emusik

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Enumeration in Musical TheoryHarald Fripertinger

Voitsberg, Graz

January 12, 1993

Abstract

Being a mathematician and a musician (I play the flute) I found it very interesting to deal with

Polyas counting theory in my Masters thesis. When reading about Polyas theory I came acrossan article, called Enumeration in Music Theory by D. L. Reiner [11]. I took up his ideas andtried to enumerate some other musical objects.

At first I would like to generalize certain aspects of 12-tone music to n-tone music, where n is apositive integer. Then I will explain how to interpret intervals, chords, tone-rows, all-interval-rows,rhythms, motifs and tropes in n-tone music. Transposing, inversion and retrogradation are definedto be permutations on the sets of musical objects. These permutations generate permutationgroups, and these groups induce equivalence relations on the sets of musical objects. The aim ofthis article is to determine the number of equivalence classes (I will call them patterns) of musicalobjects. Polyas enumeration theory is the right tool to solve this problem.

In the first chapter I will present a short survey of parts of P olyas counting theory. In thesecond chapter I will investigate several musical ob jects.

Abstract

In dieser Arbeit wird der Begriff von 12-Ton Musik aufn-Ton Musik, wobei n eine naturlicheZahl ist, erweitert. Objekte der Musiktheorie wie Intervall, Akkord, Takt, Motiv, Tonreihe, Allinter-vallreihe und Trope werden mathematisch gedeutet. Transponieren, Inversion (Umkehrung) undKrebs werden als Permutationen auf geeigneten Mengen interpretiert. Zwei musikalische Objekteheien wesentlich verschieden, falls man sie nicht durch solche Permutationen ineinander uberfuhrenkann. In die Sprache der Mathematik ubersetzt, bedeutet dies: Abzahlen von Aquivalenzklassen(von Funktionen), wobei die Aquivalenz durch eine Permutationsgruppe induziert wird. DiesesProblem wird von der Abzahltheorie von Polya und von Satzen, die in Anschlu an diese Theorieentstanden sind, gelost. Zu diesen Satzen gehoren Theoreme von N.G. de Bruijn und das PowerGroup Enumeration Theorem von F. Harary.

Im ersten Kapitel stelle ich alle grundlegenden Definitionen zusammen. Dann folgen obenerwahnte Satze, welche hier in dieser Arbeit nicht bewiesen sind. Das daran anschlieende Kapi-

tel beschaftigt sich mit den Anzahlbestimmungen musikalischer Ob jekte. Diese Satze sind nunvollstandig bewiesen.

Die Grundidee zu dieser Arbeit habe ich [11] entnommen. Daraufhin habe ich versucht dieseGedanken weiter auszubauen und die Anzahlbestimmung anderer musikalischer Objekte durchzu-fuhren. Bisher hatten Musiktheoretiker und Komponisten mit verschiedenen Methoden, oder durchAusprobieren, solche Anzahlen bestimmt. Durch Verwendung der Theorie von Polya soll ein Systemin diese Untersuchungen gebracht werden. Fur den Anwender ist es nicht notig, die Beweise in allenEinzelheiten zu verstehen. Er sollte jedoch mit mathematischen Schreib- und Sprechweisen vertrautsein. Da diese Arbeit auch von Mathematikern gelesen wird, mu sie auch allen mathematischenForderungen nach Exaktheit und Genauigkeit der Beweise gerecht werden.

The author thanks Jens Schwaiger for helpful comments.

0

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1

1 Preliminaries

There is a lot of literature about Polyas counting theory. For instance see [1], [2], [3], [9] or [10].

Definition 1.1 (Type of a Permutation) Let M be a set with |M| = m. A permutation SMis of the type (1, 2, . . . , m), iff can be written as the composition of i disjointed cycles of lengthi, for i = 1, . . . , m.

Definition 1.2 (Cycle Index) Let P be a set of |P| = n elements and let be a subgroup of SP,denoted furtheron by SP. The cycle index of is defined as a polynomial in n indeterminatesx1, . . . , xn, defined as:

CI(; x1, . . . , xn): = ||1

ni=1

xi()i .

Lemma 1.1 (Cycle Index of the Cyclic Group) Let(E)n be the cyclic group of order n generated

by a cyclic permutation of n objects, then the cycle index of (E)n is

CI((E)

n ; x1, x2, . . . , xn) =

1

nt|n

(t)xt

nt

,

where is Eulers -function.

Lemma 1.2 (Cycle Index of the Dihedral Group) Let(E)n be the dihedral group of order 2n and

degree n containing the permutations which coincide with the 2n deck transformations of a regularpolygon with n vertices.

