Chemistruck Daniel Thesis

Imposing Harmonic Restrictions on Symmetrical Scales: Creating a Tonal Center in the Half/Whole Octatonic Scale

Daniel Chemistruck

Undergraduate Thesis in partial completion of the

CCSU Honors Program

May, 2006

Advisor: Dr. Charles Menoche

Abstract

Octatonicism does not negate diatonicism. By imposing harmonic restrictions and relationships on chords and pitch class sets derived from the octatonic scale, it is possible to create a series of harmonic progressions that will establish a specific note as the tonic. This text presents a theoretical approach for incorporating references to common-practice period tonality, Jazz theory and Set theory to imply a tonal center—or at least centricity—in the octatonic scale; thus, overcoming the symmetrical nature of the scale.

Table of Contents

Introduction...................................................................................................................1 Chapter I: A Brief Overview of the Octatonic Scale ....................................................4 Chapter II: Imposing Harmonic Restrictions ............................................................14

Modal Relationships ................................................................................................14 Harmonic Relationships ..........................................................................................21

(1) Creating a Dominant to Tonic Relationship..................................................21 (2) Predominant Chords ......................................................................................22 (3) Chord Quality.................................................................................................24 (4) Root Progressions...........................................................................................25 (5) The Number of Vertical Structures in an Octatonic System ........................27

Vertical Spacing.......................................................................................................28 Intervallic Content...................................................................................................30 Voice Leading ..........................................................................................................32 Additional Considerations.......................................................................................35

(1) Subset Relations .............................................................................................35 (2) Reharmonization and Implied Harmonic Possibilities .................................36 (3) Metric and Rhythmic Considerations ...........................................................39

Chapter III: Creating a Tonal Octatonic System.......................................................41 Modal Relationships ................................................................................................43

(1) Creating a Dominant to Tonic Relationship..................................................44 (2) Predominant Chords ......................................................................................44 (3) Chord Quality.................................................................................................45 (4) Root Progressions...........................................................................................45 (5) The Number of Vertical Structures in an Octatonic System ........................46

Vertical Spacing.......................................................................................................46 Intervallic Content...................................................................................................47 Voice Leading ..........................................................................................................47

Conclusion ...................................................................................................................50 Bibliography ................................................................................................................50 Appendix A: Chord Lexicography .............................................................................53 Appendix B: Chord-Scale Relationships ....................................................................54 Appendix C: Modal Tetrachords Found in the Octatonic Scale ...............................60 Appendix D: Ron Miller’s Collated Order of All Constructed Modes......................61 Appendix E: Additional Subset Relations ..................................................................62 Appendix F: Additional Octatonic Systems ...............................................................63

1

-- Introduction --

Octatonicism does not negate diatonicism. A modified understanding of

functional harmony is needed in order to create a tonal center within the octatonic scale,

which will primarily be achieved by imposing harmonic restrictions and relationships on

existing chords and pitch class sets (pcs). Due to the symmetrical nature of the octatonic

scale, there is no way to ensure that certain scale degrees will maintain predominance

over others unless a set of guidelines is put in place to govern the construction of possible

vertical structures. Joseph Straus’s definition of traditional common-practice tonality (See

Figure i-1), the musical language of Western classical music from roughly the time of

Bach to Brahms, will serve as a reference for creating a tonal octatonic system.

1) Key. A particular note is defined as the tonic with the remaining notes defined in relation to it.

2) Key relations. Pieces modulate through a succession of keys, with the keynotes often related by perfect fifth, or by major or minor thirds. Pieces end in the key in which they begin.

3) Diatonic scales. The principal scales are the major and minor scales. 4) Triads. The basic harmonic structure is a major or minor triad. Seventh

chords play a secondary role. 5) Functional harmony. Harmonies generally have the function of a tonic

(arrival point), dominant (leading to tonic), or predominant (leading to dominant).

6) Voice leading. The voice leading follows certain traditional norms, including the avoidance of parallel perfect consonances and the resolution of intervals defined as dissonant to those defined as consonant.1

Figure i-1

When constructing a tonal octatonic system, adhering to or compensating for these six

attributes will provide the clearest relationship to tonality.

1 Joseph Straus, Introduction to Post-Tonal Theory 3rd ed. (NJ: Prentice Hall, 2005), 130.

Introduction 2

Chapter I will provide an introduction to the octatonic scale, describing the

octatonic scale’s construction and several symmetrical properties, along with an overview

of its use in musical contexts. Chapter II will provide the foundations for creating a tonal

octatonic system, while Chapter III will apply the concepts of Chapter II in the

explanation of a given octatonic system. Chapters II and III will be divided into the

following sub-sections:

• Modal Relationships will discuss the common subsets of the diatonic

collection, non-diatonic minor scales, and the octatonic collection.

• Harmonic Relationships will discuss creating the dominant to tonic relationship, predominant chords, chord qualities, and the total number of chords in an octatonic system.

• Vertical spacing will discuss the intervallic spacing and ordering of notes within a vertical structure to avoid harmonic ambiguity.

• Intervallic Content will discuss post-tonal relationships between chords.

• Voice Leading will discuss possibilities of creating new rules for harmonic progression.

• Additional Considerations will discuss additional subset relations, chord substitutions, and metric considerations.2

An explanation of chord symbols is given in Appendix A, with an explanation of

chord-scale relationships in Appendix B. Appendices C and D contain charts and

information from Ron Miller’s book Modal Jazz Composition and Harmony Volume 1,

which are referenced to throughout the text. Appendix E lists several subsets derived

from the octatonic scale, with Appendix F listing additional octatonic systems.

2 Additional Considerations is omitted from chapter III.

Introduction 3

Definition of Terms:

• Any reference to the “octatonic scale” refers to the half/whole ordering of the scale

unless otherwise noted. The preference of the half/whole ordering over whole/half is

due to the presence of the perfect fifth interval available from the tonic of the scale; a

more thorough discussion is presented in Chapter II.

• Any reference to the “octatonic collection” indicates that the ordering of the octatonic

scale is inconsequential, as the material being discussed is pertinent to both possible

orderings of the scale.

• All scales, melodic patterns, and harmonic progressions will be presented starting on

C or in relation to C unless otherwise noted.

• All discussions of scale degrees refer to the parent major scale of the given key. For

example, in the mode of C Lydian, a raised fourth scale degree indicates an F , even

though F is the naturally occurring fourth scale degree of the C Lydian mode.

• All instances referring to the “dominant” indicates the vertical structure built on the

note a perfect fifth away from the root of the scale; whereas all instances referring to

“dominant 7th” chords indicates a major-minor 7th chord built on a given note.

• The term “subdominant” refers to the vertical structure built on either the fourth or

raised fourth scale degree.

• The term “chord” is used to indicate a specific harmonic structure familiar to modern

Jazz harmony; whereas the term “vertical structure” indicates a collection of

simultaneously sounding pitches.

• The terms “scale” and “mode” are used interchangeably.

• All accidentals affect only the immediately following note.

4

-- Chapter I -- A Brief Overview of the Octatonic Scale

A brief overview of the octatonic scale is necessary to differentiate between the

concepts that will be presented herein and those of previous composers, although a

thorough discussion of the scale is beyond the scope of this text. Jazz improvisational

theory along with the works of Stravinsky and several other notable composers will serve

as the foundation for the historical use of the octatonic scale.

The octatonic scale is utilized prominently in Jazz improvisation over diminished

7th or dominant 7th chords, depending on the ordering of the scale. Due to the inherent

diminished 7th presence within the octatonic scale, since it is possible to extract two of

the three possible diminished 7th chords a minor second apart (see Figure 1-1), Jazz

terminology usually refers to the scale as the “diminished scale” in either “half/whole” or

“whole/half” ordering, referring to the order of alternating tones and semitones.3

Figure 1-14 The cycling of major and minor seconds facilitates the symmetry that is characteristic of

the octatonic scale, creating two possible orderings: alternating major and minor seconds

(whole/half), or alternating minor and major seconds (half/whole). Oliver Messiaen refers

3 Jim Hall, Exploring Jazz Guitar (Milwaukee, WI: Hal Leonard Publishing Corp., 1990), 18–21. 4 The numbers below the staff refer to the notes relationship to the parent major scale.

Chapter I – A Brief Overview of the Octatonic Scale 5

to the octatonic scale as the second mode of limited transposition because the octatonic

collection produces complete invariance at four levels of transposition and four levels of

inversion, allowing for a total of three octatonic collections, as listed in Figure 1-2.5

OCT0,1 (0134679T) OCT1,2 (124578TE) OCT2,3 (235689E0)

Figure 1-2

As a result of the symmetrical nature of the octatonic scale, Messiaen states that the

modes of limited transposition are “in the atmosphere of several tonalities at once,

without polytonality, the composer being free to give predominance to one of the

tonalities or to leave the tonal impression unsettled.”6 The predominance of a tonality

typically arises from notes that are stated frequently, sustained at length, placed in a

registral extreme, played loudly, and rhythmically or metrically stressed.7 Also,

associating a harmonically ambiguous vertical structure with a unique orchestrational

effect may serve to endow the vertical structure with an individuality that allows it to

function as a tonic sonority, at least to the extent of achieving a sense of return.8

However, each of these instances utilizes orchestrational techniques rather than

theoretical applications of harmonic progression to provide a tonal center. Chapter II

introduces several concepts on how to create a tonal center within the half/whole

octatonic scale through the use of voice leading and harmonic progression.

5 Oliver Messiaen, The Technique of My Musical Language (Paris: A. Leduc, 1966), 87. 6 Ibid., 96. 7 Straus, Introduction to Post-Tonal, 131. 8 Milton Babbitt, “The String Quartets of Bartok” Musical Quarterly 35, no. 3 (1949): 385.


