Enumeration in Musical Theory

Harald Fripertinger ^{∗} Voitsberg, Graz

Abstract

Being a mathematician and a musician (I play the flute) I found it very interesting to deal with Pólya’s counting theory in my Master’s thesis. When reading about Pólya’s theory I came across an article, called ”Enumeration in Music Theory” by D. L. Reiner [11]. I took up his ideas and tried 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 a positive 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 defined to be permutations on the sets of ”musical objects”. These permutations generate permutation groups, and these groups induce equivalence relations on the sets of ”musical objects”. The aim of this article is to determine the number of equivalence classes (I will call them patterns) of ”musical objects”. Pólya’s enumeration theory is the right tool to solve this problem.

In the first chapter I will present a short survey of parts of Pólya’s counting theory. In the second chapter I will investigate several ”musical objects”.

1 Preliminaries

There is a lot of literature about Pólya’s counting theory. For instance see [2], [1], [3],

[9] or [10]. Let M be a set with |M| = m. You should know the definition of the type (λ_{1}, λ_{2}, …, λ_{m}) of a permutation π ∈ S_{M} and the definition of the cycle index CI(Γ; x_{1}, …, x_{m}) of a permutation group Γ ≤ S_{M}. In particular we will use the cycle index of the cyclic group and of the dihedral group.

2 Applications of Pólya’s Theory in Musical Theory

Some parts of this chapter were already discussed by D.L.Reiner in [11]. Now we are going to calculate the 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].

∗The author thanks Jens Schwaiger for helpful comments.

2.1 Patterns of Intervals and Chords

2.1.1 Number of Patterns of Chords

Definition 1 (n-Scale) 1. If we divide one octave into n parts, we will speak of an n-scale. The objects 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 reason we define an n-scale as the cyclic group (Z_{n}, +) of order n.

Definition 2 (Transposing, Inversion) 1. Let us define T the operation of transposing as a permutation T: Z_{n} → Z_{n}, a ↦ T (a): = 1+a. The group ⟨T ⟩ is the cyclic group ζ_{n}^{(E)}.

2. Let us define I the operation of inversion as I: Z_{n} → Z_{n}, a ↦ I(a): = −a. The group ⟨T, I⟩ is the dihedral group ϑ_{n}^{(E)}.

Definition 3 (k-Chord) 1. Let k ≤ n. A k-chord in an n-scale is a subset of k elements of Z_{n}. An interval is a 2-chord.

2. Let G = ζ_{n}^{(E)} or G = ϑ_{n}^{(E)}. Two k-chords A_{1}, A_{2} are called equivalent iff there is some γ ∈ G such that A_{2} = γ(A_{1}).

Remark 4 1. We want to work with Pólya’s Theorem, therefore I identify each kchord A with its characteristic function χ_{A}. Two k-chords A_{1}, A_{2} are equivalent iff the two functions χ_{A_{1}} and χ_{A_{2}} are equivalent in the sense of Pólya’s Theorem.

2. Let us define two finite sets: P: = Z_{n} and F: = {0, 1}. Each function f ∈ F^{P} will be identified with A_{f}: = {k ∈ Z_{n}|f(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 ∈ F^{P} as

W (f): = ∏_{k∈Z_{n}} w(f(k)).

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

Applying Pólya’s Theorem of [1], we derive:

Theorem 5 (Patterns of k-Chords) 1. Let G be a permutation group on Z_{n}. The number of patterns of k-chords in the n-scale Z_{n} is the coefficient of x^{k} in

CI(G; 1 + x, 1 + x^{2}, …, 1 + x^{n}).

2. If G = ζ_{n}^{(E)}, the number of patterns of k-chords is \frac{1}{n} ∑_{j|gcd(n,k)} φ(j)( \frac{n}{j} ; \frac{k}{j} ), where φ

is Euler’s φ-function.

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

\frac{1}{2n}( ∑_{j|gcd(n,k)}φ(j)(\frac{n}{j} ; \frac{k}{j}) + n\binom{\frac{(n−1)}{2}}{[ \frac{k}{2} ]}) if n ≡ 1 mod 2
\frac{1}{2n}( ∑_{j|gcd(n,k)}φ(j)(\frac{n}{j} ; \frac{k}{j}) + n(\frac{n}{2} ; \frac{k}{2}))  if n ≡ 0 mod 2 and k ≡ 0 mod 2
\frac{1}{2n}( ∑_{j|gcd(n,k)}φ(j)(\frac{n}{j} ; \frac{k}{j}) + n\binom{\frac{n}{2} −1}{[ \frac{k}{2} ]}) if n ≡ 0 mod 2 and k ≡ 1 mod 2.

4. In the case n = 12 and G = ζ_{n}^{(E)}, we get the numbers in table 1.

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 ζ_{n}^{(E)}.