1. If n 1 mod 2, then

CI((E)n ; x1, x2, . . . , xn) =1

2x1x2

n12 +

1

2n

t|n

(t)xtnt .

2. If n 0 mod 2, then

CI((E)n ; x1, x2, . . . , xn) =1

4(x2

n2 + x1

2x2n2

2 ) +1

2n

t|n

(t)xtnt .

The main lemma in Polyas counting theory is

Theorem 1.1 (Lemma of Burnside) Let P be a finite set and SP. Furthermore let B be theset of the orbits of P under , then

|B| = ||1

(),

where () is defined as () := |{p P(p) = p}|.

Theorem 1.2 (Polyas Theorem) Let P and F be finite sets with |P| = n, and let SP. Fur-

thermore let R be a commutative ring over the rationals Q and let w be a mapping w: F R. Twomappings f1, f2 FP are called equivalent, iff there exists some such that f1 = f2. The

equivalence classes are called mapping patterns and are written as [f]. For every f FP we define theweight W(f) as product weight

W(f) :=pP

w

f(p)

.

Any two equivalent fs have the same weight. Thus we may define W([f]) := W(f). Then the sum ofthe weights of the patterns is

[f]

W

[f]

= CI

;yF

w(y),yF

w(y)2, . . . ,yF

w(y)n

.

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2 1 PRELIMINARIES

Theorem 1.3 (Power Group Enumeration Theorem) Let P and F be finite sets, with |P| = nand |F| = k, let SP and SF. We will call two mappings f1, f2 F

P equivalent:

f1 f2: with f1 = f2.

The equivalence classes [f] are called mapping patterns. Let w be a mapping w: F R with Q Rsuch that

W(f) :=pP

w

f(p)

is constant on each pattern. Then:[f]

W

[f]

= ||1

CI

; 1(), 2(), . . . , n()

,

wherei(): =

yF

i(y)=y

w(y) w

(y)

. . . w

i1(y)

.

This is the Power Group Enumeration Theorem in polynomial Form of [7].

Theorem 1.4 (de Bruijn) Let P and F be finite sets with |P| = n and |F| = k, let SP and SF. We will call two mappings f1, f2 F

P equivalent:


The equivalence classes [f] are called mapping patterns. The weight of a function f FP is defined as:

W(f) :=

1 if f is injective0 else.

The number of patterns of injective functions is

CI

;

x1,

x2, . . .

xn

CI(; 1 + x1, 1 + 2x2, . . . 1 + kxk)

x1=x2=...=xk=0

.

Theorem 1.5 (de Bruijn [1]) Let P and F be finite sets with |P| = n and |F| = k, let SP and SF. We will call two mappings f1, f2 F

P equivalent:


The equivalence classes [f] are called mapping patterns. Let

Y: = {[f][f] = [f]}.

Furthermore let R be a commutative ring over the rationals Q and let w be a mapping w: F R. Forevery f FP we define the weight W(f) as

W(f) :=pP

w

f(p)

.

Then [f]Y

W

[f]

= CI(; 1, 2, . . . , n),

wherei: =

yF

i(y)=y

w(y) w

(y)

. . . w

i1(y)

.

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3

2 Applications of Polyas Theory in Musical Theory

Some parts of this chapter were already discussed by D.L.Reiner in [11]. Now we are going to calculatethe number of patterns of chords, intervals, tone-rows, all-interval-rows, rhythms, motifs and tropes.

Proving any detail would carry me too far. For further information see [6].

2.1 Patterns of Intervals and Chords

2.1.1 Number of Patterns of Chords

Definition 2.1 (n-Scale) 1. If we divide one octave into n parts, we will speak of an n-scale. Theobjects of an n-scale are designated as

0, 1, . . . , n 1.

2. In twelve tone music we usually identify two tones which are 12 semi-tones apart. For that reasonwe define an n-scale as the cyclic group (Zn, +) of order n.

Definition 2.2 (Transposing, Inversion) 1. Let us define T the operation of transposing as apermutation

T: Zn Zn

a T(a) := 1 + a.

The group T is the cyclic group (E)n .

2. Let us define I the operation of inversion as

I: Zn Zn

a I(a) := a.

The group T, I is the dihedral group (E)n .

Definition 2.3 (k-Chord) 1. Let k n. A k-chord in an n-scale is a subset of k elements of Zn.An interval is a 2-chord.

2. Let G = (E)n or G =

(E)n . Two k-chords A1, A2 are called equivalent iff there is some G such

that A2 = (A1).