Due to the symmetrical nature of the octatonic scale, its subset structure is

comparably restricted and redundant.9 Like the octatonic collection itself, many of its

subsets are inversionally and/or transpositionally symmetrical, providing the ambiguity of

tonality typically associated with the scale.10 Figure 1-3 shows several possible subsets of

the octatonic scale that are combined to form intervallic sequences, taken from Jim Hall’s

book Exploring Jazz Guitar.11

Perfect 5ths followed by augmented 5ths, or minor 6ths, depending on the spelling:

Figure 1-3(a)

Major 3rds followed by Perfect 4ths:

Figure 1-3(b)

Sequence of minor and major triads:

Figure 1-3(c)

In post-tonal music, and even in earlier music, octatonic collections frequently emerge as

by-products of transposing scale fragments around an interval cycle of minor thirds,

which can also be seen in Figure 1-3, as every two measures of each example is

successively transposed up a minor third.12 In jazz improvisation, and a certain amount of

Twentieth-century Classical music, many of the intervallic sequences derived from the 9 Straus, Introduction to Post-Tonal, 144. 10 Ibid., 144. 11 Hall, Exploring Jazz Guitar, 18–21. 12 Straus, Introduction to Post-Tonal, 147.


half/whole ordering of the octatonic scale are used in conjunction with dominant 7th

chords. This arises from the fact that the upper structure of a V7 9 chord in harmonic

minor is a diminished 7th chord (see Figure 1-4).

Figure 1-4

Substituting the diminished 7th sonority for the V7 9 chord provides an increased amount

of tension, increasing the chords tendency to resolve. If the root or additional upper tones

of the V7 9 chord were added to the diminished 7th structure, it would destroy the

equidistant minor-third relationship, causing the chord to relinquish its quality to that of

an altered dominant.13 Figure 1-5 shows several common jazz “licks” used over dominant

7th chords taken from Jerry Coker’s Patterns for Jazz.14

Brooker Ervin, “No Private Income Blues,” on Mingus In Wonderland, Mingus Group

Figure 1-5(a)

David Baker, “Honesty,” on Ezz-Thetics, George Russell Sextet

Charlie Mariano, “Deep River,” on Toshiko Mariano Quartet, Toshiko Mariano Quartet

Figure 1-5(b)

13 Ludmila Ulehla, Contemporary Harmony: Romanticism through the Twelve-Tone Row (NY: The Free Press, 1966), 126. 14 Jerry Coker, Patterns for Jazz (Miami, FL: Studio Publications Recordings, 1970), 115, 131.


John Coltrane, “Straight No Chaser,” on Milestones, Miles Davis Sextet

Figure 1-5(c)

Dan Chemistruck, “The Bombing of London,” on 5/2/06, CCSU Jazz Ensemble

Figure 1-5(d)

For more insight on the use of the octatonic scale in jazz improvisation, studying one or

more of the following books will be helpful:

• Jazz Improvisation by David Baker • The Complete Method for Improvisation by Jerry Coker • Patterns for Jazz by Jerry Coker • The Lydian Chromatic Concept by George Russell • Scales for Jazz Improvisation by Dan Haerle • Thesaurus of Scales and Melodic Patterns by Nicolas Slonimsky

A significant amount of Stravinsky’s “octatonic-diatonic interaction” adheres to

the practice of utilizing the half/whole octatonic scale over dominant chords. Stravinsky’s

most octatonic works include Petroushka, The Rite of Spring, and Symphony of Psalms.15

Each of these pieces uses symmetrical constructions within juxtaposed blocks, defying

internally motivated “development” along traditional tonal lines through the static

harmony present within each block. Pieter Van Den Toorn, a scholar of Stravinsky’s

octatonic works, states that “change, progress, renewal, or development is possible only

by abruptly cutting off the deadlock and juxtaposing it with something new in the

15 Pieter C. Van Den Toorn, Stravinsky and The Rite of Spring (Berkley/Los Angeles, CA: Univ. of CA Press, 1987), 125.


collectional reference.”16 The collectional reference typically consists of either the

octatonic or diatonic collections, although a significant amount of the work can also be

analyzed using the modes of non-diatonic minor scales.17 Regardless of analytical

approach, Stravinsky’s octatonic writing is characterized by his use of tonality by

assertion, along with creating forward motion by juxtaposing blocks of octatonic material

with blocks of non-octatonic material.

The techniques of superimposition, juxtaposition and repetition are essential to

Stravinsky’s art.18 Stravinsky assigned priority to certain pitch classes by means of

doubling, metric accentuation, and persistence.19 This practice is most vividly seen in

Petroushka, The Rite of Spring, and Symphony of Psalms. Figure 1-6 shows the chord

typically associated with each of the works.

Petroushka chord:

Rite chord:

Psalms chord:

Figure 1-6

The Petroushka and Rite chords persist throughout a significant portion of their

respective works as either source material for other melodic lines or as repeating ostinato

figures. The Psalms chord, while it is not overtly octatonic itself, acts as a punctuation

16 Pieter C. Van den Toorn, Music of Igor Stravinsky (New Haven, CT: Yale Univ. Press, 1983), 328. 17 Dmitri Tymoczko, “Octatonicism Reconsidered Again,” Music Theory Spectrum 25 (2003): 188. 18 Van Den Toorn, Stravinsky, 128. 19 Ibid., 144.


mark between blocks of interacting and interpenetrating diatonic and octatonic

collectional references, as it is a subset of both references.20

The Petroushka chord is a polychord, consisting of two major triads a tritone

apart, producing pitch class set (pcs) 6-30 (013679), which is a subset of the half/whole

ordering of the octatonic collection; however, the chord can also be analyzed as a

C7 9 11 chord, although it does not function as such. The Petroushka chord is then used

as the referential source for the piano arpeggios and orchestral interactions, functioning

as it its own independent block with no tendency to resolve, despite its dominant 7th

structure.

The Rite chord can also be viewed as a polychord, consisting of an E major triad

over an E 7 chord, producing pcs 7-32 (0134689); however, this pcs is not entirely

octatonic, as pitch class 8 is not a member of the half/whole octatonic scale. The chord

can be more thoroughly analyzed as the vertical manifestation of the E Mixolydian 2 6

scale, although it does not have any of the functional implications that are typically

associated with this mode.21 “Stravinsky claims that he was not aware of Phrygian

modes, Gregorian chants, Byzantinisms, or anything of the sort in relation to his octatonic

compositions. The works were conceived intervallically, not harmonically, as two minor

thirds joined by a major third (0134).”22 Despite the foreign note found in the Rite chord,

there is no harmonic motion presented or attempted by the chord, as it serves as a pitched

rhythmic figure. Van den Toorn states, “Tonally functional schemes of modulation or

definitions of key are irrelevant here. Dominant chords are likely to be heard and

understood with reference to these prior contexts as self-enclosed blocks, not as sustained

20 Van Den Toorn, Music of Igor, 346. 21 Mixolydian 2 6 is the fifth mode of harmonic minor. See Appendix B for more information. 22 Van Den Toorn, Music of Igor, 344.


dominants awaiting an impending resolution.”23 This is characteristic of the majority of

Stravinsky’s octatonic work, typically treating a block of octatonic material as one

section of a composition to be varied with contrasting blocks of non-octatonic material,

without any intentions of creating a sense of forward motion within the octatonic block

itself.

Van den Toorn asserts that “the dominants are, octatonically, incapable of

realizing the traditional tonal escape route: there can be no resolution to a tonic.” 24 This

statement is only true if the theoretical traditions of common-practice tonality are forced

onto the octatonic collection; however, there is no need to chain modern sounds into a

system for which it was never intended.25 While Van den Toorn’s statement is true, it

does not necessitate that dominant 7ths are incapable of any resolution within an octatonic

framework. Despite the lack of traditional leading-tone elements in the octatonic scale,

principals of modern contrapuntal writing and voice leading permit all intervallic leaps

and require no resolutions.26 This concept may be further refined by altering the

definition of resolution so that it includes a defined intervallic relationship established by

the composer, rather than limiting it to the traditional concept of stepwise or perfect fifth

motion; therefore, the effect of harmonic progression is possible analogically rather than

absolutely through the transposition of a vertical structure, where the harmonic

relationship associated with the interval of transposition defines the harmonic

relationship. This type of progression is one of tonal association rather than of tonal

23 Van Den Toorn, Music of Igor, 343. 24 Ibid., 326. 25 Ulehla, Contemporary Harmony, 264. 26 Ibid., 327.


function, as the only preserved relationship to traditional tonality is the root movement of

the vertical structures.27 This concept will be discussed in greater detail in Chapter II.

Stravinsky utilizes the Dorian tetrachord (0235) as a bridge between octatonicism

and various diatonic orderings.28 The octatonic-diatonic interaction that the Dorian

tetrachord allows blurs the distinction between the two systems, but once an octatonic

framework is brought more securely into play, the authenticity of the tonal reference

fades. “It becomes a semblance, a side effect; the triads of progressions acquiring a

different feel, a different identity, owing to the symmetry of which they are now felt to be

a part.”29 Although specific instances of Stravinsky’s vertical structures can be analyzed

as subsets of either the diatonic church modes or non-diatonic minor scales, when looking

at how the structures interact with each other, the octatonic relationships become

apparent as the chords are incapable of fulfilling their traditional tonal functions. When

enough alterations take place that the accustomed Classical progressions of the major and

minor tonalities are no longer present, and no single chord takes unquestioned status as a

tonic, a solid diatonic key center is lost and we must settle for a possible, or perhaps,

probable hypothesis.30 Stravinsky is able to avoid this ambiguity of tonal center through

his blocks of static harmony, creating a tonal center simply because there are no other

choices presented.

27 Babbitt, “The String Quartets of Bartok,” 380. 28 Pieter C. Van den Toorn, “Colloquy: Stravinsky and the Octatonic, The Sounds of Stravinsky” Music Theory Spectrum Vol.25, 2003: Pg. 168. 29 Van Den Toorn, Music of Igor, 327. 30 Ulehla, Contemporary Harmony, 183.


Suggestions for Further Research:

For additional insight and musical examples of Stravinsky’s octatonic music, see

any of the books or articles used for this text written by Pieter Van Den Toorn. An

analysis of Oliver Messiaen and Béla Bartók’s music will offer additional insight into

octatonic systems that are harmonically more active than Stravinsky’s, yet maintain use

of the scale’s symmetrical qualities. Further study into the octatonic scale’s use in Jazz

will yield primarily pattern-based melodies used for improvisation, although occasionally

composers will use diminished 7th chords in unorthodox circumstances. For example,

Duke Ellington will occasionally end a piece on a diminished 7th chord to create an

ambiguous sense of tonal center and completeness.

