new tree branch enters in mm. 5–8 (e.g., m. 5.1 and 5.2;

m. 6.1; m. 7.2; and m. 8.1). More specifically, by deemphasizing the descending chromatic bass

line E–D#–D in mm. 6–7, we can produce a sense of relaxation and closure of the musical

phrase; conversely, according to the same tone-metric logic, by delaying the emphasis of m. 7.1

to 7.2 on the same major-minor seventh, the performance would generate a structural pick-up to

the downbeat of m. 8 (a need for an accentuation of m.8.1 is supported by the use of the melodic

appoggiatura-like design on the second half of m. 7.2). To conclude, tone-metric analysis, as

performance analysis, provides a reading of phrasal organization that emphasizes the perdendosi

at the end of each phrase, without denying that there are clear-cut 4-measure phrases in the first

12 measures of the piece. The existence of pivots (and related tree articulations) does not imply

the existence of performance-related articulations: pivots are algorithmically generated elements,

which emerge from the metric organization of music and are not dependent on musical

parameters. Their presence can be used to support specific performance choices (as they are

indeed indicators of moments of metric interest that can be highlighted in performance), but they

alone are not necessarily indicators of musical articulations (e.g., while tone-metric tree

articulations at a lower level necessarily happen within each piece, the actual surface details of

the music might indicate continuation, as it was shown in the analysis of K. 279/189d, iii, in

example 6.3).

Another point of interest from both performance and formal points of view is the caesura in m.

12. This caesura could be interpreted as a HC marking the end of the antecedent of the periodic

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structure of the piece (which is an unusually complex twenty-five-measure-long period: mm.1–

12 and 13–25); however, the cadential function of V in m. 12 is weakened by both the presence

of the seventh11 and by the arabesque in the right hand, which imprints a sense of motion back

towards E minor (i) in m. 13, an arrival in turn weakened (as it was at the opening) by the use of

first inversion. These harmonic and metric relationships suggest that the V7 in m. 12 functions

not so much as an ending of the first half of the piece, but as a link between the first and second

halves (between antecedent and consequent).12 In the tone-metric graph of example 6.4, this

double function of the V7 in m. 12 is highlighted through the continuation of the level 3 tone-

metric branch, which generates a link between the first 12 measures and m. 13.13

The D# in m. 12 is particularly salient from both tonal and tone-metric perspectives, as it is both

the leading tone that resolves to (or at least returns to) E minor in the subsequent measure, and

the only generator of three branches within that measure, which is significant for the following

reason.14 The farthest extending tree branch encompasses the descending triplet figure in m. 12;

the descending motion of this triplet encapsulates a D natural in chromatic dialogue with the

previous D# (the D natural effects an embellishing third motion that descends to the fifth of E

minor; and the D# of course ascends, as the leading tone, to the tonic of E minor), creating a

11 For a discussion and examples on the role of mid-19th century dominant harmonic function, see Janet Schmalfeldt,

“Towards a reconciliation of Schenkerian concepts with traditional and recent theories of form,” Music Analysis

10.3 (1991): 233–87. 12 Don McLean has noted (in a personal communication) that, from a Schenkerian perspective, the first 9 measures

of the piece might be seen as organized around a “false” register. The first B3 of the thumb pickup in m. 1 ultimately

turns out to be the structural B; the register is displaced an octave higher through the immediately following octave

leap (and again another octave higher in the expanded consequent phrase); the piece remains in the false register

until the A3 in m. 9 (left hand), which completes a structural third motion from B3 (m. 0) to A3 (m. 9) to G3 (m.

13), the last overlapped by the restatement of the false B4 at the beginning of the consequent (m. 13). 13 For a recent discussion on the difficulties often encountered in defining the phrasal articulatory functions of

caesuras such as half cadence, see Poundie L. Burstein, “The half cadence and other such slippery events,” Music

Theory Spectrum 36.2 (2014): 203–227. 14 The D#’s unique metric positioning within the piece causes the generation of the three branches, not its harmonic

and melodic functions. However, it is the specific way meter, harmony, and melody unfold throughout the piece that

makes it possible for the D# to exist in this tone-metric context; although any sonic event could potentially occur in

m. 12 instead of the D#, the musical discourse leads to the use of a dominant harmony, to which D# belongs. Among

the potential harmonic choices, Chopin decided to use the leading-tone, thus generating the specific tone-metric

situation above analyzed. This choice represents the collapses of the potentialities of sonic events into an individual

and discrete harmonic/melodic/ rhythmic function that is unique to a specific moment in the piece.