Remark 2.1 1. We want to work with Polyas Theorem, therefore I identify each k-chord A withits characteristic function A. Two k-chords A1, A2 are equivalent iff the two functions A1 andA2 are equivalent in the sense of Theorem 1.2.

2. Let us define two finite sets: P: = Zn and F: = {0, 1}. Each function f FP will be identifiedwith

Af: = {k Znf(k) = 1}.

3. Let w: F R := Q[x] be a mapping with w(1): = x and w(0): = 1, where x is an indeterminate.Define the weight W(f) of a function f FP as

W(f): =kZn

w

f(k)

.

We see that the weight of a k-chord is xk. The weight of a pattern W([f]) := W(f) is well defined.

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4 2 APPLICATIONS OF POLYAS THEORY IN MUSICAL THEORY

Theorem 2.1 (Patterns of k-Chords) 1. Let G be a permutation group on Zn. The number ofpatterns of k-chords in the n-scale Zn is the coefficient of x

k in

CI(G; 1 + x, 1 + x2, . . . , 1 + xn).

2. If G = (E)n , the number of patterns of k-chords is

1

n

j|gcd(n,k)

(j)

nj

kj

,


3. If G = (E)n , the number of patterns of k-chords is

12n j|gcd(n,k)

(j)

njkj

+ n

(n1)

2

[k2 ] if n 1 mod 2

12n

j|gcd(n,k)

(j)njkj

+ n

n2k2

if n 0 mod 2 and k 0 mod 2

12n

j|gcd(n,k)

(j)njkj

+ n

n2 1

[k2 ]

if n 0 mod 2 and k 1 mod 2.

4. In the case n = 12 and G = (E)n , we get the numbers in table 1 on page 23.

5. In the case n = 12 and G = (E)n , we get the numbers in table 2 on page 23.

Proof:

1. Application of Theorem 1.2.

2. Let us calculate the coefficient of xk in

CI((E)n ; 1 + x, 1 + x2, . . . , 1 + xn) =

=1

n

t|n

(t)(1 + xt)nt =

1

n

t|n

(t)

nt

i=0

nt

i

xti. ()

Let k: = t i, then i = kt

. With () we have:

1n

t|n

(t)

nk=0t|k

ntkt

xk = 1

n

nk=0

t|nt|k

(t) n

tkt

xk =

=

nk=0

1

n

t|gcd(n,k)

(t)

ntkt

xk.

3. Same proof as 2.

q.e.d. Theorem 2.1

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2.1 Patterns of Intervals and Chords 5

2.1.2 The Complement of a k-Chord

Definition 2.4 (Complement of a k-Chord) Let A Zn with |A| = k be a k-chord. The comple-ment of A is the (n k)-chord Zn \ A.

Remark 2.2 1. Let G = (E)n or G = (E)n be a permutation group on Zn and let 1 k < n. Thereexists a bijection between the sets of patterns of k-chords and (n k)-chords.

Proof:

The following general result holds:Let M1 and M2 be two finite sets and f a bijective mapping f: M1 M2. Furthermore let i bean equivalence relation on Mi and i the canonical projection

i: Mi Mi|i

x i(x):= [x]

for i = 1, 2. In addition to this the function f satisfies

x 1 y f(x) 2 f(y).

Then the function f: M1|1 M2|2 defined by f([x]): = [f(x)] is well defined and bijective.

In our context we have the case that M1 is the set of all k-chords, M2 is the set of all (n k)-chords, i is induced by G and f(A): = Zn \ A, then f is a bijection between the sets of patternsof k-chords and (n k)-chords.

q.e.d. Remark 2.2

2. Ifn 0 mod 2, the complement of an n2 -chord is ann2

-chord. Now I want to figure out the numberof patterns of n2 -chords [A] with the property A Zn \ A. Applying Theorem 1.5 we get:

Theorem 2.2 1. Let n 0 mod 2. The number of patterns of n2 -chords which are equivalent totheir complement, is

CI(G; 0, 2, 0, 2, . . .).

2. Ifn = 12 andG = (E)n , there are 20 patterns of 6-chords which are equivalent to their complement.

3. Ifn = 12 andG = (E)n , there are 8 patterns of 6-chords which are equivalent to their complement.

Proof:Let us define two finite sets P: = Zn and F: = {0, 1} and define a weight function by W(f):= 1 for allf FP. Each function f FP will be identified with Mf: = {k Zn

f(k) = 1} = f1({1}). Thegroup G defines an equivalence relation on P. Furthermore let := (0, 1) be a transposition in SF. Todetermine the number of patterns of n

2-chords which are equivalent to their complement, we have to

calculate the number of patterns of functions f FP which are invariant under . Using a special caseof Theorem 1.5 we get that this number is given by

CI(G; 1, 2, . . . , n)

wherei :=

j|i

j j

and (1, 2, . . .) is the type of the permutation . Since is of the type (0, 1), this is

CI(G; 0, 2, 0, 2, . . .).

q.e.d. Theorem 2.2

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2.1.3 The Interval Structure of a k-Chord

In this section we use (E)n as the permutation group acting on Zn. The set of all possible intervals

between two differnet tones in n-tone music will be called Int(n), thus

Int(n) := {x yx, y Zn, x = y} = {1, 2, . . . , n 1}.