14

-- Chapter II -- Imposing Harmonic Restrictions

Modal Relationships:

Modal scales made a profound impact on harmony during the Impressionistic

period; however, the effect that modes had on harmonic progression is more pertinent to

creating an octatonic system.31 Ludmila Ulehla points out the following characteristics of

the changes that took place between the chromatic concept of harmony and the aspects of

Impressionistic harmonization: 32

1) Lack of leading-tone. 2) Triad qualities relating to a mode rather than the major or minor scale. 3) Root progressions utilizing the full scope of chromaticism. 4) Non-traditional resolution of seventh chords. 5) Vague sense of key due to the non-diatonic effects.

These concepts are essential to creating an octatonic system because they are inherently

present within the octatonic scale (with the exception of number three). The remainder of

this section will focus on characteristic number two (triad qualities relating to a mode

rather than the major or minor scale), discussing the common subsets of the diatonic

collection, non-diatonic minor scales, and the octatonic collection. The remaining topics

will be discussed in the following sections of the chapter.

The diatonic church modes are defined by characteristic notes of the scale, allowing

for incomplete harmonies (chords missing the root, third and/or seventh chord degrees) to

maintain a tonal relationship and function. The ability to imply modal relationships

without presenting the entire mode is a convenient way to maintain associations to

31 For a thorough discussion of the influence of modes on harmony, see Ludmila Ulehla’s Contemporary Harmony: Romanticism through the Twelve-Tone Row Chapter 9: The Influence of Modes on Harmony. 32 Ulehla, Contemporary Harmony, 171.

Chapter II – Imposing Harmonic Restrictions 15

tonality, as only portions of the diatonic church modes and other non-diatonic minor

scales are found in the octatonic scale. Modal relationships should be made wherever

possible, unless it is the composer’s specific intention to avoid these relationships. Figure

2-1 shows the characteristic scale degrees of each mode of the major scale (column one

being the most characteristic and column six being the least).

Diatonic Modes Priority Table: 1 2 3 4 5 6 Lydian 4 7 3 6 2 5 Ionian (1) 7 4 3 6 2 5 Ionian (2) 7 3 2 6 5 (no 4) Mixolydian (1) 7 4 3 6 2 5 Mixolydian (2) 7 3 2 6 5 (no 4) Dorian 6 3 7 2 5 4 Aeolian 6 2 5 3 7 4 Phrygian 2 5 4 7 3 6 Locrian 5 2 7 6 3 4

[Note: The order has been adjusted to conform to “common practice.”]33

Figure 2-134

With the root being the most important note of a chord, typically the third and seventh

chord degrees are the next most important notes in terms of function, the third chord

degree providing the quality of the chord—major or minor—and the seventh chord

degree providing direction. Although the third and seventh chord degrees are usually the

most important functional notes of a chord, it does not necessitate that they are also the

most characteristic notes of a chord, which is more apparent in the Aeolian, Phrygian and

Locrian modes. The first ordering of the Ionian and Mixolydian modes show the fourth

scale degree being more characteristic than the third scale degree of the mode; this is due

to the harmonic use of the modes and their relationship to the other modes. A vertical 33 See Appendix D for an explanation of “common practice.” 34 Ron Miller, Modal Jazz Composition & Harmony Volume 1 (Rottenburg, Germany: Advance Music, 1992), 20.


structure consisting only of a major seventh interval can be interpreted as either the

Lydian or Ionian mode. Adding the third chord degree will do little in terms of aiding in

distinguishing between the two modes, as both modes contain a major third; however,

adding the fourth chord degree to the vertical structure will clearly indicate if the implied

mode is Lydian or Ionian. Unfortunately, this same logic cannot be applied to the first

ordering of the Mixolydian mode because the remaining diatonic church modes share the

first two characteristic notes of the Mixolydian mode. Figure 2-1 was taken from Ron

Miller’s book on modal Jazz composition, and it displays the book’s predisposition for

quartal voicings, as the characteristic notes of the Lydian mode and the first orderings of

the Ionian and Mixolydian modes (reversing the order of the first two characteristic notes

listed) ascend up a quartal structure (See Figure 2-2).

Figure 2-2

The second ordering of the Ionian and Mixolydian modes come from their melodic

function, aptly omitting the fourth scale degree of the two modes as it is usually

considered a melodic dissonance.35 The latter orderings are also based on tertiary

structures, with the perfect fifth being the least characteristic note of the vertical

structure, simply providing a clear root with little indication of intended mode.

The characteristic notes of a mode can be used to imply associations to traditional

Western modality. Using Figure 2-1 as a guide, it is possible to construct vertical 35 Mark Levine, The Jazz Theory Book (Petaluma, CA: Sher Music Company, 1995), 34.


structures using the most characteristic notes of a diatonic mode that are also available in

the octatonic scale. Figure 2-3 lists all the possible notes for each mode that can also be

found in the octatonic collection, with the first variation (1) indicating the half/whole

ordering and the second variation (2) indicating the whole/half ordering of the octatonic

collection. Each mode can also be described by its Forte code (Name), pitch class set

(PCS), interval vector (Vector), and prime inversion (PI).36

Mode: Scale Degrees: Name: PCS: Vector: PI: Ionian (1) 1, 3, 5, 6 4-26 0358 012120 ---- Ionian (2) 1, 2, 4, 6, 7 5-25 02358 123121 03568 Dorian (1) 1, 3, 5, 6, 7 5-25 02358 123121 03568 Dorian (2) 1, 2, 3, 4, 6 5-25 02358 123121 03568 Phrygian (1) 1, 2, 3, 5, 7 5-25 02358 123121 03568 Phrygian (2) 1, 3, 4, 6 4-26 0358 012120 ---- Lydian (1) 1, 3, 4, 5, 6 5-25 02358 123121 03568 Lydian (2) 1, 2, 4, 6, 7 5-25 02358 123121 03568 Mixolydian (1) 1, 3, 5, 6, 7 5-25 02358 123121 03568 Mixolydian (2) 1, 2, 4, 6 4-26 0358 012120 ---- Aeolian (1) 1, 3, 5 7 4-26 0358 012120 ---- Aeolian (2) 1, 2, 3, 4, 6 5-25 02358 123121 03568 Locrian (1) 1, 2, 3, 5, 7 5-25 02358 123121 03568 Locrian (2) 1, 3, 4, 5, 6 5-25 02358 123121 03568

Figure 2-3

Notice that since each mode is derived from the same collection of pitches, the sets

consisting of either tetrachords or pentachords remain the same, with only the ordering of

each set changing, as seen in the Scale Degrees column.

Since the listed pentachords in Figure 2-3 are extremely redundant, limiting them

to tetrachord formations will create more variety among the pitch class sets derived from

the octatonic collection. A possible solution for choosing which tones to select is to start

with the root, third or seventh chord degrees of a mode before referring to Figure 2-1 to

36 The “----” in the PI column indicates that the given pcs is invariant under inversion.


fill out the remainder of the tetrachord as shown in Figure 2-5(a). For example, the third

scale degree of the Ionian (2) mode is not present in the whole/half octatonic scale,

allowing for two more notes to be placed in the tetrachord. The next two most

characteristic notes available in the Ionian mode, as listed in Figure 2-1, are the fourth

and sixth scale degrees, both of which are available in the octatonic collection. Figure 2-4

shows the construction of the Ionian and Dorian tetrachords as listed in Figure 2-5(a).

Ionian (2) Tetrachord Derived From the Whole/Half Octatonic Scale:

Figure 2-4(a)

Dorian (1) Tetrachord Derived From the Half/Whole Octatonic Scale:

Figure 2-4(b)

The Dorian (1) mode provides an example that requires only one characteristic note to

complete the formation of a tetrachord, as both the minor third and minor seventh are

available in the octatonic scale. Figure 2-1 lists the major sixth as the next most

characteristic note for the Dorian mode. The Phrygian and Aeolian tetrachords listed in

Figure 2-5(a) are constructed in a similar manner; however, the Lydian, Mixolydian and

Locrian modes listed do not conform to this pattern. The Mixolydian mode uses the fifth

scale degree instead of the sixth scale degree that Figure 2-1 suggests, forming its tertiary

seventh chord, while the Locrian mode utilizes the fourth scale degree instead of the

lowered seventh scale degree to create a minor second interval between the fourth and

lowered fifth scale degrees, creating a dissonant cluster voicing. The Lydian mode uses


the fifth scale degree instead of the sixth scale degree for the same reason as the Locrian

mode.

Mode: Scale Degrees: Name: PCS: Vector: PI: Ionian 1, 4, 6, 7 4-Z29 0137 111111 0467 Dorian 1, 3, 6, 7 4-13 0136 112011 0356 Phrygian 1, 2, 3, 7 4-10 0235 122010 ---- Lydian 1, 3, 4, 5 4-Z29 0137 111111 0467 Mixolydian 1, 3, 5, 7 4-27 0258 012111 0368 Aeolian 1, 2, 3, 6 4-Z29 0137 111111 0467 Locrian 1, 3, 4, 5 4-13 0136 112011 0356

Figure 2-5(a)

Figure 2-5(b)

Figure 2-5(b) shows all the tetrachords listed in Figure 2-5(a) that are available in the

half/whole octatonic scale. Notice how each chord is available four times, with the roots

of the four chords outlining a diminished 7th chord. This is fairly characteristic of the

octatonic scale due to the symmetrical nature of the scale. The octatonic scale also


contains four major and four minor triads, along with eight diminished triads, all

behaving in similar fashions, as seen in Figure 2-9.