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powerful melodic gesture of anacrusis and generating one of the most salient moments of the

excerpt.15

The analysis of m. 12 exemplifies how tactus and musical freedom of an arabesque-like melodic

line can be integrated. By definition, tactus is a form of algorithmically generated constraint, and

arabesque, on the other hand, an expression of rhythmic (and melodic) freedom. In other words,

while the compositional principles on which an arabesque is included in a piece are based on

metric, harmonic, and melodic patterns, but its performance is not.16 A moment such as the one

in m. 12 thus represents a suspension of tactus as understood by tone-metric analysis; tone-

metric analysis is, in this sense, limited, because it shows how the measure is compositionally

related to the previous and following material, but cannot fully capture the complexity of the

sense of suspension created by the performance of the arabesque. However, because this

arabesque is indeed part of the harmonic, melodic, and rhythmic fabric of the piece, it is also

necessarily connected to its underlying tone-metric structure. For this reason, the improvisatory

character of the arabesque in m. 12 does participate in the tone-metric-based efflorescence of the

passage. The analysis in the previous paragraph has shown that the arabesque does not behave as

a simple fill-in between notes but signifies a formal break between sections; it is built on a HC

and obeys the rules of the metric organization of the piece. As pointed out before in this chapter,

it remains the task of the performer to choose an interpretation of the passage that creates

stylistic cohesion and perceptual interest for the listener.

So far, tone-metric analysis has been able to model the integration of metric and sonic

parameters for the generation of formal organization of music, providing possible performance

indications and a sense of musical narrative in repertoires heavily based on the regularity of

accentuation patterns (e.g., tactus and periodic meter). But can it be applied when these

accentuation patterns do not constitute the central principle of musical organization? In the next

15 As pointed out by Harald Krebs in his comments to this dissertation, the third motion is really a combination of

two neighbour motions—the deeper-level (and motivic) C-B, and the surface-level D-C as an additional

embellishment.” 16 This apparent conflict between rigidity of meter and freedom of melodic elements is, in fact, what, as I have been

arguing in this dissertation, generates perceptual phenomena such as the sense of, and need for, a sense of metric

freedom in performance (to counterbalance the strictness of the tone-metric texture): it is the strict metric and tone-

metric textural baseline produced by the underlying tactus (in its combinatorial structural rigidity and regularity) that

allows for freedom of composition and performance in music, and for the surfacing of phenomena such as the sense

of suspension in arabesques.

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section, I try to answer this question by analyzing the role that tone-metric analysis can have in

the understanding of the first eleven measures17 of the Sonata, Op. 1 by Berg, a composer whose

musical output is not associated with a strong sense of regularity in accentuation and measure-

long groupings.

6.4 Sonata, Op. 1, by Berg. Mm. 1–11.

Example 6.6. Score and tone-metric analysis of the Sonata by Berg, Op. 1. Mm. -1–11

17 In this example, the pickup measure is analyzed as m.1 (and not as -1), following Berg’s own indications in the

score and the traditional notation employed by Berg scholars.

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This excerpt constitutes an interesting mixture of accompanied melody and imitative-

contrapuntal writing. Within this complex texture, Berg’s own slurring practice denotes an

intention to emphasize the interplay between the contrapuntal, fragmentary, and gestural

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character of those melodic designs with a more continuous phrasal unfolding that characterizes

its first three measures, up to the V-I cadence in m. 4.1.

The analysis of this cadence, for example, well summarizes this interplay. The concluding

function of this cadence appears to be both confirmed and negated at the same time by the

presence of, respectively, the G–F# anticipation in the inner voice (indicating continuation), and

the separation between F# and B in the bass across mm. 3 and 4 (indicating separation across the

cadence). In this context, tone-metric analysis reveals a pivot over the same V–i cadential

motion, thus coming in support of the notion that the B in m. 4.1 functions more as the beginning

of the new phrase than an end to the previous one. A reading of this passage according to the

combined melodic-harmonic and tone-metric views, indicates that a performance of the V–i

motion might be interpreted as a kind of poetic or rhetorical afterthought of the same gesture

built over the dominant on G-F# in m. 3.1–3.3, and could be performed as an echo to it, through

a perdendosi. But if we base the study of this cadence on tone-metric considerations, then the

same tone-metric analysis seems to stall in mm. 1–2, when a simple pivot (suggesting

separation) appears across a held chord. How does this make any sense?

As we have recalled a number of times in previous chapters, pivots and note-content of pieces

are not necessarily paired, in the sense that pivots appear as a result of algorithmically conceived

elements generated by the regularity of the tactus’ unfolding. Often composers such as Berg use

tools like accentuation markings to generate a seemingly metrically counterintuitive point of

attraction within the texture of the piece in order to highlight a specific element (the crescendo

and stringendo in mm. 6–8 in example 6.6); in other words, it is not uncommon for Berg and

others to displace the natural accentuation pattern for rhetorical reasons. Mm. 1–2 seem

constitute one of those examples: in them, the composer deliberately holds a chord over a weak-

to-strong accentuation pattern (generator of a tone-metric pivot) with the intent of capturing the

attention of the listener and emphasizing the opening statement of the piece. In this case, the

pivotal separation is nullified by the composer’s desire to open the piece with a held dissonant

chord. This interplay between the natural unfolding of meter vs. the conscious alteration of it,

makes the music of Berg particularly challenging (from any analytical point of view), and highly

improvisational in character.