Definition 2.5 (Interval Structure) On Zn we define a linear order 0 < 1 < 2 < . . . < n 1. LetA := {i1, i2, . . . , ik} be a k-chord. Without loss of generality let i1 < i2 < . . . < ik. The intervalstructure of A is defined as the pattern [fA], wherein the function fA is defined as

fA: {1, 2, . . . , k} Int(n)

fA(1) := i2 i1,

fA(2) := i3 i2,

. . .

fA(k 1) := ik ik1,

fA(k) := i1 ik,

and two functions f1, f2: {1, 2, . . . , k} Int(n) are called equivalent, iff there exists some (E)k such

that f2 = f1 . The group (E)k is generated by T and I with T(i) := i + 1 mod k and I(i) := k + 1 i

for i = 1, . . . , k. The differences ij+1 ij must be interpreted as differences in Zn. They are the intervalsbetween the tones ij and ij+1.

Theorem 2.3 LetA1 := {i1, i2, . . . , ik} and A2 := {j1, j2, . . . , jk} be two k-chords with i1 < i2 < . . . n f1(n2 ) we calculate

E(f)

(i)

E(f)

n f1(n

2)

+

i(nf1(n2 ))j=1

E(f)

n f1(n

2) +j

=

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= 0 +

i(nf1(n2 ))j=1

f(j) = (f)

i n + f1(n

2)

.

Thus everything is proved.

7. The following formulas hold: E I = I E, E Q = Q E, E R = R E and E2 = id.

8. Let us define three permutation groups on Allint(n).G1 := I, R, G2 := I, R, E und G3 := I, R, E , Q. For G2 we must assume n 6, andfor G3 we must assume n = 12. We calculate that |G1| = 4, |G2| = 8, |G3| = 16.

Remark 2.8 (Counting of All-Interval-Rows) Let

x1, x2, . . . , xn1, y1, y2, . . . , yn1, z1, z2, . . . , zn1

be indeterminates over Q. Furthermore let f be a mapping f: {1, 2, . . . , n 1} Int(n). We define:

R := Q[x1, x2, . . . , xn1, z1, z2, . . . , zn1]

and

W(f) :=n1i:=1

wi

f(i)

.

The functions wi are defined as

wi: Int(n) R

j wi(j) := zj

n1:=i

xj .

After calculating W(f) you have to replace terms of the form xj by yj mod n. Then you get W(f)

Q[y1, y2, . . . , yn1, z1, z2, . . . , zn1]. According to Theorem 2.7 f is an all-interval-row, if and only if,

W(f) =

n1i=1

yizi.

Proof:f is an all-interval-row, if and only if, f and (f) are injective. The function f is injective, iff W(f) is

divisible byn1i=1 zi. According to the construction ofW(f) the power ofxi is

ij=1 f(j) (f)(i) mod

n. Thus

W(f) =n1i=1

zf(i)y(f)(i)

and the function (f) is injective, iff W(f) is divisible by n1i=1 yi. Consequently the number of all-

interval-rows in n-tone music is the coefficient ofn1i=1 yizi in

n1i=1

n1j=1

zj

n1k=i

xkj

xj=yj mod n.

q.e.d. Remark 2.8

Remark 2.9 For G1 or G2 or G3 we want to calculate

() := |{f Allint(n)

(f) = f}|.

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2.4 Patterns of Rhythms 17

After some calculations we can derive that there are only 4 permutions such that () = 0. In

Remark 2.8 we calculated (id). The value of (I R) is the coefficient ofn1i=1 yizi in

n2 1i=1

n1j=1

j= n2

zjznj

n1k=i

xkj

n1k=ni

xknj

zn2

n1k=n2

xkn

2xj=yj mod n .

Now let n 6. The value of (I V) is the coefficient ofn1i=1 yizi in

n2 1i=1

n1j=1

j= n2

zjznj

n1k=i

xkj

n1k=(n2 +i)

xknj

zn2

n1k=n2

xkn2

xj=yj mod n

.