Disregarding Figure 2-1 completely, there is still another alternative to consider

when constructing chords. It is possible to use the tetrachords that form the scales upon

which the modal chords are based to create vertical structures.37 Each mode of the major

scale is created by combining two tetrachords, with at least one of the tetrachords

providing the unique color tones of the scale. The tetrachord formations are as follows:

Tetrachords of the Diatonic Church Modes: First Tetrachord: Second Tetrachord: Mode: Name: PCS: Vector: PI: Name: PCS: Vector: PI: Ionian 4-11 0135 121110 0245 4-11 0135 121110 0245 Dorian 4-10 0235 122010 ---- 4-10 0235 122010 ---- Phrygian 4-11 0135 121110 0245 4-11 0135 121110 0245 Lydian 4-21 0246 030201 ---- 4-11 0135 121110 0245 Mixolydian 4-11 0135 121110 0245 4-10 0235 122010 ---- Aeolian 4-10 0235 122010 ---- 4-11 0135 121110 0245 Locrian 4-11 0135 121110 0245 4-21 0246 030201 ----

Figure 2-6 The only shortcoming with using the tetrachords listed in Figure 2-6 is that the only listed

tetrachord that is also available in the octatonic scale is pcs 4-10 (0235), which is

typically referred to as the Dorian tetrachord. Therefore, using this method with only the

diatonic church modes will allow only one modal inference; however, it is also possible

to use the tetrachords shared by the non-diatonic minor scales and the octatonic scale,

which are listed in Appendix C.

37 See Appendix B for chord-scale relationships.


Harmonic Relationships:

Due to the lack of leading-tone resolution in the octatonic scale, traditional

concepts of functional harmony cannot be realized. Although certain tonal root

progressions are still available in the octatonic scale, the vertical structures built over

these roots and their interactions with each other will be inherently different; thus, it is

still possible to maintain a tonal association through root progressions despite the lack of

tonal function. Before codifying a system of vertical structures to be used as an octatonic

system, harmonic relationships should be established by expanding upon common-

practice tonality. Without a tonal basis, the elements of chromatic harmony can

overwhelm the octatonic scale, producing confusion of key center and direction. Thus,

the creation of an octatonic system should attempt to preserve or compensate for as many

harmonic relationships found in traditional common-practice tonality as possible in order

to provide a clear tonal center. There are five main factors to consider when constructing

harmonic relationships: (1) creating a dominant to tonic relationship, (2) predominant

chords, (3) chord quality, (4) root progressions, and (5) the number of vertical structures

in the system.

(1) Creating a Dominant to Tonic Relationship:

The half/whole ordering of the octatonic scale was chosen over the whole/half

ordering because the former contains the common-practice period dominant (G) to tonic

(C) relationship. When constructing a dominant to tonic relationship, the two chords must

be given vertical structures and/or intervallic content that are unique from the other

vertical structures in the system. This will provide a distinct contrast between the tonic


and dominant chords, and the other subordinate sonorities. The presence of additional

unique vertical structures in the system is inconsequential as the main purpose for

creating the disparity between the tonic and dominant from the remainder of the system is

to prevent the symmetrical structure of the octatonic scale from overwhelming all of its

vertical manifestations, as the octatonic scale contains numerous symmetrical

constructions as seen in Figure 2-5(b). Therefore, making the intervallic content of all the

vertical structures mutually exclusive within a given system will aid in the possibility of

inferring additional tonal centers within the system, whereas permitting a degree of

recursion among the subordinate sonorities will place a greater distinction on the tonic

and dominant chords.

(2) Predominant Chords:

In addition to establishing the dominant to tonic relationship, it is also important

to designate one or more chords as predominant chords. The predominant chords of

common-practice tonality typically consist of a vertical structure built on the

subdominant or supertonic; however, neither of these chords is present in the octatonic

scale. The most practical alternative for replacing the subdominant chord is a diminished

structure built on the raised fourth scale degree, functioning as a secondary leading-tone

chord in traditional harmony as seen in Figure 2-7(a). Just as practical a substitution for

the subdominant is the tritone dominant (a dominant chord built on the raised fourth, or

tritone, of the scale), allowing for the subdominant to also resolve to the tonic as seen in

Figure 2-7(b).38

38 Ulehla, Contemporary Harmony, 213.


Secondary Leading-Tone: Tritone Dominant:

Figure 2-7(a) Figure 2-7(b) Another way to analyze the two possible chords presented in Figure 2-7 is to label the

tritone dominant as the subdominant of the octatonic system, with the diminished 7th

chord functioning as either a common-tone diminished 7th (the root of the chord

providing the common-tone) or as a secondary leading-tone chord, seen in Figure 2-8.

I IV13 VIIdim7/V Vdim addM7 I

Figure 2-8 This interpretation offers the most flexibility with chord progressions, as it allows both

dominant 7th and diminished 7th chords to be built on the raised fourth scale degree. Due

to the nature of the octatonic scale, it is possible to build a common-tone diminished 7th

chord from any note within the scale without adding foreign notes to the scale.


(3) Chord Quality

When choosing the chord qualities for an octatonic system, it is important to

consider the structure and intervallic content of the other chords present in the system.

For example, maintaining the same triadic quality for the tonic, subdominant and

dominant harmonies will provide a distinguishable relationship to the major scale as each

of these harmonies consists of major triads in the major scale, although this is only

possible with the diminished triad in the octatonic scale. Despite the limited choices for

maintaining consistent triadic qualities of the tonic, subdominant and dominant chords, it

is possible to construct either a major, minor, or diminished triad on any of the notes

found in the C diminished 7th chord (See Figure 2-9). This characteristic of the octatonic

scale has potential for exploiting the various types of mediant relationships to allow a

sense of modulation to other tonal centers without introducing foreign notes to the scale.

Major Triads Available in the Octatonic Scale:

Minor Triads Available in the Octatonic Scale:

Diminished Triads Available in the Octatonic Scale:

Figure 2-9


The chords presented thus far are largely based on tertiary triads, presenting only a

handful of the total possible trichords, tetrachords, etc that can be found in the octatonic

scale.39 Additional concerns regarding similarities between chords will be discussed in

the Intervallic Content section.

(4) Root Progressions

Root progressions in the octatonic scale are not limited to those arising from the

diatonic church modes. Although it is possible to reharmonize familiar tonal progressions

(see the Additional Considerations section), chromatic additions and mixture of modes

must prevail if a composition is not to become overwhelmed by its own unique

qualities.40 Chord progressions involving root movement of a minor second (leading-tone

resolution), major or minor third (mediant relationship), and perfect fourth or fifth

(cadential resolution) are all possible from several starting points of the octatonic scale,

as seen in Figure 2-10. Despite the lack of clear functional harmony, the root progression

accompanied by smooth voice leading provides forward motion until the progression

cadences on the C major triad.

I IIIdim VImin VIdim7 IV VImin7 5 III7 I

Figure 2-10

39 See Appendix E for a complete listing of additional subset relations. 40 Ulehla, Contemporary Harmony, 180.


Figure 2-10 makes use of several inverted chords to provide a smooth bass line that

moves predominately by either semitone or perfect fifth motion. In addition to inversions,

it is also possible to use root progressions involving implied harmonies, but this will be

discussed in the Additional Considerations section.

A unique feature of the octatonic scale is the possibility of using interval cycles or

intervallic sequences. An interval cycle is created by starting on any pitch and moving

repeatedly by any interval.41 Although it is only possible to have an intervallic cycle

consisting of minor thirds or tritones in the octatonic scale, it is possible to create

intervallic sequences by cycling through a pattern of intervals. The first two staff systems

of Figure 1-3 show two basic intervallic sequences, the first consisting of up a perfect

fifth – down a tritone – up a minor sixth – down a tritone, and the second consisting of up

a major third – down a minor third – up a perfect fourth – down a minor third. Although

these sequences are presented in a melodic context, it is also possible to use them as root

progressions. There are also less complicated sequences that will go through a complete

sequence in fewer notes than the patterns listed in Figure 1-3. Figure 2-11 shows an

intervallic sequence consisting of up a perfect fifth and down a major third.

Figure 2-11 Likewise, it is also possible to create more intricate patterns than the ones shown here.

For an exhaustive listing of possible symmetrical progressions in the octatonic scale, see

41 Straus, Introduction to Post-Tonal, 154.


Nicolas Slonimsky’s chapter on the Sesquitone Progression in his Thesaurus of Scales

and Melodic Patterns.42

(5) The Number of Vertical Structures in an Octatonic System:

The number of vertical structures that are present in a given system will affect the

total possible achievable tonal centers and will play a factor in the degree of relationships

occurring between each chord. Fewer chords will facilitate a higher degree of recursion

between vertical structures and will provide the clearest sense of tonal center due to the

limited amount of vertical structures being presented. Contrarily, additional chords will

increase the possibilities for creating alternative approach chords, tonal centers and

unique vertical structures, although it is still possible to have a high degree of recursion

due to the symmetrical nature of the octatonic scale’s subsets.

42 Slonimsky, Nicolas, Thesaurus of Scales and Melodic Patterns (NYC, NY: Schirmer Books, 1975), 51– 73.


Vertical Spacing:

Equal in importance to choosing chord tensions is determining the vertical

spacing of the chord. The categories of chord spacing are:

1) Tertiary – The adjacent notes are of a major or minor third interval. 2) Cluster – The adjacent notes are of a major or minor second. 3) Quartal – The adjacent notes are of a perfect or augmented fourth. 4) Mixed – The adjacent notes are of a combination of seconds, thirds, and fourths.43

The most apparent relations to common-practice tonality will occur with the use of

tertiary structures, although the uses of the other possible spacings are equally practical.

Regardless of the chord spacing used, principles of orchestration will help clarify chord

qualities and intended roots. There are several concerns to consider avoiding any

additional harmonic ambiguity than is already present in the octatonic scale. According to

Ulehla, reasons for harmonic ambiguity that arise from poor orchestration and/or chord

voicings can generally be found in one or more of the following situations:

1) The lack of a prominent bass tone. 2) A melodic movement in the bass register. 3) Abnormal order of intervals in the vertical structure which negate the strength

of the overtone series. 4) A conspicuous use of the tritone, melodic or harmonic, in which either tone

may claim its right as a root. 5) An absence of the cadential root progression by which an association of tonal

movement could either anticipate a root or recognize it following the chord’s resolution.44

Adhering to these guidelines will provide a clear sense of tonal function. These rules are

essential to securing the tonal underpinnings of an octatonic framework due to the

octatonic scale’s inherent tendency to lend itself to a number of these situations.