The improvisatory aspects of Berg’s opening measures of this piece are discussed by

Schmalfeldt in her analysis of the piece in the chapter “Berg’s path to atonality: the piano

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Sonata, Op. 1.”18 In it, Schmalfeldt provides a sophisticated harmonic and melodic reading of the

measure just analyzed from a tone-metric perspective. Schmalfeldt discusses a variety of

harmonic, motivic, and formal elements throughout the piece; among them, she focuses on the

relationship between those elements in the opening 4 measures of the piece, pointing out that

those measures are based on three motives (example 6.7), organized over a “sub-

dominant/dominant/tonic” incomplete harmonic progression (example 6.8 a and b) that is

structurally unfolding around the chord C#–G–B–F# at the end of m. 1 (see example 6.7 c).

Example 6.7. The motivic annotations of Schmalfedt on the sonata, Op. 1. Mm. 1–4.19

18 Janet Schmalfeldt, “Berg’s path to atonality,” 79–109. 19 Schmalfeldt, “Berg’s path to atonality,” 86.

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Example 6.8. The three-level reduction by Schmalfeldt of the Sonata, Op. 1. Mm. 1–11.20

The comparison of Schmalfeldt’s motivic, harmonic, and structural analyses (which includes: (a)

the underlying structural harmonies; and (b) and (c) two levels of Schenkerian reductions),

highlight a complex relationship between F# and G in the upper voice in mm. 1–2. From a

motivic point of view, F# and G belong to two distinct motives, F# being the ending of the first

motive (a), and G the beginning of the second (b). Moreover, from a harmonic-reductive

perspective the dissonant chord underneath F# (set-class 4–16) is less structurally relevant than

the notional half-diminished ii7 supporting G (see example 6.8, levels a and b). This implies a

tendency (at least from a local harmonic perspective) of the chord supporting F# to resolve

20 Schmalfeldt, “Berg’s path to atonality,” 87.

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forward onto the half-diminished ii7 (anticipated by the motivic G in the upper part) on the

second beat of m. 1. However, from a reductive-structural viewpoint, F# is more significant than

G (see example 6.8c). This generates a sense of stability for the sonority underlying F#, which is

highlighted by Schmalfeldt through her motivic analysis (motive ‘a’ being the F# with its

preceding pickup).

A comparison between tone-metric and Schmalfeldt’s analysis of the excerpt highlights a

dichotomy between metric and melodic-harmonic designs. In particular, this difference has been

analyzed in the previous paragraphs for the opening and closing measures. Ultimately, it remains

the task of performers and listeners to judge whether these views can be integrated, or whether a

choice has to be made between them. Those choices have to be made by taking into account the

very distinct nature and generating principles of the different analytical approaches, and the

emphasis that each of them brings to the table for the discussion.

In this sense, it is worth noting Adorno’s comments on the nature of the musical work itself and

the relationship with interpretive claims. In his Alban Berg, he states that “the work itself, by

virtue of its inherent logic, imposes certain features upon its author, its executor, without his

even having to reflect upon it.”21 Taking Adorno’s statement a step further, tone-metric analysis

explains the integration of musical parameters into a unified musical continuum based on the

premise that the positioning of sonic events within tone-metric structure can tell us how musical

features unfold. This interrelation between the positioning of sonic events and the unfolding of

musical features generates a basis for a unique analytical algorithm and related range of readings

of the piece. This algorithmic basis informs the reading of the score through its “inherent logic;”

the interpretation of the emergent features of that logic remains, however, the core of the

analytical work once the basic algorithm is in place, as waves describe features of the piece that

cannot be made manifest only by the dry algorithm generating them, and are, therefore, emergent

forms of investigation. The performance and analytical insights provided by wave analysis

become emergent features of the composition itself, in a way similar to Schenker’s

understanding of the relation between the composition and his analytical sketches of it.22

21 Adorno, Alban Berg, 16. 22 Matthew Brown, Explaining tonality: Schenkerian theory and beyond (Rochester, NY: University of Rochester

Press, 2005).

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6.5 Conclusions

In this chapter, I showed how tone-metric analysis can be used in conjunction with other

analytical approaches to model tonal relationships (Lerdahl and Jackendoff’s A generative

theory); formal articulations (Caplin’s form-functional analysis); phrase structures (Schachter’s

Schenkerian-reductive approach); and the relationship between analysis and performance

(Schmalfeldt’s integration of post-tonal pitch-class set taxonomy and Schenkerian analysis, as

well as Adorno’s view on the relationship between the musical work and its analysis). More

specifically, tone-metric analysis understands sonic events as discrete objects occupying a

determined position within a piece. This uniqueness is expressed in the form of binary (thesis-

arsis) organization that can be algorithmically modeled and is not based on specific stylistic,

harmonic, or melodic parameters. This independence from any particular musical syntax and

style allows for the application of tone-metric analysis to any tactus-based repertoires, and

potentially provides novel insights into both analysis and performance of music.

With Chapter 6, I conclude the analytical work of this dissertation. In this chapter, I have

analyzed short excerpts by Bach, Mozart, Chopin, and Berg, and situated those analyses as

exploratory extensions of the tone-metric methodology presented in the previous chapters.

Although tone-metric analysis was born as a method for the analysis of North German Baroque

organ music, it became evident that the methodology can be used to study any repertoires that

can be reduced to a tactus-based form of metric organization. For this reason, I expanded the

analysis to a brief consideration of later repertoires. Because those repertoires are also much

more studied, I decided to compare my analytical approach with pre-existing ones, to show how

tone-metric analysis can inform our understanding of this music and its performance.

