Facts and Counter Facts

8/12/2019 Facts and Counter Facts

1/28

Facts and CounterfactsMathematical Contributions to Music-theoretical Knowledge

Thomas Noll

Technische Universitat Berlin

Escola Superior de Musica de Catalunya, Barcelona

[email protected]

Abstract

The present article argues in favor of a discipline of MathematicalMusic Theory (see [18], [20] for earlier attempts). By reviewingand re-interpretating known results, we draw further conclusions andformulate working hypotheses. Especially, we recapitulate a known

fact about diatonic triads and seventh chords in connection withan analogous fact about pentatonic and diatonic scales in the 12-tone system. Putting these facts together on the basis of a canonicaltheory of scale generation we discuss some open questions concerningtraditional music-theoretical agreements and disagreements.

Triads and Seventh Chords understood as third chains in the 7-tone diatonic system have the property that diatonic steps of theupper or lower voice correspond to third transpositions of the en-tire triads. The pentatonic and the diatonic scales as fifth chains inthe 12-tone system have the similar property that certain semitonesteps correspond to fifth transpositions of the entire scales. The com-monality between both situations can be more generally describedas a P1-cycle and occurs systematically in the generation of well-formed scales. On the basis of this theory we attempt to combineseveral music-theoretical concepts in a unified picture. Besides thatwe describe pseudo-diatonicsystems as instances of a model class ofthe proper diatonic and discuss the counter-factual construction ofa faithful recomposition.

Acknowledgments: Im grateful to Bernd Mahr for many inspiring discus-

sions about the epistemology of mathematical modeling, to David Clampitt and

Norman Carey for an inspiring correspondence about the interpretation of their

theory as well as to my colleagues in Barcelona for their feedback.

335


2/28

336 Thomas Noll

1 Abstract Music Theory

Many traditional music-theoretical concepts, such as diatonic scale, thirdchain, chromatic alteration are seemingly anchored in the realm of mu-sical notation. Graphemic elements, such as staff, relative note head po-sitions, accidentals etc. help to communicate these concepts among mu-

sicians and theorists. The double articulation of music in terms of scoresand performances is mirrored in the common practice of musical analy-sis. The study of musical scores on the basis of the above-mentioned typeof concepts constitutes the core of music-analytical and music-theoreticalknowledge. Even pure theoretic concepts like Rameaus fundamental basscannot be understood without the concepts of diatonic third chains andthe diatonic fifth progressions. The methodological difficulties to grasp themusic-theoretical meanings of these concepts beyond their graphemic an-chors caused serious doubts about their scientific value within systematicmusicology. We may distinguish two main attitudes or strategies to facethis problem. From a pragmatic point of view one may simply do musictheory without considering as a hard science. Many music theorists con-

ceive their work as hermeneutic, philological or pedagogical contributionsto a general discourse about music. Accordingly does the enrichment ofmusic-theoretical knowledge not need to be measured in evident facts, butrather in terms of plausible modes of access to music, in terms of philologicalconnections (i.e. knowledgeabout knowledge, with responsibility only forthe former one). The second strategy aims at grasping musical facts alongthe ontological stratification of musical reality and to formulate them onacoustical, psycho-acoustical, cognitive, phenomenological, social and otherlevels of description. These attempts rely on the modes of access that othersciences -such as physics, psychology, philosophy, sociology, etc. have de-veloped. The coexistence of the complementary strategies has a remarkablehistory on its own and includes numerous expressions of skepticism on bothsides, inspirations as well as attempts to build bridges between them.

It would be misleading, however, to consider the ontologically sensitive ap-proaches as scientific ones and to oppose them to a mere pragmaticallyoriented unscientific music theory. Instead we propose the term abstractmusic theory in order to refer to attempts which aim at grasping music-theoretical contents within consistent frameworks of abstract musical con-cepts. In this article we discuss particular approaches to abstract musictheory, where mathematics plays a major role in the reformulation of tra-ditional concepts as well as in the investigation of dependencies between


3/28

Facts and Counterfacts 337

them. Many fruitful contributions from the last three decades indicate, thata successful mathematisation of music-theoretical knowledge is by no meansrestricted to the domain of acoustics or to the quantitative investigation ofexperimental data (such as in computational psychology). The strengthen-ing of abstract music theory by means of mathematical conceptualizationcontributes in a threefold way to an enrichment of the music-theoreticalknowledge.