43 Miller, Modal Jazz Composition, 20. 44 Ulehla, Contemporary Harmony, 219.


Using inversions is possible, although they should be handled with care as the

symmetrical structure of the scale may lead to the confusion of the intended root. Figure

2-10 provides an example that successfully utilizes chord inversions. The G /B chord

can be analyzed as either a first inversion G Maj chord or as a B min 6 chord. The given

voicing doubles both the G and the B , but the interval of the perfect fifth between G

and D firmly establishes the G as the root of the chord. Additionally, the root of a

major triad sounds more clearly than the root of a minor triad because the overtone series

produces the major triad in lower partials.45 The chord must also be considered in its

contextual use. The preceding Adim7 chord can also be labeled as G dim7/A,

functioning as a common-tone diminished 7th chord to the G Maj chord. Figure 2-12

shows an example using the same chord with a different voicing and contextual use,

where labeling the chord G Maj would be wrong.

Figure 2-12

The root progression of B min 6 – E 7 functions as a IImin 6 – V7 in A minor, but is

deceptively resolved to Amin7/E. While the bass movement of the harmonic progression

follows the traditional deceptive cadence by having the dominant 7th chord resolve up a

semitone, the resolution chord is not the expected EMaj chord, but a second inversion

Amin7 chord. The Amin7 functions as an inverted chord because the only doubled note is

the root, with no functional choices for an alternative chord label. 45 Ulehla, Contemporary Harmony, 329.


Intervallic Content:

In addition to the surface construction of the vertical structures, it is also

important to consider their intervallic content and pitch class relationships, as it will

allow additional relationships and distinctions to form between the various vertical

structures. Two intervallic relationships that will be discussed are the R relationships and

the Z relationship. There are many other possible relationships, but they are beyond the

scope of this text.

The R relationship exists in four forms: Rp, R0, R1, and R2. Allen Forte’s

description of each relationship is given in Figure 2-13.46

Relation: Interpreted As: Rp Maximum similarity with respect to pitch class. R0 Minimum Similarity with respect to interval class. R1 Maximum Similarity with respect to interval class.

Interchange feature. R2 Maximum Similarity with respect to interval class.

Without interchange feature.

Figure 2-13

To be in the relation Rp, two sets, S1 and S2, of cardinal number n must have at least one

common subset of cardinal number n-1.47 As Forte explains on page 47 of his book The

Structure of Atonal Music, this relationship is not especially distinctive since many sets

are related to a large number of other sets. For two sets to be in relation R0, two vectors

must have no entries the same. For two sets to be in relation R2, two vectors must have

four similar interval-class entries. Figure 2-14 compares the vectors of pcs 5-10 (01346)

and 5-Z12 (01356), indicating that entries for ic1, ic2, ic4, and ic6 correspond.

46 Allen Forte, The Structure of Atonal Music (New Haven, CT: Yale University Press, 1973), 49. 47 The “cardinal number” is the amount of different pitch classes in given a set.


5-10 [ 2 2 3 1 1 1 ] 5-Z12 [ 2 2 2 1 2 1 ]

Figure 2-14

For two sets to be in relation R1, two vectors must contain the same digits and have four

corresponding interval-class entries, as seen in the comparison of pcs 4-2 (0124) with 4-3

(0134) shown in Figure 2-15.

4-2 [ 2 2 1 1 0 0 ] 4-3 [ 2 1 2 1 0 0 ]

Figure 2-15

As seen in Figure 2-15, the entries for ic2 and ic3 are interchangeable. In this sense, R1

provides a closer relation than R2 does.

Any two sets related by transposition or inversion must have the same interval-

class content; however, the converse is not true.48 Sets that have the same interval-class

content but are not related by transposition or inversion are called Z-related sets, and the

relationship between them is the Z-relation. Two prominent subsets of the octatonic

collection that are in the Z-relation are pcs 4-Z15 (0146) and pcs 4-Z29 (0137), which

also have the property of being the two “all-interval” tetrachords because their interval-

class vectors consist of 111111.

48 Straus, Introduction to Post-Tonal, 91.


Voice Leading:

Due to the structure of the half/whole octatonic scale, it is not possible to strictly

adhere to common-practice tonality standards of voice leading. Therefore, certain rules

will have to be developed for each individual system with reference to concepts of

traditional voice leading, focusing primarily on stepwise motion and the cadential root

movement of down a perfect fifth. This paper focuses on the half/whole ordering of the

octatonic scale primarily because of the presence of the perfect fifth occurring between

the tonic and dominant of the scale, which is absent in the whole/half ordering of the

scale. The interval of a perfect fifth is essential to establishing a sense of tonality because

in harmonic contexts the perfect fifth establishes the lower tone as a root so strongly that

it will retain its tonal predominance through a significant amount of contrapuntal activity

until another combination of strong intervals seize harmonic priority.49 Specific

guidelines in relation to octatonic systems will be discussed in Chapter III. The remainder

of this section will focus on the limitations of the octatonic scale in relation to traditional

functional harmony.

The octatonic scale allows the construction of dominant 7th and secondary

leading-tone structures, but both are unable to realize their traditional tonal resolutions

without adding foreign notes to the octatonic scale. A significant amount of octatonic

literature is used in an octatonic-diatonic interaction because the octatonic scale adds

chromaticism to the dominant chord, increasing its tendency to resolve. For example,

Figure 2-7(b) shows that it is possible to construct an F 7 chord, but it cannot be resolved

down a perfect fifth to BMaj because the B is absent in the C half/whole octatonic scale;



however, it is possible to imply this resolution because the upper structure of BMaj is

available in the octatonic scale. The concept of implied harmony will be discussed in

greater detail in the Additional Considerations section. The F 7 chord has two other

functional possibilities for resolution: CMaj and Gmin. The resolution of F 7 to CMaj is

shown in Figure 2-7(b) and is referred to as the tritone dominant. The second possible

resolution is up a semitone to Gmin, which will function as a deceptive cadence but the

fifth of the chord will have to be omitted, as D is not present in the C half/whole

octatonic scale and using the available fifth will produce a diminished triad, significantly

weakening the cadence. Figure 2-16 shows a possible voicing for this progression.

. Figure 2-16

Secondary leading-tone chords will function in a similar manner to the deceptive cadence

shown in Figure 2-16, as the only functional resolution available in the octatonic scale is

up a semitone to a minor triad with the fifth of the chord omitted to avoid creating a

diminished triad. Although neither of the choices provides as distinct a cadence as their

traditional tonal counterparts offer, each example resolves the dissonance of the given

vertical structure to a consonant tertiary sonority.

Following the guidelines of smooth voice leading, additional resolutions can be

justified by melodic movement. For example, Figure 2-10 shows an E 7 resolving down

a minor third to CMaj with all the notes of the chord either resolving through stepwise


motion or sustaining, with the exception of the bass. To further simulate a cadential

progression, the E 7 could have been voiced in first inversion to create bass movement

from G to C to provide a dominant to tonic bass motion. If this harmonic motion, or any

other harmonic motion that the composer desires, is used consistently to designate phrase

endings within a piece, it will assume a cadential function as the piece progresses. It is

also important to remember that all intervals which have a closer placement to the bass or

lowest tone, have a greater harmonic contribution than do intervals rising above them;

therefore, it is necessary to maintain as strict a sense of harmonic continuity in the lowest

sounding register as the octatonic scale permits.50

When creating an octatonic system, the best results will occur by establishing

relations between a series of vertical structures that will be used consistently throughout a

piece. Whether this process occurs prior to the beginning of a composition or allowing

the relations to establish themselves as the piece develops is irrelevant, as long as there is

an underlying system of voice leading in place. While this text has a predisposition for

the use of smooth voice leading to provide possible alternatives for the lack of traditional

functional harmonic resolutions, more angular approaches towards voice leading are

equally as plausible, as long as the composer is consistent with the methods used.



Additional Considerations:

Although the focus of this text is harmonic motion, it is not the only factor in

creating a tonal center. Several other musical aspects that have only been mentioned thus

far include (1) subset relations, (2) reharmonization and implied harmonic possibilities,

and (3) metric and rhythmic considerations. Each of these concepts is as potentially

useful as any of the other concepts presented in this chapter and will serve to further

solidify the intended tonal center.

(1) Subset Relations:

The Modal Relationships section showed that it is possible to imply a specific

vertical structure that is not entirely present in the octatonic scale by using the

characteristic notes of the related mode; however, it is also possible to imply additional

harmonies by using common subsets. Figure 2-17 displays a series of possible harmonic

implications using dyads over notes from the octatonic scale.51

Dyads:52 C C C C C C C C Dyads: D D E F G A B C

C7 9 Cmin CMaj C7 4 C5 C6 C7 C C Maj7 C Maj9 C mM7 C M7sus4 C M7 5 C M7 5 C Maj13 C Maj7 D 13 D 6 D 13 9 D dim7 D 13 D dim7 D 6 D 6 E13 5 EMaj7 5 E7 13 E9 13 Em7 6 E7 6sus4 E7Alt5 E7 13 F 7 11 F dim7 F m7 5 F 7 5 F 7 9 4 F dim F 7 5 F 7 4

Gdimsus4 G+sus4 G6sus4 GM7sus4 Gsus4 Gsus Gmin11 G7sus4A7 9 Adim Amin Adim7 Amin7 Amin Amin7 9 Amin

Implied Chords:

A min9 A sus4 A 9 5 A 9 13 A dim7 A Maj9 A 9 A 9

Figure 2-17

51 See Appendix D for Ron Miller’s Collated Order of All Constructed Modes. 52 The dyads are presented vertically, with the bass note from the octatonic scale and the implied chord label from the resulting trichord listed underneath each dyad in accordance to Ron Miller’s Collated Order of All Constructed Modes.