5. In the case n = 12 and G = ϑ_{n}^{(E)}, we get the numbers in table 2.

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 ϑ_{n}^{(E)}.

2.1.2 The Complement of a k-Chord

Definition 6 (Complement of a k-Chord) Let A ⊆ Z_{n} with |A| = k be a k-chord. The complement of A is the (n − k)-chord Z_{n} \ A.

Remark 7 1. Let G = ζ_{n}^{(E)} or G = ϑ_{n}^{(E)} be a permutation group on Z_{n} and let 1 ≤ k < n. There exists a bijection between the sets of patterns of k-chords and (n − k)-chords.

2. If n ≡ 0 mod 2, the complement of an \frac{n}{2}-chord is an \frac{n}{2}-chord. Now I want to figure out the number of patterns of \frac{n}{2}-chords [A] with the property A ∼ Z_{n} \ A. Applying the Theorem of [2] we get:

Theorem 8 1. Let n ≡ 0 mod 2. The number of patterns of \frac{n}{2}-chords which are equivalent to their complement, is CI(G; 0, 2, 0, 2, …).

2. If n = 12 and G = ζ_{n}^{(E)}, there are 20 patterns of 6-chords which are equivalent to their complement.

3. If n = 12 and G = ϑ_{n}^{(E)}, there are 8 patterns of 6-chords which are equivalent to their complement.

2.1.3 The Interval Structure of a k-Chord

In this section we use ϑ_{n}^{(E)} as the permutation group acting on Z_{n}. The set of all possible intervals between two differnet tones in n-tone music will be called Int(n), thus

Int(n): = {x − y|x, y ∈ Z_{n}, x ≠ y} = {1, 2, …, n − 1}.

Definition 9 (Interval Structure) On Z_{n} we define a linear order 0 < 1 < 2 < … < n − 1. Let A: = {i_{1}, i_{2}, …, i_{k}} be a k-chord. Without loss of generality let i_{1} < i_{2} < … < i_{k}. The interval structure of A is defined as the pattern [f_{A}], wherein the function f_{A} is defined as

f_{A}: {1, 2, …, k} → Int(n)
f_{A}(1): = i_{2} − i_{1}, f_{A}(2): = i_{3} − i_{2}, …, f_{A}(k − 1): = i_{k} − i_{k−1}, f_{A}(k): = i_{1} − i_{k},

and two functions f_{1}, f_{2}: {1, 2, …, k} → Int(n) are called equivalent, iff there exists some φ ∈ ϑ_{k}^{(E)} such that f_{2} = f_{1} ∘ φ. The group ϑ_{k}^{(E)} is generated by T̃ and Ĩ with T̃(i): = i + 1 mod k and Ĩ(i): = k + 1 − i for i = 1, …, k. The differences i_{j+1} − i_{j} must be interpreted as differences in Z_{n}. They are the intervals between the tones i_{j} and i_{j+1}.

Theorem 10 Let A_{1}: = {i_{1}, i_{2}, …, i_{k}} and A_{2}: = {j_{1}, j_{2}, …, j_{k}} be two k-chords with i_{1} < i_{2} < … < i_{k} and j_{1} < j_{2} < … < j_{k}. Furthermore let f: = f_{A_{1}} and g: = f_{A_{2}}: {1, 2, …, k} → Int(n) be constructed as in Definition 9. Then

[f] = [g] ⟺ [{i_{1}, i_{2}, …, i_{k}}] = [{j_{1}, j_{2}, …, j_{k}}].

I omit the proof of this theorem.

Remark 11 If the permutation group acting on Z_{n} is the cyclic group ζ_{n}^{(E)}, then the interval structure of A: = {i_{1}, i_{2}, …, i_{k}} must be defined as the pattern [f_{A}] in regard to ζ_{k}^{(E)}: = ⟨T̃⟩ with T̃(i): = i + 1 mod k. The function f_{A} is defined as in Definition 9.

Remark 12 Let f be a function f: {1, 2, …, k} → Int(n). The pattern [f] is the interval structure of a k-chord, iff ∑_{i=1}^{k} f(i) = n. One must interpret this sum as a sum of intervals, thus as a sum of positive integers.

Remark 13 Let x, y_{1}, y_{2}, …, y_{n} be indeterminates over Q and let R be the ring R: = Q[x, y_{1}, y_{2}, …, y_{n}]. Now I want to define a weight function w: Int(n) → R, i ↦ w(i): = x^{i}y_{i}. The weight of a function f: {1, 2, …, k} → Int(n) is the product weight

W (f): = ∏_{i=1}^{k} w(f(i)) = ∏_{i=1}^{k} x^{f(i)}y_{f(i)} = x^{∑_{i=1}^{k} f(i)} ∏_{i=1}^{k} y_{f(i)}.