1. The elaboration of a consistent conceptual network and its practicalusage in musical analyses adds new insights to the traditional knowl-edge. In the context of this article we show this in the case of centralideas of Rameaus music theory. We argue that Rameaus originalplan to explain various music-theoretical facts as a consequence ofchoosing the fifth and third as generating intervals can indeed berealized to a remarkable degree.

2. Conceptual bridges between abstract music theory and acoustical,

psychological, and other levels of description can be based on inner-mathematical translations between mathematical models for the cor-responding structures and processes. TheRameau Diagram(see Sec-tion 5) can be seen as a refinement of Fred Lerdals model oftonalpitch space, which was developed as a bridge between Music Theoryand Cognitive Psychology. Furthermore we formulate some ideas,how abstract music theory can be a source for investigations into amathematical theory of the phenomenology of the mind.

3. Mathematical formulations of musical facts can be relativized by con-structing musical counterfacts, i.e. by constructing alternative modelswhich are not exemplified by any music. These counterfacts either in-

dicate an insufficiency of the mathematical characterizations and/orthey may inspire musicians to test them out in practice, and to even-tually turn them into musical facts. Our discussion is inspired byEytan Agmons counterfactual recompositions of Schumanns AmKamin from Kinderszenen (c.f. [2]) which intend to test analyticalassertions by changing the music in such a way, such that the asser-tion still holds. It is also inspired by Gerard Balzanos [3] suggestionto experiment with generalized diatonic systems, which share certainproperties with the familar one.


4/28

338 Thomas Noll

2 Smooth Transpositions

Several authors studied a music-theoretically interesting type of double-relation between tone sets. Phenomenologically one may characterize thesedouble-relationss as a solidarity between a transformation of the entire ob-ject on the one hand and a minimal change of a detail an the other.1 In

our discussion we depart from two prominent examples, where the trans-formation in question is a transposition and the minimal change of a detailis a scale step, namely pentatonic and diatonic scales within the chromatic12-tone scale, and triads and seventh chords within the diatonic scale.

Figure 1 displays the situation in the case of the diatonic within the 12-tonescale. The fifth-transposition of the diatonic scale is in solidarity with asingle chromatic alteration. This fact is known to every musician. However,instead of taking it for granted Gerard Balzano [3] made clear that this isa very special property, which is in accordance to the generation of thediatonic by fifths.2

Figure 1: Balzano Solidarity between fifth-transposition and minimal voice-leading for diatonic scales within the 12-tone system.

Similarly, it is wellknown that a linear sequence as displayed in Figure2 is in solidarity with a fifth fall F B E A D G C F. The reason for this possibility has been systematically discussedby Eytan Agmon [1] and is illustrated in the rightmost circle of Figure 3: In

1c.f. Eytan Agmon [1], Gerard Balzano [3], Norman Carey and David Clampitt [5],[4], [8], Richard Cohn [10], John Douthett, David Lewin, John Rahn [25] and others.

2or equivalently, in accordance to the special interval content (2,5,4,3,6,1) of this scaleas stated by John Rahn [25].


5/28


Figure 2: Solidarity between linear voiceleading in two voices and fifth-fallin the fundamental bass

diatonic seventh chords we find a solidarity between downwards steps of onevoice in solidarity with a diatonic transposition of the entire chord down athird. Consequently do two steps such as in Figure 2 correspond to diatonicfifth transpositions, and yield the fifth fall sequence in the fundamentalbass. Similarly have triads this solidarity property between an upwardsmovement of one voice and a diatonic transposition down a third.

Figure 3: Right: Agmon Solidarity between third-transposition and mini-mal voiceleading for triads and seventh chords within the third-generateddiatonic 7-tone system. Left: Similar smooth transpositions occur for thepentatonic scale within the fifth-generated 12-tone and 7-tone systems.