Notice that the chords listed in Figure 2-17 are derived from the diatonic church modes or

non-diatonic minor modes.53 As all the dyads are formed with the note C as a bass, they

can be transposed to any of the other three notes of the Cdim7 chord to produce

additional harmonic implications. Although there is a large amount of harmony that can

be implied through dyads, if overused or similar harmonies are transposed symmetrically,

it will ultimately detract from a clear tonal center. Additional listings of harmonic

implications, including trichords and tetrachords, are given in Appendix E.

(2) Reharmonization and Implied Harmonic Possibilities:

Reharmonizing traditional tonal progressions using those available in the

octatonic scale will provide a way of working in and out of an octatonic system. It is not

necessary for an entire composition to strictly adhere to the guidelines and restrictions of

an octatonic system, as most of the octatonic literature to date consists of octatonic-

diatonic interaction. Figure 2-7(a) illustrates a IVdim7 – Vdim addM7 - I progression,

which can serve as a replacement for the traditional I – IV – V – I progression. If a

IVmin7 5 was used instead of the fully diminished 7th, it would serve the same purpose,

except that with the exception of the root, the remaining chord tones would be invariant

with the original IVMaj7 chord found in the major scale. As for the V dim addM7 chord,

as long as it is presented in root position it will function as a dominant structure because

the dominant to tonic root progression will remain intact as intervals that are placed in the

lowest register have a greater harmonic contribution than intervals rising above them.

Jazz theory refers to this practice as “modal interchange,” since the root of the chord

remains the same, while the chord-scale relationship changes. 53 See Appendix B for a list of chord-scale relationships.


Implying harmony is another way to simulate functional harmony without

presenting a chord in its entirety; however, if done improperly, implied chords may lead

to the confusion of the intended chord and harmonic progression. Ulehla states, “Any

chord in which the root tone is omitted is less securely situated than if the root tone were

present and especially if located in the bass. Enharmonic changes may invite surprising

resolutions and thereby alter an analysis.”54 If an implied harmony does not use its

traditional tonal function, whether it is cadential resolution or simply a passing chord, the

implied harmony is likely to be seen with a different function and is more likely to be

mislabeled. Evaded progressions, or those involving root movements other than the

cadential dominant to tonic progression, are feasible but less convincing when using

implied roots.55 Figure 2-18 shows four different possible harmonizations of a 12 bar

blues. Figure 2-18(b) is an octatonic reharmonization of Figure 2-18(a) using implied

harmony on the subdominant chord in measures 2, 5, and 10 because the root F is not

present in the octatonic scale. Also, the dominant chord in measures 9 and 12 is

reharmonized as a diminished triad, since the typical V7 chord is not available in the

octatonic scale.

Figure 2-18(c) presents a more complex version of Figure 2-18(a), reharmonized

with a series of IImin7 - V7 progressions. This new progression is then reharmonized in

Figure 2-18(d) using only chords or implied harmony derived from the octatonic scale.

The only new implied harmony that is not present in Figure 2-18(b) is the D7 9 in

measures 9 and 12, serving as a secondary dominant chord to replace the Dmin7 in

Figure 2-18(c). However, the D7 9 is an example of an implied harmony that has several

54 Ulehla, Contemporary Harmony, 127. 55 Ibid., 129.


12 Bar Blues:

Figure 2-18(a)

12 Bar Blues Reharmonized with the Octatonic Scale:

Figure 2-18(b)

12 Bar Bebop Blues:

Figure 2-18(c)

12 Bar Bebop Blues Reharmonized with the Octatonic Scale:

Figure 2-18(d)


different ways of being analyzed. By omitting the root on the D7 9 chord, the four

remaining notes form an F dim7 chord, which can also be labeled as a secondary leading-

tone chord to Gdim addM7.56 Regardless of the label, the function of both chords is the

same; yet, if Figure 2-18(c) is played prior to Figure 2-18(d), the chord in question will

likely be heard, or at least understood, as a rootless D7 9 chord. Similar chord

substitutions are possible with any chord whose upper structure is found in the octatonic

scale, despite the lack of a root. In certain instances, simplifying a chord progression will

provide a clearer relation to the intended harmonies simply because there are fewer

harmonies to imply; this is seen as the basic progression of Figure 2-18(a) requires only

one reharmonization in Figure 2-18(b) compared to the more complex progression of

Figure 2-18(c) that requires four reharmonizations, seen in Figure 2-18(d). Since using

implied chords opens an analysis to several different interpretations, placing incomplete

chords on weak beats will help maintain their secondary function.

(3) Metric and Rhythmic Considerations:

Rhythmic and metric placement is an essential part of dictating harmonic

function, having an equally influential impact on a chord progression as voice leading or

harmonic relationships. Regardless of interval, rhythmic stress dictates the hearing of

harmonic versus non-harmonic tones.57 Ulehla makes the following statement about the

use of rhythm in non-diatonic settings,

Rhythmic stresses give attention to select groups of pitches. The pitch contour formed by the phrase produces some tones of prominence and others which serve in a supporting rhythmic capacity. Harmonies contain roots, without requiring a

56 In measure 12 of Figure 2-18(d), the Gdim addM7 chord is abbreviated as Go7 because any note a whole step above a chord tone in a diminished 7th chord is a possible tension. 57 Ulehla, Contemporary Harmony, 322.


diatonic tertiary order. Those are the new developments recognized today. The tonality today is not one that necessarily centers on one central tonic for the entire composition. It shifts tonal centers at will. All twelve tones may take on the equivalent role of the former tonic. But they will always assume a position that governs twelve notes, each of which may hold reign above the others at any time. Tones which start a phrase, climax the contour of a phrase, become part of a cadence; all contribute towards the movement within the phrase. They lead somewhere. The recipient of that motion has more tonal power than the insignificant motivic assortment of tones which are heard ‘en route’. Tonality is fleeting, but it is there. It is not in the form of one key dominating all, but a transient assortment which may include all twelve tones in rotation.58

Despite the importance of rhythmic and metric placement, these considerations are listed

as a subsection of the Additional Considerations section rather than in their own section

because it is possible to establish a vertical structure as a tonic through sheer repetition,

rather than through the use of harmonic progression or other traditional tonal concepts,

effectively circumventing the focus of this text; therefore, only a limited discussion of

rhythmic and metric considerations is presented.

Similar to traditional common-practice tonality, placing the primary harmonies of

an octatonic system—typically the tonic, dominant, and possibly subdominant—on the

strong beats of a measure will help separate the primary chords of a system from the

subordinate sonorities, providing a clear sense of forward motion. Also, in terms of

melodic line, the tones which occupy the highest or lowest position of each small “peak”

are more prominent than the connecting tones between them.59 Therefore, another

possibility to assert harmonic importance on a vertical structure is to place it at the peak

of a melodic line. A vertical structure can also gain importance by sustaining longer than

the other structures surrounding it. For a more thorough discussion of rhythm in non-

diatonic settings, see the Linear Roots section of Ulehla’s Contemporary Harmony.

58 Ulehla, Contemporary Harmony, 322. 59 Ibid., 304.

41

-- Chapter III -- Creating a Tonal Octatonic System

This chapter will be limited to the naturally occurring tetrachords found in the

octatonic scale. There are twelve prime forms and an additional six prime inversions for a

total of eighteen separate tetrachords, which are listed in Figure 3-1.

Tetrachords in the Octatonic Scale: Name: PCS: Vector: PI:

4-3 0134 212100 ---- 4-9 0167 200022 ----

4-10 0235 122010 ---- 4-12 0236 112101 0346 4-13 0136 112011 0356

4-Z15 0146 111111 0256 4-17 0347 102210 ---- 4-18 0147 102111 0367 4-26 0358 012120 ---- 4-27 0258 012111 0368 4-28 0369 004002 ----

4-Z29 0137 111111 0467

Figure 3-1 The concepts discussed in Chapter II will help limit the eighteen choices of tetrachords to

eight or fewer, so that there is no more than one tetrachord per scale degree of the

octatonic scale. This chapter will only be divided into five of the six sections found in

Chapter II:

• Modal Relationships • Harmonic Relationships • Vertical Spacing • Intervallic Content • Voice Leading

The Additional Considerations section has been omitted from this chapter because it dealt

with more compositional concerns rather than specific theoretical applications.

Chapter III – Creating a Tonal Octatonic System 42

This chapter will relate the concepts presented in Chapter II to the octatonic

system presented in Figure 3-2(a), derived from System 2 of Appendix F. The octatonic

system is presented alongside the major scale to provide a comparison between the two

systems for later discussions.

Octatonic Scale:

[Note: The chord built on the fourth scale degree of the octatonic scale has been omitted.]60

Figure 3-2(a)

Major Scale:

Figure 3-2(b)

Figure 3-2(c)

60 See the Harmonic Relationships - subsection (5) of this chapter for an explanation.

Octatonic Scale: Major Scale: Chord: Name: PCS: Vector: Chord: Name: PCS: Vector:

I7 4-27 0258 012111 IMaj7 4-20 0158 101220 IIo7 4-28 0369 004002 IImin7 4-26 0358 012120 IIIMin7 4-26 0358 012120 IIImin7 4-26 0358 012120 IVMin7 5 4-27 0258 012111 IVMaj7 4-20 0158 101220

V dim addM7 4-18 0147 102111 V7 4-27 0258 012111 VImin7 4-26 0358 012120 VImin7 4-26 0358 012120 VIIo7 4-28 0369 004002 VIIMin7 5 4-27 0258 012111


Modal Relationships: This system was created with the intention of forming relationships to the major

scale on the raised fourth and sixth scale degrees; several other modal inferences are

made on other scale degrees as well. A description of each relationship is given below.

• The tonic chord of the octatonic system is a dominant 7th structure, implying the

Mixolydian mode. The decision to use a dominant 7th structure as a tonic was derived

from a 12-bar blues progression—see Figure 2-18(a)—which also utilizes a dominant

7th as tonic. Also, the possibility of omitting the seventh chord degree will provide the

same tonic triad as the major scale.

• The lowered third chord of Figure 3-2(a) is completely foreign to the major scale

presented in Figure 3-2(b), yet the chord quality of the third chord in both systems is

the same, implying the Phrygian mode in the given context.