Now let n = 12. In order to calculate (Q V R) you must compute

5

i=1z6

11

j=2i

xj6z3z9

11

j=i

xj3

11

j=i+6

xj9 +

11

j=i

xj9

11

j=i+6

xj3

i1j=1

n1k=1

k{3,6,9}

zkz5k mod 12

11l=j

xlk

11l=2ij

xl5k mod 12

i+5

j=2i+1

n1k=1

k{3,6,9}

zkz5k mod 12

11l=j

xlk

11l=12+2ij

xl5k mod 12

.

Then substitute yj mod 12 for xj and find the coefficient of

11i=1 yizi.

Theorem 2.8 (Number of Patterns of All-Interval-Rows) The number of patterns of all-inter-val-rows in regard to Gi for i = 1, 2, 3 is

1. 14

(id) + (I R)

for i = 1.

2. 18

(id) + (I R) + (I V)

for i = 2.

3. For i = 3 we calculate

1

16

(id) + (I R) + (I V) + (Q R V)

=

=1

16(3 856 + 176 + 120 + 120) = 267.

Proof:Application of the Lemma of Bunside, Theorem 1.1.

q.e.d. Theorem 2.8

2.4 Patterns of Rhythms

Definition 2.8 (n-Bar, Entry-time, k-Rhythm) An important contribution in a composition is abar. Usually a lot of bars of the same form follow one another. If you know the smallest rhythmicalsubdivision of a bar, you can figure out how many entry-times (think of rhythmical accents played ona drum) a bar holds. If there are n entry-times in a bar, I call it an n-bar. In mathematical terms ann-bar is expressed as the cyclic group Zn. We can define cyclic temporal shifting S as a permutation

S: Zn Zn

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t S(t): = t + 1.

Retrogradation R (temporal inversion) is defined as

R: Zn Zn

t R(t): = t.

The group S is (E)n and S, R =

(E)n . A k-rhythm in an n-bar is a subset of k elements of Zn. The

permutation groups (E)n or

(E)n induce an equivalence relation on the set of all k-rhythms. Now we

want to calculate the number of patterns of k-rhythms. We get the same numbers as in Theorem 2.1.

Theorem 2.9 (Patterns of k-Rhythms) 1. Let G be a permutation group on Zn. The numberof patterns of k-rhythms in the n-bar Zn is the coefficient of xk in

CI(G; 1 + x, 1 + x2, . . . , 1 + xn).

2. If G = (E)

n , the number of patterns of k-rhythms is

1

n

j|gcd(n,k)

(j)

nj

kj

,


3. If G = (E)n , the number of patterns of k-rhythms is

12n

j|gcd(n,k)

(j)njkj

+ n

(n1)2

[k2 ]

if n 1 mod 2

12n

j|gcd(n,k)

(j)njkj + n

n2k2 if n 0 mod 2 and k 0 mod 2

12n

j|gcd(n,k)

(j)njkj

+ n

n2 1

[k2 ]

if n 0 mod 2 and k 1 mod 2.

2.5 Patterns of Motifs

Definition 2.9 (k-Motif) 1. Now I want to combine both rhythmical and tonal aspects of music.

2. Assume we have an n-scale and an m-bar, then the set M

M := {(x, y)x Zm, y Zn} = Zm Zn

is the set of all possible combinations of entry-times in the m-bar Zm and pitches in the n-scale

Zn. Furthermore let G be a permutation group on M. In Remark 2.11 we are going to study twospecial groups G. The group G defines an equivalence relation on M:

(x1, y1) (x2, y2): g G with (x2, y2) = g(x1, y1).

In addition to this we have |M| = m n.

3. Let 1 k m n. A k-motif is a subset of k elements of M.

Remark 2.10 Let f be a mapping f: M {0, 1}. Now we identify f with the set

Mf := {(x, y) M

f(x, y) = 1}.

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2.5 Patterns of Motifs 19

This means: f is the characteristic function of Mf. The function f is the characteristic function of ak-motif |Mf| = k. Two functions f1, f2: M {0, 1} are defined equivalent

f1 f2: g G with f2 = f1 g.

Let fi be the characteristic function of the k-motif Mi for i = 1, 2, then we have:

f1 f2 g G with g(M2) = M1.

Now w(0) := 1 and w(1) := x define a weight function w: {0, 1} Q[x], where x is an indeterminateover Q. In addition to this let

W(f) :=pM

w

f(p)

.

Now we can say: f is the characteristic function of a k-motif W(f) = xk.

Theorem 2.10 (Number of Patterns ofk-Motifs) The number of patterns of k-motifs in an n-scale and in an m-bar is the coefficient of xk in

CI(G; 1 + x, 1 + x2, . . . , 1 + xmn).