Chapter 7

Conclusions

7.1 Summary

The relationship between the apparently unrestrained freedom of melodic and harmonic patterns

and the evidently underlying strict metric regularity that characterizes North German Baroque

organ music is a source of interest and fascination for many performers and scholars of early

music. Since my first steps into the sonic world of North German music as a performer, I have

been fascinated by how elegantly the variety of soundscapes generated by the combination of

melodic and harmonic patterns could interact with the steady and almost-mechanical unfolding

of tactus; by how this apparently melodic and harmonic freedom, the formal structures that ensue

from it, and the mechanical unfolding of strong and weak accents blend so smoothly to generate

the improvisational character typical of the style; and by how this interaction can be translated

into engaging and meaningful performances. In light of these observations, I started questioning

how we might better understand and describe this interaction from both music-theoretical and

perceptual perspectives, and whether it would be possible to model this interaction in a

systematically organized way. This dissertation was undertaken to try to answer these questions.

In North German Baroque organ music, as in most Western repertoires of the past, melodic and

harmonic elements are functions of the underlying metric regularity: harmonic progressions,

phrasal shapes, and gestural patterns are organized and led by the articulation of the strong and

weak beats of the underlying tactus, and hierarchies of sonic events are established primarily

based on that metric foundation. What makes tactus so important as the substrate within which

sonic events unfold? How is the link between tactus and the unfolding of musical events

generated? Do tactus and sonic events share a common conceptual space? If so, what is this

space and how can it be modeled?

Historical sources and secondary literature about tactus (studied in Chapter 2) indicate that tactus

is an alternation of an initial strong (S) accent (corresponding to T, thesis) followed by one or

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more weak (W) accents (corresponding to A, arsis). This suggests that tactus can be understood

as a specific combination among the many permutations in which S and W accents can be found

(e.g., in 2/2 there are two permutations of accents, SW and WS; and SW is the only tactus-like

combination among them). I decided to investigate this combinatorial specificity of tactus

through tools used in combinatorics: Pascal’s triangle, and its geometric isomorphic counterpart,

Sierpinsky’s triangle. In Chapter 3, I showed how each tactus designs a specific arrangement of

numerical values on Pascal’s triangle, and how this arrangement provides information about both

the articulation of tactus as a unique combination of accents (Pascal’s triangle) and its nature as a

fractal space (Sierpinsky’s triangle).

Chapter 3 is the most conceptual chapter in the dissertation. The level of abstraction reached in

Chapter 3 was necessary in order to study the combinatorial aspects of tactus: by detaching

accent organization from its music-semantic role, and by studying it in its most general

(mathematical-geometrical) form, I was able to identify properties of tactus that emerge only

when looked at from a combinatorial perspective: the geometric patterns obtained through its

mapping onto Pascal’s and Sierpinsky’s triangles. By mapping those patterns onto the musical

score (a process described in Chapter 4), it became possible to graphically describe the unfolding

of sonic events in a piece as a function of the geometry of accents of tactus, and to graphically

quantify and represent that unfolding as a wave-like structure. Those waves, which I call tone-

metric waves (they are discussed in Chapters 5 and 6), describe the unfolding of the piece as a

function of strong and weak beats.

Sonic events generate ripple-like disturbances in the regularity of the underlying tactus, like

stones thrown into a pond generate ripples on the surface of the water. These ripples interfere

with both the underlying tactus and with each other, creating a variety of shapes visible

analytically as the individual crests of tone-metric waves. I call this disturbance efflorescence

and I describe it and its musical significance through the analyses carried out in Chapters 5 and

6. Efflorescence can be used to describe and quantify the sense of acceleration and deceleration

perceived when listening to music. In fact, efflorescence is not a one-dimensional projection of

rhythmic unfolding (e.g., a line of sonic events arranged as a sequence of rhythmic impulses),

but a two-dimensional phenomenon resulting from the interaction between tactus and sonic

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events.1 The ability of tone-metric analysis to graphically describe and quantify efflorescence

can be used not only to visualize phrasal and formal boundaries and articulations in music, but

also to describe and quantify the perception of those boundaries as they unfold to generate a

sense of time compression and expansion.2 In other words, efflorescence, because it is built on

the interaction of sonic events and tactus, is also a descriptor of phrasal and formal structures as

they unfold in time; as such, it affords a baseline of reference for analytical, perceptual, and

performance considerations.

7.2 An essay

My doctoral research is an attempt to model and explain the influence of the combinatorial

properties of tactus and rhythm on the analysis and performance of musical scores. As such, this

dissertation can be seen as a form of Versuch (Essay): a trial, an experimental effort, or, through

an associative and alliterative connection with the word “assay,” an evaluative test.