Remark 1 It is puzzling that in many cases of smooth transpositions mu-sic theorists tend to invest considerable effort in order to argue in favorof minimal voiceleadings and to explicitly dismiss the consideration of atransposition (e.g. in terms of fundamental bass progression). We suggestto draw two different conclusions on two levels of investigation. On the onehand we agree with Eytan Agmon [2] that the very possibility of such pref-

erences is interesting as such and does indeed demand more attention inmusical analysis. On the other hand there seems to be a strong phenomeno-logical reason not to ignore the phenomenon of a passing chord. In otherwords, the double phenomenon of smooth transposition is mirrored by a dou-ble phenomenon of ignoration in music theory, namely fundamental-bass-ignorance vs. passing-chord-ignorance. On a second level of descriptionit seems therefore reasonable to interpret this problem as a genuine musicalphenomenon. In the context of the theory of canonical scaling (see Section4) we show a strong duality between melody and harmony in terms of a


6/28

340 Thomas Noll

literal parallelism in the processes of scale-step generation and harmonic-generator generation. But the non-commutativity of the canonical groupsuggests to consider the conjugation of both processes as an extra transfor-mative effort and may thus explain the uncertainty in the apperception ofmelody and fundamental bass.

The general case of a smooth transposition in a finite cyclic tone system iscovered by the following definition and proposition:

Definition 1 On the subsets of Zn we consider the binary relation ofsmooth transposition given by

X Y iff(k Znandx Xsuch thatY =X+ k= X\{x} {x 1}).

Its equivalence closure is called P1-relation. Those P1-equivalence classeswith more than one element are called P1-Cycles.

P1-cycles are defined analogously to Richard CohnsP-cycles, which includeinversions as transformations (see [9], [10], [8]).

Proposition 1 A subset X Zn has a smooth transposition iff it is ak-chainX={t k modn| t = 0, . . . , m 1}such thatk m= 1 modn. Inthis case it generates aP1-cycle of lengthn.

Both of our examples have an additional property which is the possibilityof a double employmentof smooth transpositions. The third transpositionof a triad is smooth in a regular 4-tone cycle Z4 as well as in a 7-tone cycleZ7. Both can be studied together if one restricts to the two triads within an

embedded 4-cycle (i.e. a seventh chord) within the diatonic. Similarly hasthe pentatonic two double employments of smooth transpositions within aa 7-tone cycle which is a diatonic embedded into a 12-tone system.

Sets with smooth transpositions have further properties which qualify themaswellformed scales. It is therefore useful to study them in the full contextof Norman Carey and David Clampitts theory of well-formed scales.3

3For a systematic and detailed introduction see [4].


7/28


3 Wellformed Scales

Suppose we are given a finite subset X = {x0,...,xN1} R\Z of realnumbers modulo 1. On R\Z we inherit the total order from the semi-open interval [0, 1), but all arithmetic operations are meant modulo 1. Thesequence

(x0,...,xN1)is said to be in scale order for X, if x0 < x1 < . . . < xN1. The set(X) := {xixi1| i= 1,...,N 1} {1 xN1} of differences betweenadjacent elements is called the set of step intervals ofX. Two cases are ofparticular importance:

One step interval only: (X) ={ 1N

}. In this case X= x0+ 1NZN,

i.e. X is completelyregular.

Two step intervals only: (X) = {s, t}. In this case X is said tohaveMyhill Property.

For a moment we consider the case x0 = 0. An element y X is said togenerateXstarting fromx0= 0 ifX={0, y, 2y, ..., (N 1)y}. In this casethe permuted sequence

(0, y, 2y, ..., (N 1)y) = (x(0), x(1),...,x(N1))

(for some permutation : ZN ZN) is said to represent the generationorder of X with respect to y. Generation order and scale order can becompared by counting the number of elements ofX within each segment[iy, (i+ 1)y) spanned by two successive multiples of the generator. Thiscan also be done for the gap segment [(N1)y, 0). A generated scaleX(starting from from x0 = 0) is called a wellformed scale, if the numberof elements in each such segment including the gap segment is the

same. This means that the scale X has to be generated by an elementy = xk X in such a way that the permutation : ZN ZN is amultiplication modulo N, i.e. (k) = m k mod N . More generally, aset Xshall be considered to be a well-formed scale, if it has an elementx0 X, such that the transposition X x0=Xis wellformed in the aboverestricted sense. Norman Carey (c.f.[4]) shows that wellformedness of ascaleXis equivalent to either being regular or having Myhill property. Inthe former case the scale is also called degenerate and in the latter non-degenerate. A further result of Carey and Clampitt ([5]) is the connection


8/28


9/28


Figure 4: Wellformed scales with irrational generator log2(3/2) (outer cir-cles) and rational approximations 23 ,

35 ,

47 ,

712 ,

1017 ,

1729 (inner circles). Corre-

sponding tone are connected. The meaning of the 22-matrices is explainedtowards the end of Section 4.