Proof:This completely follows Polyas Theorem 1.2.

q.e.d. Theorem 2.10

Remark 2.11 (Special Permutation Groups) Now I want to demonstrate two examples for groupG.

1. In Definition 2.2 we had a permutation group G2 = (E)n or G2 =

(E)n acting on the n-scale

Zn. Moreover in Definition 2.8 there was a permutation group G1 = (E)m or G1 =

(E)m defined

on the m-bar Zm. For that reason, we define the group G as G := G1 G2. Two elements(x1, y1), (x2, y2) M are called equivalent with respect to G, iff there exist G1 and G2,with

(x2, y2) = (, )(x1, y1) = ((x1), (y1)).

Because of the fact that we know how to calculate the cycle index of G1 G2, we can computethe number of patterns of k-motifs.

2. In the case m = n, we can define another permutation group G, as it is done in [8]. The group Gis defined as

G := T , S , AA Gl(2, Zn),

withT: M M

x

y

T

x

y

:=

x

y + 1

S: M Mx

y

S

x

y

:=

x + 1

y

A: M M

x

y

A

x

y

:= A

x

y

.

The multiplication A

x

y

stands for matrix multiplication. The set Gl (2, Zn) is the group of

all regular 2 2-matrices over Zn.

You can easily derive the following results:

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(a) Tn = Sn = idM and Tj = idM and S

j = idM for 1 j < n.

(b) T S = S T. In addition to this T S and S T.

(c) Let 0 i,j < n, then: Ti Sj AA Gl(2, Zn), iff i = 0 or j = 0.(d) Let A :=

a bc d

, then: A Tk Sl = T(cl+dk) S(al+bk) A.

(e) G is the group of all affine mappings Zn2 Zn

2.

Although we know quite a lot about the group G, I could not find a formula for the cycle indexof G for arbitrary n.

Example 2.2 Let us consider the case, that n = m = 12.

1. IfG is defined as G := (E)n

(E)n , then we derive

CI(G; x1, x2, . . . , x144) =

= 1576 (x1441 + 12x241 x602 + 36x41x702 + 147x722 + 8x483 + 24x83x206 + 60x364 + 96x246 + 192x1212).

By applying Theorem 2.10, the number of patterns of k-motifs is the coefficient of xk in1 + x + 48x2 + 937x3 + 31261x4 + 840 006x519392669x6 + 381561 281x7 + 6532510709x8 +98700483548x9 + 1 332 424 197 746x10 + . . ..

2. IfG := T , S , AA Gl(2, Zn), I computed the cycle index of G with a Turbo Pascal program

asCI(G; x1, x2, . . . , x144) =

= 1663552

(x1441 + 18x721 x

362 + 36x

481 x

482 + 24x

481 x

323 + 72x

361 x

542 + 48x

361 x

182 x

184 + 648x

241 x

602 +

432x241 x122 x

163 x

86 + 192x

181 x

92x

274 + 9x

161 x

642 + 72x

161 x

162 x

166 + 54x

161 x

324 + 108x

161 x

168 + 2 592x

121 x

662 +

1728x121 x302 x

184 + 1 728x

121 x

182 x

83x

126 + 1 152x

121 x

62x

83x

64x

46x

412 + 128x

91x

453 + 384x

91x

93x

186 +

162x81x682 + 1 296x81x202 x166 + 972x81x42x324 + 1 944x81x42x168 + 6 912x61x152 x274 + 4 608x61x32x43x94x26x612 +648x41x