The earliest Essays, whose literary origins date back to Michel de Montaigne’s Essais (1570-92)3

and Francis Bacon’s Essayes: religious meditations. Places of perswasion and disswasion. Seene

and allowed (1597),4 were comparatively short and often highly personal treatments of a broad

range of subjects. However, with John Locke’s An essay concerning human understanding

(1690)5 the essay became a more sophisticated and systematized literary device employed to

share the results of personal intellectual inquiries into the nature of things. My interpretation of

this dissertation ultimately is close to this kind of notion. For example, similarly to how Locke’s

(pre-genomics) view of the initial human state as a tabula rasa on which subsequent

experimentation and experience yields understanding, I position the tactus substratum as a

1 This difference can be described in mathematical terms as the difference between summation and multiplication. It

is, however, outside of the scope of this dissertation to explore the mathematical implications of these findings,

which remain the subject for further research. 2 See tone-metric analyses of Durch Adams Fall by Buxtehude (example 5.1) and fugue on the Magnificat primi toni

by Pachelbel (example 5.10). 3 John Holyoake, Montaigne essais, (London: Grant and Cutler, 1983). 4 Francis Bacon, Essayes, religious meditations, places of perswasion & disswasion, (Haslewood books, 1924). 5 John Locke, and Phemister, P. An essay concerning human understanding, (Oxford, England: Oxford University

Press, 2008).

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neutral phenomenon that remains as such until the emergence of its efflorescent interaction with

the piece’s sonic content. The study of that interaction yields new understanding into the nature

of tactus and, more broadly, music performance. In a way, the interpretive interactions of my

approach perhaps lie even closer to David Hume’s approach in his An enquiry concerning human

understanding (1748),6 a reworking of his A treatise of human nature (1739-40),7 with its

subtitle Being an attempt to introduce the experimental method of reasoning into moral subjects.

My research is an attempt to introduce the experimental method of reasoning not into moral

subjects but into musical thinking, via the study of quantitative mathematical underpinnings as a

basis for musical interpretation.

The title Versuch continued to be used in German philosophical writings throughout the

eighteenth, nineteenth, and twentieth centuries,8 becoming a literary hallmark of philosophical

inquiry, and a tool for a deeper understanding of reality. To quote Wittgenstein: “My

propositions serve as elucidations in the following way: anyone who understands me eventually

recognizes them as nonsensical, when he has used them—as steps—to climb beyond them. (He

must, so to speak, throw away the ladder after he has climbed up it.) He must transcend these

propositions, and then he will see the world aright.”9 In this dissertation, the algorithmic

procedure itself is mathematical and interpretively neutral. It is the ladder which, once climbed,

can be thrown away. “Hearing the world aright” involves the subsequent analytical and

performative interpretation of the musical surface as it emerges from the underlying scaffold of

the tactus. Versuch/Essay/ Treatise/Enquiry/Attempt are all concepts that assist in the positioning

of this dissertation within the experimental essay tradition.

In music, three essays were of inspiration to my work: Carl Philip Emmanuel Bach’s great

Versuch über die wahre Art das Clavier zu spielen (1753), usually translated as Essay on the true

art of playing keyboard instruments;10 Johann Joachim Quantz’s Versuch einer Anweisung die

Flöte traversiere zu spielen (1752), which is usually translated as On playing the flute although,

6 David Hume, An enquiry concerning human understanding (Oxford: Oxford University Press, 2014). 7 David Hume, Norton, D., and Norton, M., A treatise of human nature (Oxford, England: Oxford University Press,

2005). 8 Elmar Waibl, and Herdina, P. Dictionary of philosophical terms: German-English/English-German. Vol. 1:

German-English. (Berlin: De Gruyter Saur, 1997). 9 Proposition 6.54, in Ludwig Wittgenstein, Tractatus logico-philosophicus, (London: Routledge, 2014). 10 Carl P. E. Bach, Essay on the true art of playing keyboard instruments. 1st ed. (New York: W.W. Norton &

Company, 1949). ). Although the standard English translation of C.P.E. Bach’s treatise is “The art,” the German

word “Art” might be better translated as the “manner” or “way.”

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more literally, the title means An attempt/essay at a method/instructions for playing the

transverse flute;11 and Leopold Mozart’s Versuch einer gründlichen Violinschule (1756), or A

treatise on the fundamental principles of violin playing, although, again, more literally An

attempt/essay at a fundamental school of violin).12

The three works are ostensibly an effort to thoroughly treat their subjects of keyboard playing,

flute playing, and violin playing, respectively: Bach’s wahre Art (true way) as a remarkable

catalogue of information (i.e., fingering, embellishments, performance, figured bass signs,

thoroughbass, accompaniment, and improvisation) with much advice and admonishment about

best practice versus broadly witnessed ineptitude; Quantz’s Anweisungen (instructions) as a

methodological book with copious additional stylistic commentaries; and Leopold Mozart’s

Violinschule (violin school) as a treatise on basic instrumental technique, valued now as a

compendium of historical performance practice and literary gift.

Overall, my approach emerges from the idea of Versuch as an essay or attempt to assess the

results of a single experimental topic—in the case of this dissertation, a new formulation of the

concept and consequence of a mathematical (binary and fractal) understanding of tactus. The

approach is less inductive, in the sense of a compilation of data and analytics, and more

deductive, in the sense of an algorithmic procedure – namely, the articulation of the premise of

the underlying tactus substratum, its interaction with the musical surface, and consequent

generation of tone-metric waves (and trees), and the subsequent analysis and interpretation of the

musical results, including their implications for performance.