• With the exception of the root, the subdominant chord in both systems is invariant.

Using a rootless or implied harmony instead of the provided chord of the octatonic

system will maintain a closer relation to the major scale; however, the presence of the

root will provide the given chord with a higher degree of functionality.

• The dominant chord of the octatonic system has no direct correlation to a diatonic

church mode, as it was chosen for its intervallic qualities. Although it is also possible

to analyze the dominant chord as a first inversion G major triad over a G bass note,

the chord’s functional use within a composition will dictate the chord’s label.

• The VI chord in both systems is invariant.

• The remaining two chords of the octatonic system are diminished 7th structures,

functioning as subordinate sonorities due to their recursive qualities.


Harmonic Relationships:

(1) Creating a Dominant to Tonic Relationship:

The dominant chords in both systems have unique chordal tensions allowing for

chromatic resolution to the tonic chord along with the root movement of down a perfect

fifth. In order to create a clear distinction in sonority, the dominant chord of the octatonic

system contains a unique pcs from the other chords present in the system. Similarly, the

tonic chord of the octatonic system has a unique chordal structure, although it shares its

pcs with one of the other vertical structures in the system: the subdominant.

(2) Predominant Chords:

This octatonic system attempts to preserve the similarities present between the

tonic and subdominant chords that are found in the major scale. Within each system, the

tonic and subdominant chords are derived from the same pcs, despite the differing

chordal structures in the octatonic system. Although it is possible to have dominant 7th

structures built on both the tonic and subdominant chords of the octatonic system, the

Modal Relationships section of this chapter states that the subdominant chord in the

octatonic system is trying to create maximum invariance with the subdominant of the

major scale.

To compensate for the lack of an F in the octatonic scale, the root of the

subdominant chord has been replaced with an F to create a secondary leading-tone chord

to the dominant to provide a suitable predominant chord. Of the four chords that are built

on diminished triads in the given octatonic system, the subdominant chord is the only one

capable of functioning in a traditional tonal setting by resolving up a half step.


(3) Chord Quality:

The chord qualities of the given octatonic system have been chosen to provide a

significant amount of recursion without becoming overtly redundant, imitating the

structure of the major scale. The major scale has diatonic minor 7th chords built on the

second, third and sixth scale degrees; the octatonic system maintains a similar

consistency of tertiary structures on the lowered third and sixth scale degrees. Also, the

octatonic system contains diminished 7th structures built on the lowered second and

lowered seventh scale degrees.

(4) Root Progressions:

Since this system omits one of the possible chords in the octatonic system (see the

next subsection for a more thorough explanation), the possible root progressions will be

more restricted than normal. A prominent disadvantage to this system is not having the

perfect fifth root movement available between the third and sixth scale degrees, as the

chord built on the third scale degree has been omitted; however, through inversions it is

possible to utilize a bass progression of E to A, as seen in Figure 3-3.

Figure 3-3


(5) The Number of Chords in an Octatonic System:

To imitate the major scale as closely as possible, this particular system consists of

seven chords rather than eight. Omitting the fourth chord of the octatonic system allows

the intervallic content to more accurately reflect that of the major scale by reducing the

total unique tetrachords in the given system to four, compared to the three unique

tetrachords found in the major scale. An alternate version of this system includes an

additional chord built on the fourth scale degree of the octatonic scale, labeled

IIIminMaj7 5. 61 This additional chord would introduce the previously foreign tetrachord

4-17 (0347) into the given octatonic system. Although the composer may desire a variety

of unique vertical structures, it will ultimately detract from isolating the dominant and

tonic relationship by placing an equal amount of importance on what was previously a

subordinate sonority.

Vertical Spacing:

This system makes use of tertiary harmony because it will provide the most

immediately recognizable foundation to traditional tonality. The tertiary underpinnings

will also aid in emphasizing the overtone series, providing a clearer sense of intended

harmonic structure and function. There are several untraditional occurrences of the tritone

that should be handled with care, particularly on the dominant chord to avoid the

analyzation of the chord as G Maj/G. For the best results, the root of the chord should

always be placed in the lowest voice, and when that is not possible, the root of the chord

should be doubled.

61 See System 2 of Appendix F for this alternative octatonic system.


Intervallic Content:

In this particular octatonic system, the tonic, 4-27 (0258), and dominant, 4-18

(0147), chords are in the relationship R1 to form a stronger association with each other, as

seen in Figure 3-4.

4-27 [ 0 1 2 1 1 1 ] 4-18 [ 1 0 2 1 1 1 ]

Figure 3-4

Although this relationship is not present in the major scale, it is an attempt to compensate

for the lack of leading-tone resolution and other relationships that are incapable of being

reproduced in the octatonic scale without introducing foreign notes to the scale. Instances

of pcs transposition occur between the lowered third and sixth chords, the lowered second

and lowered seventh chords, and also the tonic and subdominant chords.

Voice Leading:

The half/whole ordering of the octatonic scale does not contain the leading-tone

found in the major scale, thus necessitating the need to redefine the voice leading rules

for the dominant to tonic resolution. Within the octatonic system, every note in the tonic

chord is approachable by half step, but two of the notes (G and B ) are already embodied

in the dominant chord, leaving the D to resolve down to C and the F to resolve either up

to G or down to E, as seen in Figure 3-5.


Figure 3-5

Although there is a tritone present in the dominant chord, it exists between the root and

lowered fifth chord degrees rather than between the third and lowered seventh chord

degrees. For the resolution of this tritone, rather than having each note expand or contract

by half step—as is common-practice for traditional Western harmony—the lowered fifth

chord degree (D ) will resolve down a half step while the root (G) can either resolve

down a perfect fifth or sustain. Another possible resolution is to anticipate the D

resolution to C, creating another tritone between C and F and allowing the F to resolve

to G, seen in Figure 3-6.

Figure 3-6

Another possibility to consider is including both altered fifths (D and D ) in the

dominant chord, producing a pentachord. This will introduce additional voice leading

possibilities, as the D would resolve up by half step to the E of the tonic chord, as seen in

Figure 3-7, creating smoother resolution than the doubled F resolving to G and E in

Figure 3-6. Adding an additional note to the vertical structure also serves the purpose of

differentiating the sonority from the other vertical structures present in the system.


Figure 3-7

An analogous process may be applied to the tonic chord by reducing the dominant 7th

structure to a major triad, providing the only traditional common-practice period

consonance of the system and establishing it as the tonic of the scale. Similar procedures

may be applied to other chords in the system, to create larger, more complex chord

progressions.

50

-- Conclusion --

Through the examination of harmonic restrictions and relationships on previously

existing vertical structures and pitch class sets found in the octatonic scale, this text

demonstrated that it is possible to create a series of harmonic progressions that will

establish a specific note as the tonic within the octatonic scale. Chapter I presented a brief

overview of the octatonic scale as used in both Jazz and Twentieth-century Classical

music. The intent of this overview was to establish a historical foundation of the

octatonic scale to be contrasted with the material presented in Chapters II and III. Chapter

II introduced theoretical applications for establishing a tonal center within the octatonic

scale, with Chapter III applying the concepts of Chapter II to a given octatonic system.

The main function of Chapter III is to provide the reader with an example of how theory

can become music. Breaking up chapters II and III into the six sub-sections—listed

below—aided in the organization and cohesion of the material presented in this text.

• Modal Relationships discussed common subsets of the diatonic collection, non-diatonic minor scales, and the octatonic collection.

• Harmonic Relationships discussed creating the (1) dominant to tonic relationship, (2) predominant chords, (3) chord qualities, and (4) the total number of chords in an octatonic system.

• Vertical spacing discussed the intervallic spacing and ordering of notes within a vertical structure to avoid harmonic ambiguity.

• Intervallic Content discussed post-tonal relationships between vertical structures.

• Voice Leading discussed the need for establishing new rules of harmonic progression.

• Additional Considerations discussed (1) additional subset relations, (2) chord substitutions, and (3) metric considerations.

Conclusion 51

Additional research possibilities include creating additional octatonic systems,

creating a more thorough subset relations chart (Appendix E), applying the concepts of

this text to other symmetrical scales (whole-tone, augmented/hexatonic, etc.), and

utilizing the presented guidelines in composition. Although it is not directly pertinent to

this text, establishing a standard for chord symbols and chord-scale relationships would

aid all Jazz literature, and would have facilitated the creation of the musical examples

used in this text.

Had more time been allotted for a composition to be completed utilizing principles

presented in this text, it would have aided in presenting the material of Chapter III by

drawing upon a single source rather than fragments of a composition that have not yet

been completely worked out. Additional examples of musical literature would have

improved the text as a whole; however, studying the footnoted references will provide a

wealth of musical examples and detailed analyses that will aid in understanding the

material presented, but are beyond the scope of this text. It is important to keep in mind

that this text would not have been possible without combining aspects of common-

practice period tonality, Jazz theory, and Set theory. As the distinctions between genres

of music continue to blur, it allows for new musical styles to emerge from the fusion of

previous styles; however, this fusion does not mean that one system must dominate over

the other. While this text does not explore the full range of possibilities of the octatonic

scale, I hope that it does convey at least one critical concept: octatonicism does not

negate diatonicism.