10/28

344 Thomas Noll

The step sizes of the 2-tone-scale are fifth and fourth. In the the 3-tone-scalewe have major second and fourth, in the 5-tone scale we have major secondand minor third, etc. A dual family to these scales is a family of minimalnon-degenerate scales which are generated by the rational semiconvergents.These are exactly the sets with smooth transpositions and finiteP1-cycles,such as studied in the previous section. In Figure 4 we compare bothfamilies of scales. The duality will be analyzed in Section 4.

4 Canonical Scaling

Departing from the correspondence between wellformed scales and the con-tinued fraction developments of their generators we study a binary treewhich embodies the whole generative structure of continued fractions. Fig-ure 5 displays theStern-Brocot-Treein the traditional manner.5 The nodesof the tree are ratios k

n of coprime positive integers k andn.

01

10

11

12

21

13

23

32

31

14

25

35

34

43

53

52

41

15

27

38

37

47

58

57

45

54

75

85

74

73

83

72

51

RL

RR LL

RRRR LLLL

RRRRRRRR LLLLLLLL

Figure 5: Stern-Brocot Tree. Each path on this binary tree yields a wordin letters R and L, in accordance with the labels on the edges.

The uppermost ratios 01

and 10

play an extra role asinfinitesimal generatosas we will see. We use them together with the proper node 11 in order toproduce themediants 1

2 = 0+1

1+1and 2

1 = 1+1

1+0as new nodes in a new horizon-

tal row, 12

being horizontally adjusted between 01

and 11

and analogously, 21

5It was independently invented by Moriz Stern (1858) and Achille Brocot (1860)


11/28


being adjusted between 11

and 10

, respectively. Their connecting edges withthe root 1

1are labeled L and R. Recursively, given an already produced

ith row (such as 12 , 21 fori = 2), we proceed as follows, in order to produce

an (i + 1)th row with two successors for each node in theith row: To eachpair k1

n1and k2

n2of adjacent nodes (from left to right) in the ith row there

is a node lm

somewhere higher in the tree which is horizontally adjusted

in the middle between

k1

n1 and

k2

n2 . Construct the right successor node ofk1n1

as the mediant k1+ln1+m

and adjust it in the (i+ 1)th row midways be-

tween k1n1

and lm

. The connecting edge with k1n1

is labeled R. Likewise,

construct the left successor node of k2n2

as the mediant l+k2m+n2

and adjust it

between lm

and k2n2

. The connecting edge with k2n2

is labeled L. In order to

construct the missing outermost nodes construct the mediants 1i+1

= 0+11+i

and i+11

= i+11+0

and adjust them horizontally between their predecessors

and the infinitesimal generators, i.e between 01

and 1i

as well as betweeni1 and

10 respectively. The nodes of the tree are in 1-1-correspondence to

the positive rational numbers and the finite (and infinite) pathways fromthe root downwards along the edges of this tree correspond to the semi-convergents of finite (and infinite) continued fractions. Positive rationalnumbers x thus correspond to finite words (x) encoding the pathwaysfrom the root 11 to the nodes x. For example,

47 corresponds to the word

(47

) =LRLL {R, L}.

Definition 2 A positive rational numberx = 1, 12 ,21 shall be called chro-

matic, if the two rightmost letters of its word (x) are different, i.e. if(x)ends either withLRor withRL. In the first casex is called a Careynumber and in the latter case aClampitt number.