The results presented in this dissertation are not meant to show that music is an algorithmic

phenomenon (I strongly believe it is not), but that the algorithmic basis for tactus can have an

impact on the way we perceive and analyze music. The use of an algorithmic basis founded on

combinatorial patterns is, to my knowledge, a novelty within the field of music theory, and the

ability of that basis to model surface details and large-scale formal structures in music is a

strength of the tone-metric approach. More specifically, tone-metric analytical tools provide a

novel reading of musical form in motion, and set the stage for the development of future

11 Joachim Quantz, and Reilly, E., On playing the flute (2nd ed.) (New York: Schirmer Books, 1985). 12 Leopold Mozart, and Einstein, A., A treatise on the fundamental principles of violin playing (2nd ed.) (London:

Oxford University Press, 1951).

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approaches to the understanding of cadences, hypermetric organization, and formal structures in

a variety of repertoires based on mathematical principles and relationships.

The algorithmic basis of tone-metric analysis produces a relationship that should be

problematized. If, on the one hand, the algorithm provides us with a clear analytical foundation,

on the other hand it also introduces the risk of understanding analysis as a mechanical exercise.

The algorithmic nature of the system is indeed mechanical at its core, but it is meant to result in

an efficient scaffold and process for communicating musical ideas and exchanging interpretive

information in compelling ways. That is, it is not to be construed as a way to downplay the

dialogic nature of analytical work, which remains fundamental for any analytical discourse about

music, but rather to provide a new platform for that discourse. In other words, tone-metric

analysis is not meant to provide an entirely automated modeling of music, but to generate a

secure platform for discussions about its structure, performance, and deeper appreciation.

North German Baroque organ music has been the main focus of this dissertation, but the method

was also shown to be of potential value for other eighteenth- to twentieth-century pieces. This

modeling is a useful counterpart to aspects of existing analytical theories (e.g., roman numeral

analysis, Caplin’s form-functional analysis, and Lerdahl and Jackendoff’s generative theory).

The tone-metric analytical method creates a dynamic and interactive understanding of form, and

provides analysts and performers with a platform for the discussion of multiple performance-

related issues, strengthening the connection between the theoretical-analytical and performance-

perceptual aspects of music-making through the modeling of the relationship between the

discrete positioning of notes within the piece, and the sense of continuous motion springing from

the tension and relaxation produced by their intervallic relationships. In tone-metric analysis, this

modeling is achieved through the use of a single graphic representation (i.e., the tone-metric

wave), which can be used to describe both the discrete positioning of sonic events and their

continuous flow through time. Interestingly, tone-metric analysis’ ability to model the

relationship between the discrete and the continuous through tone-metric waves is heavily based

on twentieth-century fractal principles; these, in turn, are connected to the mathematical work of

Pascal (i.e., Pascal’s triangle, see Chapter 3), who is also a crucial figure in establishing calculus,

the mathematical discipline that studies the relationship between instant (discrete) and gradual

(continuous) movement in curvilinear motion.

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This analytical approach, which, to my knowledge, has never been applied to music theory

research before, aims at linking a theoretical background with analysis and performance of

music. This bridge is obtained through the concept of efflorescence (amply discussed in Chapter

3), which is both a quantitative way to understand tone-metric waves (i.e., it can be defined as

the number of tone-metric layers that make up the waves), and a descriptor for the sense of

expansion and contraction in the perception of musical flow across time. Ultimately, this

dissertation’s goal is to provide analysts, performers, and listeners with experiential-analytical

guidance for a fuller understanding of the intricacies of music flow beyond the readings of the

same proposed by traditional syntactical-categorical labels (i.e., sentence or period).

7.3 How does tone-metric analysis fit into the field of music theory?

Tone-metric analysis studies the relationship between tactus and surface rhythms based on

accentuation patterns and their combinatorial organization, thus inextricably relating the

flexibility of metric organization to its more rigid tactus-based substratum. Thanks to its

mathematical underpinnings, this methodology can be seen as an alternative or complement to

current analytical approaches such as those of Hasty, Lerdahl and Jackendoff, and Schachter

(studied in Chapters 2 and 6) on the one hand, and a bridge between music analysis and music

perception/cognition on the other hand.

For example, unlike Hasty’s approach, tone-metric analysis does not conceive of music’s metric

unfolding as a purely moment-to-moment feedback-based form of perceptual adjustment, but as

based on tactus as a phenomenon existing independently of our moment-to-moment perception.

Tone-metric analysis is also unlike Lerdahl and Jackendoff’s method, as it understands tactus as

an algorithmically conceived and predictive constant of any metric relationships within a

composition whose effect cannot be weakened by any lack of its perceptual impact. Moreover,

tone-metric analysis is also different from Schachter’s approach, as it springs from the principle

that music-unfolding cannot be reduced to an analytical concatenation of harmonic scale-steps

and voice-leading principles without privileging the metric relationships that underly, govern,

and hierarchize those relationships. Finally, music cognition has shown that the regularity of the

beat is understood via a mechanism of perceptual feedback; however, music as an artistic

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product is dependent on fixed metric relationships; i.e., scores present a distribution of rhythmic

values that remain fixed independently of how we perceive those relationships.