702 + 432x

41x

342 x

184 + 5 184x

41x

222 x

166 + 3 456x

41x

102 x

64x

86x

412 + 3 888x

41x

62x

324 + 7 776x

41x

62x

168 +

2592x41x22x

344 + 5 184x

41x

22x

24x

168 + 4 608x

31x

32x

153 x

156 + 13 824x

31x

32x

33x

216 + 3 072x

31x

473 +

9216x31x113 x

186 + 1 728x

21x

172 x

274 + 13 824x

21x

52x

94x

46x

612 + 10 368x

21x2x

354 + 20736x

21x2x

34x

168 +

1152x1x42x

53x

206 + 3 456x1x

42x3x

226 + 9 216x1x2x

53x

216 + 27 648x1x2x3x

236 + 6 912x1x

53x

24x

1012 +

13 824x1x53x8x

524 + 20 736x1x3x

24x

26x

1012 + 41472x1x3x

26x8x

524 + 3 174x

722 + 2 208x

362 x

184 +

6624x242 x166 + 4 608x

122 x

64x

86x

412 + 3 726x

82x

324 + 7 452x

82x

168 + 2 592x

42x

344 + 5 184x

42x

24x

168 +

7224x483 + 1 008x243 x

126 + 9 288x

163 x

166 + 25 536x

123 x

186 + 2 688x

123 x

66x

612 + 1 296x

83x

206 +

10 752x63x36x

912 + 32 832x

43x

226 + 3 456x

43x

106 x

612 + 13824x

23x

56x

912 + 38 400x

364 + 36 864x

124 x

812 +

41 472x44x168 + 8 832x

246 + 6 144x

126 x

612 + 39 936x

188 + 18432x

68x

424 + 49 152x

1212 + 24 576x

624).

By applying Theorem 2.10, the number of patterns ofk-motifs is the coefficient ofxk in 1+x+5x2+

26x3

+216x4

+2024x5

+27806x6

+417209x7

+6345735x8

+90590713x9

+1190322956x10

+ . . ..For k = 1, 2, 3, 4 these numbers are the same as in [8]. In the case k = 5 however, it is stated thatthere exist 2 032 different patterns of 5-motifs, while here we get 2 024 of these patterns.

2.6 Patterns of Tropes

Definition 2.10 (Trope) 1. If you divide the set of 12 tones in 12-tone music into 2 disjointedsets, each containing 6 elements, and if you label these sets as a first and a second set, we willspeak of a trope. This definition goes back to Josef Matthias Hauer. Two tropes are calledequivalent, iff transposing, inversion, changing the labels of the two sets or arbitrary sequences ofthese operations transform one trope into the other.

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2.7 Special Remarks on 12-tone music 21

2. For a mathematical definition let n 4 and n 0 mod 2. A trope in n-tone music is a function

f: Zn F: = {1, 2} such that |f1({1})| = |f1({2})| =

n

2.

f(i) = k is translated into: The tone i lies in the set with label k. Furthermore T and I arepermutations on Zn as in Definition 2.2. The group T, I is

(E)n . Two tropes f1, f2 are called

equivalent, if and only if,

(E)n S2 such that f2 = 1 f1 .

3. Let x and y be indeterminates over Q. Define a function w: F Q[x, y] by w(1):= x andw(2): = y. For f FZn the weight of f is defined as product weight

W(f): =xZn

w

f(x)

.

A function f: Zn F: = {1, 2} is a trope, iff W(f) = xn

2 y

n

2 .

Theorem 2.11 (Patterns of Tropes) Let be Eulers -function. The number of patterns of tropes

in regard to (E)n is

14

1n

t|n2

(t) n

tn2t

+

t|n

t0 mod 2

(t)2nt

+n

2n4

+ 2

n2 1

if n 0 mod 4

14

1n

t|n2

(t) n

tn2t

+

t|n

t0 mod 2

(t)2nt

+n2

2n2

4

+ 2

n2 1

if n 2 mod 4.

In 12-tone music there are 35 patterns of tropes. (See [5].) Hauer himself calculated that there are 44

patterns of tropes, because in his work the permutation group acting on Zn was the cyclic group T.

Proof:We want to use Theorem 1.3 which says:

[f]

W([f]) =1

|S2|

S2

CI

(E)n ; (1, ), (2, ), . . . , (n, )

,

where

(m, ): =yF

m(y)=y

w(y) w

(y)

. . . w

m1(y)

.

The number of patterns of tropes is the coefficient of xn2 y

n2 in

1

2

CI((E)n ; x + y, x

2 + y2, . . . , xn + yn) + CI((E)n ; 0, 2xy, 0, 2x2y2, . . . , 0, 2x

n2 y

n2 )

.

This can be transformed into the formula above.q.e.d. Theorem 2.11

2.7 Special Remarks on 12-tone music

In addition to the operations of transposing T and of inversion I we can study quartcircle- and quintcirclesymmetry in 12-tone music.

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Remark 2.12 (Quartcircle Symmetry) The quartcircle symmetry Q is defined as

Q: Z12 Z12

x Q(x) := 5x.Q is a permutation on Z12, since gcd(5, 12) = 1. Furthermore

1. Q I, T.

2. Q T = T5 Q.

3. Q I = I Q = 7x, which is called the quintcircle symmetry.

4. Q2 = idZ12 .

5. Let G be G: = I , T , Q. Each element G can be written as

= Tk Ij Ql

such that k {0, 1, . . . , n 1}, j {0, 1}, and l {0, 1}.