A numberx is chromatic if the predecessor of its predecessor on the Stern-Brocot tree is a convergent for it. The generating third 27 of the diatonicwith word( 2

7) = LLLRand the generating fifth 7

12of the 12-tone system

with word(

7

12) = LRLLRare both Carey numbers. The generating fifth35

of the regular pentatonic with word( 35

) =LRL is a Clampitt number.The generating fifth 4

7 of the diatonic is not chromatic. A number x is

chromatic if and only if the smaller of the two wellformed scales generatedby it allows double employment of smooth transpositions.

Figure 6 displays the same tree in another way, such that nodes kn

corre-

spond to integral 2-vectors v=

kn

Z2 R2 with coprime coordinates


12/28

346 Thomas Noll

1 1

1 2

1 3

1 4

1 5

1 6

2 1

2 3

2 5

2 7

2 9

3 1

3 4

3 7

3 10

4 1

4 5

4 9

5 1

5 6

6 1

3 2

3 5

3 8

3 11

5 3

5 8

5 13

7 4

7 11

9 5

4 3

4 7

4 11

5 2

5 7

5 12

7 5

7 12

8 3

8 11

10 7

11 45 4

5 9

7 2

7 9

9 7

11 3

6 5

7 3

7 10

8 5

8 13

9 2

12 5

13 8

9 4

10 3

11 8

12 711 7

13 5

Figure 6: Stern-Brocot-Tree (5 generation steps)

k and n. In this representation we may give a geometric meaning to thetree generation. The free monoid{R, L} of (finite) words in the two let-ters R and L is isomorhphic to the monoid SB = R,L SL(2,Z)

generated from

L=

1 01 1

and R=

1 10 1

.

SL2(Z) denotes theSpecial Linear Groupof 22-matrices with determinant1 and integer coefficients. The isomorphism: : {R, L} R,Lis simplygiven by substituting each letter R with (R) = Rand each letter Lwith(L) = L and by multiplying the obtained matrices in the correct order. If


13/28


we restrict the group action

: SL(2,Z) Z2 Z2 with (, v) :=(v) := v

to the monoid S Bwe see that the nodes of the Stern-Brocot Tree become

SB e=

11

| SB

, where e =

11

and the two successors ofv = e (i.e. the end nodes of the two edges,departing from a common node v arevL = L e and vR = R e.

We go one step further in order to make the geometric meaning moreexplicit. Each positive rational number x Q+ corresponds to a uniqueelement x SB, such that we may indeed identify the two. This suggeststo study the entire picture in association with the group action

: SL(2,Z) M2(Z) M2(Z) with (, ) := () := .

Here,M2(Z) denotes the full algebra of all integral 2 2-matrices. In otherwords, we now realize our tree in a four-dimensional space. In analogyto vL and vR above we have L = L I and vR = R I, where

I=

1 00 1

denotes the identity matrix. By identifying 22 matrices

with pairs (v, w) Z2 Z2 we may express the transition from the Stern-Brocot-Tree in M2(Z) to the one in Z

2 by saying that the map plus :M2(Z)= Z2 Z2 Z2 with plus(v, w) =v +w is an equivariance of thetwo group actions and . So, no matter at which node we pass from thetree inM2(Z) to the tree in Z2. The pathways are the same. A comparisonof the Figures 6 and 7 shows this correspondence.

The two successor functions on matrices, i.e. multiplication of a matrix with L or R from the right can be simply understood as adding theright column to the left (v, w)L = (v, w) L = (v+ w, w) and (v, w)R =(v, w) R = (v, w+ v) respectively.