In other words, tone-metric analysis proposes a form of analytical description for the unfolding

of meter in music that is consistent with the notion of tactus as a fixed algorithmic entity; within

this framework, it allows for an interpretation of the unfolding process that relates tactus with the

surface rhythmic distribution of sonic events. As such, thanks to its mathematical underpinnings

tone-metric analysis situates itself not beside, but as a bridge between, a purely music cognitive

theory, and more traditional analytical approaches like those of Hasty, Lerdahl and Jackendoff,

and Schachter.

Why is this bridging role of tone-metric analysis important? As a performer, I consider myself a

messenger, caught between the work of art and the listener. In order to convey the message of

the score, a thorough awareness of a piece’s compositional and analytic-theoretical intricacies is

necessary. This awareness is best built through the application of different music-analytical

approaches to the music, paired with an understanding of how listeners perceive that sonic world

through time and space. In this complex process of information processing, tone-metric analysis

provides a reading of meter’s unfolding that bridges traditional analytical-theoretical and

cognitive approaches to music: it can both enrich our perception of musical flow through the

notion of efflorescence, which allows us to quantify the metric flow as a layering of metric

information, and enter into a dialogue with various music-theoretical approaches to achieve a

deeper understanding of melodic, harmonic, rhythmic, and formal features.

The algorithmic approach of tone-metric analysis has the ability to model both the minutiae of

note-to-note concatenations and the large-scale organization of pieces. The approach has proven

useful in describing notions such as efflorescence and modeling the improvisatory character of

music. For example, the approach allows for the description of kinesis as tone-metric

wavelengths, and efflorescence as tone-metric wave amplitude; gestures can be modeled as

specific configurations of branches in tree structures; and the improvisational character of the

music can be visually described as the general flow of waves. As a result, the notion of flow of

information in music can be described as the interaction between the basic algorithm-generating

rules and the actual occurrence of sonic events. What is modeled by, and emerges from, this

interaction is the elusive improvisational character of music. Detailed research into

quantification processes and the mathematical properties of efflorescence are beyond the scope

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of this dissertation but will be a subject of my future research. In particular, I plan to investigate

how we can quantify the values indicated by tone-metric waves, and articulate the connections

that link the conceptual aspects of the tone-metric approach, its music-analytical realizations, and

its mathematical underpinnings.

7.4 Personal remarks

I have always considered this dissertation, at its core, a study on the nature of tactus—a largely

overlooked element in music theory literature. What is tactus? How does it affect composition

and performance? Why do we so often refer to the notion of beat to define harmonic priorities

and, often, melodic and formal designs, and yet do not account for its algorithmic foundations

and accentuation patterns? As a performer and a theorist, I am often faced with questions such

as: how fast should the beat be in this piece? How much emphasis should we put on the main

beat of a new measure? Answers to these questions, at least among performers, most often

require a musicological knowledge of tactus or beat, not a music-theoretical one. In fact, music

theory has probably never focused on the study of tactus and beat in terms of their foundational

(i.e., accentuatual) aspects; for this reason, I decided to investigate tactus from a purely

conceptual point of view, to see whether such an investigation could reveal new insights into

metric relationships in music. Indeed, my research has revealed aspects of tactus that were

previously overlooked or unknown: in particular, its combinatorial nature, and the possibility to

read music through the sieve of its mathematical features. This has opened the door to a new way

of looking at, and as a result a new way of listening to music’s unfolding, the relationship

between algorithmic and interpretive aspects of music, and, more broadly, the understanding of

music analysis in computational terms.

Telling the story of how and why mathematical tools (e.g., Pascal’s and Sierpinsky’s triangle)

and geometric sequences can help us analyze music has presented a number of challenges. For

example, finding a satisfying graphic representation of waves, or making all examples

understandable to an audience that had never seen such an analytical approach before, was

probably among the biggest challenges throughout the editing process. However, exploring an

uncharted territory such as the mathematical aspects of tactus has also been a wonderful

intellectual journey that I will cherish for the rest of my life. I look forward to continuing this

178

adventure by exploring new repertoires, expanding my understanding of how meter can affect

music-making, and promoting this knowledge within and beyond the field of music theory.

7.5 What next?

In this dissertation, I laid out the foundations of tone-metric analysis and discussed its role in the

study of music. However, I did not present the reader with sound samples to show how a piece

can be performed following the indications provided by tone-metric analysis. I am planning to

compensate for this absence of audio material by designing a lecture recital in which I will

demonstrate (in a sort of live version of this dissertation) how tone-metric analysis can provide

insights into the interpretation of music. A lecture-recital would constitute a step closer towards

bridging the gap between tone-metric analysis and performance, and would fully realize one of

this dissertation’s primary goals, which is providing analytical insights for performers.