52

-- Bibliography --

Babbitt, Milton. “The String Quartets of Bartok.” Musical Quarterly 35, no. 3 (July 1949). Baker, David. How To Play Bebop. Bloomington, IN: Alfred Publishing Company, 1987. Coker, Jerry. Patterns for Jazz. Miami, FL: Studio Publications Recordings, 1970. Forte, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973. Hall, Jim. Exploring Jazz Guitar. Milwaukee, WI: Hal Leonard Publishing Corp., 1990. Levine, Mark. The Jazz Piano Book. Petaluma, CA: Sher Music Company, 1989. Levine, Mark. The Jazz Theory Book. Petaluma, CA: Sher Music Company, 1995. Liebman, David. A Chromatic Approach to Jazz Harmony and Melody. Rottenburg, Germany: Advance Music, 1991. Messiaen, Oliver. The Technique of My Musical Language. Paris: A. Leduc, 1966. Miller, Ron. Modal Jazz Composition & Harmony Volume 1. Rottenburg, Germany: Advance Music, 1992. Slonimsky, Nicolas. Thesaurus of Scales and Melodic Patterns. NYC, NY: Schirmer Books, 1975. Straus, Joseph. Introduction to Post-Tonal Theory. 3rd ed. NJ: Prentice Hall, 2005. Tymoczko, Dmitri. “Octatonicism Reconsidered Again.” Music Theory Spectrum 25 (2003). Ulehla, Ludmila. Contemporary Harmony: Romanticism through the Twelve-Tone Row. NY: The Free Press, 1966. Van den Toorn, Pieter C. “Colloquy: Stravinsky and the Octatonic, The Sounds of Stravinsky.” Music Theory Spectrum 25 (2003). Van den Toorn, Pieter C. Music of Igor Stravinsky. New Haven: Yale University Press, 1983. Van Den Toorn, Pieter C. Stravinsky and The Rite of Spring. Berkley: University of California Press, 1987.

53

-- Appendix A -- Chord Lexicography

All chord labels adhere to the following rules:

• Roman numerals do not indicate major or minor triadic quality. Roman numerals only

indicate the root’s relation to the tonic chord of the parent major scale.

• A major triad is indicated by “Maj” or just by the chord’s root. (i.e., CMaj; C)

• A minor triad is indicated by “min”. (i.e., Cmin)

• A dominant 7th chord is notated by “7”. (i.e., C7)

• If both the third and seventh chord degrees are minor, it is notated “min7” or abbreviated as “m7”. (i.e., Cmin7; Cm7)

• If both the third and seventh chord degrees are major, it is notated “Maj7” or abbreviated as “M7”. (i.e., CMaj7; CM7)

• If the triad of a chord is minor and a major seventh interval is present, it is notated “minMaj7” or abbreviated “mM7”. (i.e., CminMaj7; CmM7)

• If a triad has the sixth chord degree present, while omitting the seventh of the chord, the label “6” replaces the position of the “7”. (i.e., C6 is a C major triad with an A above the root; Cmin6 is a C min triad with an A above the root; Cmin 6 is a C minor triad with an A above the root.)

• If higher scale degrees of a chord are present, they replace the position of the “7” in the chord label. (i.e., C13 indicates a C7 chord with an additional 13th; CMaj13 indicates a CMaj7 with an additional 13th)

• Additional alterations to a chord are listed after the first chord tension. (i.e., C7 9)

• Alterations to dominant 7th chords are listed as compound intervals, altering the 9th, 11th, and 13th chord degrees. Alterations to the 5th indicate a change in triad quality.

• The term “Alt” indicates that the 5th and 9th chord degrees are raised and lowered.

• The term “sus” indicates a tertiary structure that substitutes the third of the chord for the chord degree following the “sus”. (i.e., Csus2; C7sus4)

• The term “add” indicates additional tones are added to the preceding chord, without requiring any further alteration to the chord. (i.e., Cadd9 indicates the notes C, E, G, and D)

54

-- Appendix B -- Chord-Scale Relationships

Modes of the Major Scale:

Mode: Altered Scale Degrees: Tonic 7th Chord: Ionian: N/A Maj7

Dorian: 3, 7 min7

Phrygian: 2, 3, 6, 7 min7 9

Lydian: 4 Maj7 4

Mixolydian: 7 7

Aeolian: 3, 6, 7 min7 6

Locrian: 2, 3, 5, 6, 7 min7 5

Appendix B: Chord-Scale Relationships 55

Modes of the Harmonic Major Scale:

Mode: Altered Scale Degrees: Tonic 7th Chord:

Ionian 6: 6 Maj7, Maj7 6

Dorian 5: 3, 5, 7 min7 5

Phrygian 4: 2, 3 4, 6, 7 7Alt

Lydian 3: 3, 4 minMaj7

Mixolydian 2: 2, 7 7 9

Lydian 5 2: 2, 4, 5 Maj7 5, Aug

Locrian 7: 2, 3, 5, 6, 7 dim7


Modes of the Harmonic Minor Scale:

Mode: Altered Scale Degrees: Tonic 7th Chord: Aeloian 7: 3, 6 m7, mM7

Locrian 6: 2, 3, 5, 7 min7 5

Ionian 5: 5 Maj7 5, Aug

Dorian 4: 3, 4, 7 min7 4

Mixolydian 2 6: 2, 6, 7 7 9, 7 13

Lydian 2: 2, 4 Maj7 9

Altered 7: 2, 3, 4, 5, 6, 7 7Alt


Modes of the Ascending Melodic Minor Scale:


Dorian 7: 3 m7, mM7

Phrygian 6: 2, 3, 7 sus 9

Lydian-augmented: 4, 5 Maj7 5

Lydian-dominant: 4, 7 7 11

Mixolydian 6: 6, 7 mM7 in 2nd inversion

Aeolian 5: 3, 5, 6, 7 min7 5

Altered: 2, 3, 4, 5, 6, 7 7Alt


Modes of the Ascending Melodic Minor 5 Scale:


Melodic Minor 5: 3, 5 minMaj7

Phrygian 6 4: 2, 3, 4, 7 min7 5, dim7

Lydian 3 5: 3, 4, 5 Maj7, aug

Mixolydian 2 4: 2, 4, 7 7 11

Altered 6 7: 2, 3, 4, 5, 6, 7 dim7, 7Alt

Aeolian 5 7: 3, 5, 6 minMaj7

Altered 6: 2, 3, 4, 5, 7 7Alt


Diatonic Church Modes Priority Table:62 1 2 3 4 5 6 Lydian 4 7 3 6 2 5 Ionian (1) 7 4 3 6 2 5 Ionian (2) 7 3 2 6 5 (no 4) Mixolydian (1) 7 4 3 6 2 5 Mixolydian (2) 7 3 2 6 5 (no 4) Dorian 6 3 7 2 5 4 Aeolian 6 2 5 3 7 4 Phrygian 2 5 4 7 3 6 Locrian 5 2 7 6 3 4

Harmonic Minor Modes Priority Table: 63 1 2 3 4 5 6 Aeloian 7 6 7 2 3 5 4 Locrian 6 5 6 2 7 3 4 Ionian 5 4 5 7 3 2 6 Dorian 4 6 4 3 2 7 5 Mixolydian 2 6 2 3 5 7 6 4 Lydian 2 4 2 7 3 6 2 Altered 7 4 7 2 5 6 3

Melodic Minor Modes Priority Table: 64 1 2 3 4 5 6 Lydian-augmented 5 7 3 4 6 2 Lydian-dominant 4 7 3 6 2 5 Mixolydian 6 6 7 3 2 5 4 Dorian 7 7 3 6 2 5 4 Aeolian 5 5 3 7 6 2 4 Phrygian 6 6 2 4 7 3 5 Altered 4 7 6 3 5 2

[Note: At least two of the tones must be used to get sufficient modal definition in the non-diatonic minor modes.]

62 Miller, Modal Jazz Composition, 20. 63 Ibid., 90. 64 Ibid., 33.

60

-- Appendix C -- Modal Tetrachords Found in the Octatonic Scale

The following tetrachords are used in the construction the non-diatonic minor

scales and are also present in the octatonic scale. The tetrachords are taken from page 130 of Ron Miller’s book, Modal Jazz Composition & Harmony Volume 1. Each tetrachord is labeled with its name and intervallic construction.

Spanish (121): Dorian (212):

Hungarian Minor (213): Hungarian Major (312):

Hungarian Pentatonic (231): Hungarian Phrygian (123):

Hungarian Spanish (132): Blues (321):

61

-- Appendix D -- Ron Miller’s Collated Order of All Constructed Modes

Using the major scale as a reference, by considering the raising of a scale degree as a brightening and the lowering of a scale degree as a darkening, the resulting order of brightest to darkest is:

1. Lydian 5 3 2. Lydian 5 3. Lydian 2 4. Lydian 5. Lydian 3 6. Ionian 5 7. Ionian 8. Ionian 6 9. Mixolydian 2 4 10. Mixolydian 4 11. Mixolydian 6 12. Mixolydian 13. Mixolydian 2 14. Dorian 7 5 15. Dorian 7 16. Dorian 7 5 17. Dorian 4 18. Dorian 19. Aeolian 7 20. Aeolian 7 5 21. Aeolian 22. Aeolian 5 23. Phrygian 7 5 24. Phrygian 6 4 25. Phrygian 6 26. Phrygian 3 27. Phrygian 28. Locrian 6 29. Locrian 6 30. Locrian 7 31. Locrian 4 32. Locrian 33. Altered 6 34. Altered 7 35. Altered 6 7

62

-- Appendix E -- Additional Subset Relations

Trichords: Cdim Cmin Cmin6 Cmin7 C7Alt9 C7 9 C7+4 CMaj

C C C C C C C C E E E E E E E E G G A B D E G G

Trichords Continued:

C6 C7 C7 9 C7 9 C dim C min7 6 C dim7 C minMaj7 C C C C C C C C E E E E E E E E A B D D G A B C Trichords Continued: C min9 C min11 C C E E D F

Tetrachords: Cmin7 5 Cmin7 C7 5 C7 C7Alt9 5 C7Alt9 C7 9 5 C7 9

C C C C C C C C E E E E E E E E G G G G G G G G B B B B D D D D

Tetrachords Continued:

C7 9 5 C7 9 C7 9 4 C7 4 Cdim7 Cmin6 Amin6 C6 C C C C C C C C E E E E E E E E

G G G G G G G G D D F F A A A A

Tetrachords Continued: C dim7 C dim7 5 C mM7 5 C mM7 5 C min9 5 C m9 C m11 5 C m11 5

C C C C C C C C E E E E E E E E G A G A G A G A B B C C D D F F

63

-- Appendix F -- Additional Octatonic Systems

Note the following distinctions of the given octatonic systems:

• System 3 uses implied harmony on the D9 and F9 • System 4 uses quartal harmony • System 5 uses mixed voicings

1

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3

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