This has the following geometric interpretation: The two exotic ratios 01

and 10 in Figure 5 can be associated with the matrices

l=

0 01 0

and r=

0 10 0

,

being infinitesimal generators for R = exp(r) and L = exp(l). Together

with their commutator s = [r, l] = r l l r =

1 00 1

they form a


14/28

348 Thomas Noll

1 00 1

1 01 1

1 02 1

1 03 1

1 04 1

1 05 1

1 10 1

1 11 2

1 12 3

1 13 4

1 14 5

1 20 1

1 21 3

1 22 5

1 23 7

1 30 1

1 31 4

1 32 7

1 40 1

1 41 5

1 50 1

2 11 1

2 13 2

2 15 3

2 17 4

2 31 2

2 33 5

2 35 8

2 51 3

2 53 8

2 71 4

3 12 1

3 15 2

3 18 3

3 21 1

3 24 3

3 27 5

3 42 3

3 45 7

3 51 2

3 54 7

3 72 5

3 81 3

4 13 1

4 17 2

4 31 1

4 35 4

4 53 4

4 71 2

5 14 1

5 22 1

5 27 3

5 33 2

5 38 5

5 41 1

5 72 3

5 83 5

7 23 1

7 32 1

7 45 3

7 54 3

8 35 2

8 53 2

Figure 7: The Stern-Brocot-Tree expressed in 2 2 matrices. It can beunderstood as the Cayley Graph of the monoid L, R SL2(Z) (com-pare with Figure 6: The sum of the columns of each matrix yields thecorresponding ratio on the traditional Stern-Brocot Tree).

basis forsl2(R), whose elements can be interpreted geometrically as tangentvectors to the identity matrix I SL2(R). The transport of these tangentvectors to another matrix is given by multiplying from the left:

r= r =

a bc d

0 10 0

=

0 a0 c

l= l =

a bc d

0 01 0

=

b 0d 0

.


15/28


In other words the successors are obtained by simply adding the corre-sponding tangent vectors to the base point :

L := + l and R := + r.

The same holds for any multiples tlortr. This means that the geodesics exp(tl) and exp(tr) are straight lines. This is different in the case

of their commutator exp(ts) = et 0

0 et

where the correspondinggeodesics are hyperbolas instead.

Remark 2 The discrete subgroup SL2(Z) and its conjugates are nicelyembedded into the (differentiable) Lie-group SL2(R), such that it is use-ful to consider the geometric picture, where an algebraic property such asnon-commutativity corresponds to the geometric property of curvature. Wedeparted from the distinguished Cayley Graph L,R. But as soon as twoor more scale generators come into play we are dealing with a small for-rest of conjugated and shifted binary trees, as we will see below in the caseof melodic and harmonic generation.

Remark 3 We emphasize this particular view on the continued fractionsin connection with a hypothesis on the phenomenology of the mind andrefer to [23] for details. This interpretation goes beyond abstract musictheory and basically consists in a canonical reformulation of Fechners law.This law characterizes the way, how the experiencing mind has access toits own activity. The basic idea is, that mental activity involves simula-tion of dynamics in terms of canonical transformations and has access tothis activity in terms of their infinitesimal generators. The action is thesimplest situation in which one may study Hamiltonian dynamics. It doesfurthermore literally correspond to the situation in linear optics. The pointis that SL2(R) acts as a group of canonical transformations on the planeR2 which can be seen as a two-dimensional phase space, e.g. as the cotan-gent bundle TR over a one dimensional linear configuration space. The

pathways down the internal tree are piecewise geodesics with respect to theinvariant Pseudo-Riemannian structure on SL2(R), which are the flowsis this special situation. The following arguments of about the duality ofmelodic and harmonic generation contribute to this phenomenological dis-cussion. But even within the narrow scope of abstract music theory thesearguments are interesting as such.

After this lengthy mathematical and metaphysical preparation we can nowformulate the idea of canonical scaling and explain the dualityof melodic


16/28

350 Thomas Noll

and harmonic scale generation as shown in Figure 4. Consider any realnumberr R(0 < r


17/28


5 What is Harmonic Tonality?

Many theorists tend to assume that there is a common ground to a largerepertoire of music written since the beginning of the 17th century, whichis often labeled as harmonic tonality. This subject domain is approachedalong several levels of description, among which we emphasize (in alpha-betical order) counterpoint, harmony, melody and tonality. It is still amatter of debate and of ongoing research to understand the interplay ofthe corresponding musical elements and phenomena. A common anchorfor many later developments and ramifications are the theoretical writ-ings of Jean Philippe Rameau. However, besides several other weaknessesRameau leaves us with a missing explanatory link between the generatingrole of the intervals of the abstract perfect triad of the one hand and hisconcrete treatment of triads and seventh-chords, fundamental bass progres-sions, double emploi etc. on the other (see [13]). A particularly sensitiveand interesting part is chapter nine of his Treatise on Harmony (see [24]),where he qualifies triads and seventh chords as the only conceivable har-monic units and treats them in a rather homogenous way. We argue thata missing piece of puzzle can be added to Rameaus argumentation in the

context of the above discussion.