In this dissertation, I studied primarily German Baroque organ music; however, in the future I

intend to study how the tone-metric approach works in, and investigate its applicability to, a

larger set of pre- and post-Baroque repertoires, formal designs, and instrumentations. More

specifically, I plan to study in greater detail how tactus might inform us not only about triadic

and tonal music but also about extended-tonal, twelve-tone, and atonal music where these remain

tactus-based. In other words, I plan to investigate whether tone-metric analysis, which has

already been briefly shown (in Chapter 6) as a valuable tool for the modeling of several post-

Baroque and twentieth-century excerpts, might produce interesting results in applications to

other repertoires as well. The application of tone-metric theory to more recent repertoires also

entails the analysis of much more complex metric patterns than those presented and studied in

this dissertation. More complex meters would generate different (although not necessarily more

complex) wave structures than simpler meters, due to the different originating sequences

(similarly to how 2/2 is different from 3/8 because it is generated by a different sequence).

In some cases, tone-metric analysis might even have to be reshaped at its core in order to

function as a modelling technique. For example, in works such as Stravinsky’s Rite of Spring, in

which constant changes of metric texture occur, and in Messiaen’s Quatuor pour la fin du temps,

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which uses sophisticated retrograde rhythmic relationships, tone metric analysis might be of

limited use in its current state. My educated guess is that Stravinsky’s complex metric textures

would be better served through a primary investigation of the originating combinatorial

sequences (similar to the ones used in this dissertation for binary and ternary meters), analyzed

in mathematical terms to create a sort of combinatorics-based metric variant of Babbitt-Forte set

theory combinatorics; those sequences could be fed into the algorithm to generate a resultant

complex sequence highlighting metric properties shared between the individual meters. Unlike

Stravinsky’s work, the metric language of Messiaen in his famous Quatuor and other works is,

for the most part, not reducible to a basic beat, as its rhythmic language often inserts small

“added values” to a prevalent rhythmic pattern, and is often cumulatively retrogradeable, each

metric phrase being built as a summation of different rhythmic values to generate palindromic

phrase structures. In such cases, tone-metric analysis might have to be reformulated, not as tactus

but as specific metric patterns to be conceived and analyzed in combinatorial terms before being

fed back into the algorithmic mapping.

Although I am excited about the prospect of continuing my research in several directions, at this

stage tone-metric graphs are produced by hand and quite slowly, and it would be very difficult to

use tone-metric analysis to analyze large-scale works. In order to improve both speed and

efficiency of production, it is planned that the algorithm and its related visual components

(waves and trees) be encoded as computer software. Part of my future research will be geared

towards generating such software. A consequence of the ability to quickly produce the

algorithmic basis for the analytical discussion would be to open up the possibility of broader

distribution of the analytical system, especially for theory and performance courses in academic

settings. I also speculate that this approach might provide analysts with useful insights for the

understanding of what makes certain features of some pieces more appealing and successful than

others: how and why (once broader samplings are obtained) certain tone-metric structures are

more frequently employed than others, and in what ways they have been affecting listeners’

perceptions of music. For example, tone-metric analysis could be used to study the interaction

between metric (beat and rhythm), melodic, and harmonic patterns in a variety of repertoires,

such as contemporary pop music;13 describe trends and discover potential preferences of

13 My hypothesis is that tone-metric analysis might provide interesting insights about the interplay of meter and

melodic-harmonic textures in pop music, because it would be able to indicate how metric patterns unfold in relation

180

audiences associated with specific patterns; and provide a way to better understand audience

preferences and tastes.

The study of the algorithmic organization of tactus and its mapping onto a graphic representation

able to model and quantify efflorescence shows that the musical score can be considered as a

unified continuum of mathematical (i.e., combinatorial and fractal) relationships. As such, the

analytical process becomes a vehicle for the understanding of tactus, beyond its role as a musical

phenomenon, and of music as a sonic expression of mathematical relationships. Can these

mathematical relationships tell us something about the nature of tactus? More broadly, is our

perception of tactus (not only in the context of the musical score) dependent on the combinatorial

and fractal features analyzed in this dissertation? Can we apply the principles underlying the

analytical process of tone-metric analysis to the study of sound beyond the musical score? It is

nowadays recognized that sound and rhythm provide useful insights into the structure of

molecules (e.g., photo-acoustic spectroscopy and tomography);14 affect patterns of behavior in

debilitating conditions (e.g., overcoming freezing of gait in Parkinson’s patients);15 and influence

our biological states and psychological well-being through the release of hormones (e.g.,

endorphins).16 My hope is that the principles of tone-metric analysis will find applications within

and beyond music research and performance, through an interdisciplinary dialogue between arts

and sciences for the study of the fundamental principles that govern our understanding of the

sonic world that surrounds us.

to melodic-harmonic phrases (for an example on how tone-metric analysis can be used to analyze relatively regular

metric and melodic-pattern, see example 1.6 and related analysis). 14 Yao Xia, J., and L. V. Wang, “Photoacoustic tomography: principles and advances,” Electromagnetic waves 147

(2014): 1–22. 15 S. Ghai, G. Schmitz, and A. O. Effenberg, “Effect of rhythmic auditory cueing on parkinsonian gait: a systematic

review and meta-analysis,” Scientific reports 8.1 (2018): 506. 16 C. McKinney, and T. Honig, “Health outcomes of a series of Bonny method of guided imagery and music

sessions: a systematic review,” Journal of Music Therapy 54.1 (2017): 1–34.

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