6. The cycle index of G: = I , T , Q is

CI(G; x1, x2, . . . , x12) =

=1

48

t|12

(t)xt12t + 2x61x

32 + 3x

41x

42 + 6x

21x

52 + 11x

62 + 4x

23x6 + 6x

34 + 4x

26

.

This group G is an other permutation group acting on Z12 with a musical background. The questionarises, how to generalize the quartcircle symmetry of 12-tone music to n-tone music. Should we takeany unit in Zn or only those units e such that e

2 = 1?

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2.7 Special Remarks on 12-tone music 23

k 1 2 3 4 5 6 7 8 9 10 11 12# of patterns 1 6 19 43 66 80 66 43 19 6 1 1

Table 1: Number of patterns of k-Chords in 12-tone music with regard to (E)n .

k 1 2 3 4 5 6 7 8 9 10 11 12# of patterns 1 6 12 29 38 50 38 29 12 6 1 1

Table 2: Number of patterns of k-Chords in 12-tone music with regard to (E)n .

k 2 3 4 5 6 7

# of patterns 6 30 275 2 000 14060 83280k 8 9 10 11 12

# of patterns 416 880 1 663 680 4 993 440 9 980 160 9 985 920

Table 3: Number of patterns of k-rows in 12-tone music.

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24 LIST OF TABLES

References

[1] N.G. De Bruijn. Color Patterns that are Invariant Under a Given Permutation of the Colors.Journal of Combinatorial Theory, 2: pages 418 421, 1967.

[2] N.G. De Bruijn. Polyas Theory of Counting. In Applied Combinatorial Mathematics, chapter 5,pages 144 184, Beckenbach, E.F., Wiley, New York, 1964.

[3] N.G. De Bruijn. A Survey of Generalizations of Polyas Enumeration Theorem. Nieuw Archiefvoor Wiskunde (2), XIX: pages 89 112, 1971.

[4] H. Eimert. Grundlagen der musikalischen Reihentechnik. Universal Edition, Wien, 1964.

[5] H. Florey. Analytische Bemerkungen zu Josef Matthias Hauers letztem Zwolftonspiel. Appendixto a record published by Hochschule fur Musik und darstellende Kunst in Graz, 1988.

[6] H. Fripertinger. Untersuchung uber die Anzahl verschiedener Intervalle, Akkorde, Tonreihen undanderer musikalischer Objekte in n-Ton Musik. Masters thesis, Hochschule fur Musik und Darstel-

lende Kunst, Graz, 1991.

[7] F. Harary and E.M. Palmer. The Power Group Enumeration Theorem. Journal of CombinatorialTheory 1, pages 157 173, 1966.

[8] G. Mazzola. Geometrie der Tone. Birkhauser, Basel, Boston, Berlin, 1990. ISBN 3-7643-2353-1.

[9] G. Polya. Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen und chemische Verbindun-gen. Acta Mathematica, 68: pages 145 254, 1937.

[10] G. Polya and R.C. Read. Combinatorial Enumeration of Groups, Graphs and Chemical Compounds.Springer Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1987. ISBN 0-387-96413-4or ISBN 3-540-96413-4.

[11] D.L. Reiner. Enumeration in Music Theory. Amer. Math. Monthly, 92: pages 51 54, 1985.

List of Tables

1 Number of patterns ofk-Chords in 12-tone music with regard to (E)n . . . . . . . . . . . 23

2 Number of patterns ofk-Chords in 12-tone music with regard to (E)n . . . . . . . . . . . 23

3 Number of patterns ofk-rows in 12-tone music. . . . . . . . . . . . . . . . . . . . . . . . 23

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CONTENTS 25

Contents

1 Preliminaries 1

2 Applications of Polyas Theory in Musical Theory 32.1 Patterns of Intervals and Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Number of Patterns of Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 The Complement of a k-Chord . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 The Interval Structure of a k-Chord . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Patterns of Tone-Rows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Patterns of All-Interval-Rows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Patterns of Rhythms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Patterns of Motifs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Patterns of Tropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Special Remarks on 12-tone music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Working for my doctorial thesis I succeeded in constructing complete lists of orbit representativesof k-motifs for k = 1, 2, . . . , 8 under the action of the permutation group of the second item of example2.2.

Address of the author:Harald FripertingerHohenstrae 8A-8570 Voitsberg

Institut fur MathematikUniversitat GrazHeinrichstrae 36/4A-8010 Graz

This paper was published in Beitrage zur Elektronischen Musik, volume 1, of the Hochschulefur Musik und Darstellende Kunst, Graz in 1992. A shorter version can be found in the Actes 26e

Seminaire Lotharingien de Combinatoire, 476/S-26, 29 42, 1992.

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