Let us suppose that Agmon-Solidarity for triads and seventh chords withinthe third-generated diatonic and the Balzano-Solidarity for the pentatonicand diatonic within the fifth-generated 12-tone system are both of music-theoretical relevance in the sense that smooth transpositions exemplify akind of fundamental type of contiguity. This is to say, that among otherfactors the scale degree connections between triads and seventh chordsas well as the modulatory connections between diatonics are based on thiscommon principle of smooth transposition. In addition let us remind aboutthe theoretical fact this is a structural consequence of the wellformedness ofthese chords and scales. This links them directly to Carey and Clampittstheory of scale generation including the major result which explains the

prominent tone systems : structural, pentatonic, diatonic, 12-tonesystem,... as being determined from the perfect fifth as a generator. The followingconsiderations are inspired by the idea to put the two lines of argumenttogether into one network of arguments, which includes convincing argu-ments in support of the central elements of Rameaus theory and adds thementioned missing piece of puzzle to his argumentation. First we have acloser look on the semiconvergents of the major- and minor thirds:

y= log2(5/4) with semiconvergents 1

1,1

2,1

3,1

4,2

7,

3

10,

4

13,

5

16,

6

19,

7

22, . . .


18/28

352 Thomas Noll

y= log2(6/5) with semiconvergents 1

1,1

2,1

3,1

4,2

7,

3

11,

4

15,

5

19,

6

23,11

42, . . .

The two thirds are indistinguishable along their first 5 semiconvergents:11

, 12

, 13

, 14

, 27

. In the sequel we choose the minor-third as a second generatorbesides the fifth. The reason will be discussed in argument 7 below. Figure8 is called the Rameau Diagram. It displays a first attempt to localizeprominent music-theoretical concepts along the pathways for the perfect

fifth and the minor third. This is driven by the followingWorking Hypothesis: All generation steps on the two pathways of thetree including the node of the fifth-generated 12-tone system and the minor-third-generated 11-tone system as well as the two concordances betweenthese pathways at the levels of 3 tones and 7 tones have musical meaning.

Figure 8: RameauDiagram

The arguments below form a first attempt to make this working hypothesisplausible. However, the present music-theoretical interpretations have stillto be dealt with caution. A detailed discussion of possible alternatives andcounter-arguments is subject of a future study.


19/28


1. Concordances The two pathways correspond to the two genera-tors fifth and (minor) third. The common scale impulsesn= 3 andn = 7 support different concordances between them. In both casesthe third-generated fifth is composed of two thirds: 21 = 2 mod 3and 2 2 = 4 mod 7, but in the case n = 3 the fifth-generated thirdis composed of two fifths (2 2 = 1 mod 3) while for n = 7 the fifth-generated third is composed of four fifths (4 4 = 2 mod 7).6 We

discuss the possible musical meanings of these concordances sepa-rately:

2. Stumpf-Concordance. The third generated 3-tone system remindsof an abstract regular triad, because it becomes later specialized in4- and a 7-tone-system in order to yield the ordinary diatonic triad.If we establish a concordance between the third-generated fifth (twothirds) and the generating fifth, we have to assume a concordancebetween the double-fifth and the third as well. This embodies a plau-sible way (among others, see [22]) to mathematically formalize CarlStumpfs [26] idea to view the triad as a concordance of consonantintervals, rather than as a rigid segment from the overtone-series.

3. Rameau-Concordance The fact that 27 is a common semi-convergentof both thirds and 4

7a semiconvergent of the fifth supports Rameaus

principle to consider these intervals as the only conceivable funda-mental bass progressions within the diatonic. Furthermore is theRameau-Concordance a bridge between Agmon and Balzano Solidar-ity.

4. Agmon and Balzano SolidarityThe smooth transpositions of tri-ads and seventh chords within the third-generated diatonic (AgmonSolidarity) as well as the smooth transpositions of the pentatonicand diatonic within the fith generated 12-tone-system are part of theRameau-Diagram. Thus we may include most of Section 2 into this

argumentation.

5. Double Employment Two prominent music-theoretical conceptscan be interpreted as instances of a double employment of smoothtranspositions. Firstly, we argue that Rameausdouble emploiindeedcorrespond to the double-smooth transpositions. The transpositionof (F,A,C) to (D,F,A) (or vice versa) is smooth within (D,F,A,C)

6It is desirable to formulate the concordances in terms of canonic transformations,but we leave that for a further study.


20/28

354 Thomas Noll

and within the diatonic, because it is in solidarity with the step fromC to D (or vice versa). This step exchanges an external dissonanttone with an internal one. At the same time (because of the sol-idarity) does the corresponding transposition substitute the funda-mental bass. This means that Rameaus usage of double emploiis very close to our mathematical definition. Secondly, can the dou-ble employment of the two smooth transpositions of the pentatonic

be interpreted in the sense of Riemanns tonal functions Dominantand Subdominant. The former is a fifth transposition in solidaritywith a leading tone exchange between C and B, while the other isa fifth transposition in solidarity with a quasi-leadingtone exchangebetween E and F. Daniel Harrison [12] suggests to consider thefunctional agents and functional bases as independent constituents oftonal functions, which may or may not occur together (especially inchromatic music). In the light of our discussion one may refine hispoint by saying, that the double employment of the solidarity betweenfifth transposition and semitone step might indeed explain, that oneof the two phenomena could imply the other for very systematic rea-sons. A possible objection against our interpretation is nevertheless

the fact that a tonal function is seldom expressed in terms of an entirepentatonic. Therefore we provide also the following argument whichmay eventually relativize this objection.

6. Tonic Triads There are two qualitatively different ways of embed-ding the fifth-generated 3-tone system into the regular 5-tone-cyclewith regard to its further embedding as a pentatonic in the fifth-generated diatonic and 12-tone system. The sensitive embeddingchooses a fifth-chain which remains a fifth-chain in the embeddedpentatonic like (C,G,D). This is what Carey and Clampitt call thestructural system. The ignorant embedding - however - goes acrossan interval which becomes a third rather than a fifth. We call this

thetonic triad. In the case of a pentatonic (C,G,D,A,E) these tonictriads would be (E,C,G) and (A,E,C). The mathematical reasonto take this possibility into consideration is the fact that in any case(sensitive or ignorant) the 3-chain is not(!) smooth within the fifth-generated diatonic and does therefore not allow double employment.In other words, there is no serious reason to discard the ignorantembedding from the outset. However, we nevertheless consider thisas the most arguable construction in this list. To include the phe-nomenon oftonalityis desirable, but has to be done with caution.


21/28


7. Harmonic Minor Analogue A very surprising interpretation comesfor free, if we construct the canonical chromatic completion of thethird-generated diatonic, i.e. a tone system in which the seventhchords allow double employment. In analogy to the Carey number712

which represents the fifth-generated 12-tone system, we have theClampitt number 311 which represents a minor-third-generated 11-tone system. The right successor of 2

7on the Stern-Brocot tree is 3

10

and represents a major-third-generated 10-tone system. However inthe light of double employment it is preferable to choose the minorthird as a generator for the chromatic completion. Each minor thirdhas 3 chromatic steps. The surprising fact is the following: There isonly one possibility to embed the diatonic such that the steps fromBtoCand fromEto Fbecome semitones. But in this case the stepfrom G to A becomes also a semitone! This may shed new light onthe raised leading tone in minor or the lowered sixth in major. Inthe 11-tone system there has nothing to be raised or lowered, becausethe ordinary diatonic tonesGandAare already at semitone distance.The augmented second between F andG# appears merely as a resultof a lacking concordance between the two chromatic completions.

6 Pseudo-Diatonic Systems

Instead of interpreting the tree down to the root, we may fix a non-rootedconcept of a concordant diatonic, namely the third-generated Agmon sys-tem being linked to the fifth generated Balzano system (cf. Figure 6).

This concept can be studied in terms of its extension, i.e. by exploring itsmodel class. The fact, that we removed the root of the tree in Figure 6,implies that there are infinitely many models for this concept, which we

call pseudo-diatonic systems. For each chromatic number 0

Facts and Counter Facts

Documents

Transcript of Facts and Counter Facts