Now we can define W ([f]): = W (f). According to Remark 12 the pattern [f] is the interval structure of a k-chord, iff ∑_{i=1}^{k} f(i) = n.This is true, iff W (f) = x^{n} ∏_{i=1}^{k} y_{f(i)}. The indices of the y’s in W (f) show, which intervals occur in the k-chord.

An Application of Pólya’s Theorem of [1] is

Theorem 14 The inventory of interval structures of k-chords in n-tone music is the

coefficient of x^{n} in CI(ϑ_{k}^{(E)}; ∑_{i=1}^{n−1} x^{i}y_{i}, ∑_{i=1}^{n−1} x^{2i}y_{i}^{2}, ∑_{i=1}^{n−1} x^{3i}y_{i}^{3}, …, ).

Example 15 The inventory of the interval structures of 3-chords in 12-tone music is the coefficient of x^{12} in

CI(ϑ_{3}^{(E)}; ∑_{i=1}^{11} x^{i}y_{i}, ∑_{i=1}^{11} x^{2i}y_{i}^{2}, ∑_{i=1}^{11} x^{3i}y_{i}^{3}).

This is

y_{1}^{2}y_{10} +y_{1}(y_{2}y_{9} +y_{3}y_{8} +y_{4}y_{7} +y_{5}y_{6})+y_{2}^{2}y_{8} +y_{2}(y_{3}y_{7} +y_{4}y_{6} +y_{5}^{2})+y_{3}^{2}y_{6} +y_{3}y_{4}y_{5} +y_{4}^{3}

If you are interested in the number of patterns of 3-chords with intervals ≥ k, then put y_{1}: = y_{2}: = …: = y_{k−1}: = 0 and y_{k}: = y_{k+1}: = …: = y_{n}: = 1. In the case k = 2 there are 7 patterns of 3-chords with intervals greater or equal 2.

2.2 Patterns of Tone-Rows

Definition 16 (Tone-Row, k-Row) 1. Arnold Schönberg introduced the so called tone-rows. Here I am going to give a mathematical form of his definition. Let n ≥ 3. A tone-row in an n-scale is a bijectiv mapping f: {0, 1, …, n − 1} → Z_{n}, i ↦ f(i). f(i) is the tone which occurs in ith position in the tone-row.

2. Let n ≥ 3 and 2 ≤ k ≤ n. A k-row in n-tone music is an injective mapping f: {0, 1, …, k − 1} → Z_{n}.

Remark 17 1. A k-row with k = n is a tone-row.

2. Two k-rows f_{1}, f_{2} are equivalent if f_{1} can be written as transposing, inversion, retrogradation or an arbitrary sequence of these operations of f_{2}.

Transposing of a k-row f is T ∘ f, Inversion of f is I ∘ f. According to Definition 2, we know that T and I are permutations on Z_{n}, and that ⟨T, I⟩ = ϑ_{n}^{(E)}. Actually inversion of a k-row f should be defined as T^{f(0)} ∘ I ∘ T^{−f(0)} ∘ f. Retrogradation R, is a permutation R ∈ S_{{0,1,...,k−1}} defined as:

R: = \{ \begin{matrix}(0, k − 1) ∘ (1, k − 2) ∘ … ∘ (\frac{k}{2} − 1, \frac{k}{2} ) & if k ≡ 0 mod 2 \\ (0, k − 1) ∘ (1, k − 2) ∘ … ∘ (\frac{k−3}{2}, \frac{k+1}{2}) ∘ (\frac{k−1}{2}) & if k ≡ 1 mod 2. \end{matrix}

Let Π: = ⟨R⟩ ≤ S_{{0,1,...,k−1}}, then |Π| = 2. Retrogradation of a k-row f is defined as f ∘ R.

3. Since Π: = ⟨R⟩, the cycle index of Π is

CI(Π; y_{1}, y_{2}, …, y_{k}) = \{ \begin{matrix}\frac{1}{2}(y_{1}^{k} + y_{2}^{\frac{k}{2}} ) & if k ≡ 0 mod 2 \\ \frac{1}{2}(y_{1}^{k} + y_{1}y_{2}^{\frac{k−1}{2}} ) & if k ≡ 1 mod 2. \end{matrix}

Thus two k-rows f_{1}, f_{2} are equivalent iff ∃φ ∈ ϑ_{n}^{(E)}∃σ ∈ Π such that f_{1} = φ ∘ f_{2} ∘ σ.

Applying Theorem 5.2 of [1], we get

Theorem 18 (Number of Patterns of k-Rows) The number of patterns of k-rows in

Z_{n} is CI(Π; \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{k}} ) CI(ϑ_{n}^{(E)}; 1 + x_{1}, 1 + 2x_{2}, …, 1 + nx_{n})|_{x_{1}=x_{2}=...=x_{n}=0}.

This is 1.

\frac{1}{2}(\frac{1}{4}((2)_{k} + 2^{\frac{k}{2}} (\frac{k}{2} )!((\frac{n}{2} ; \frac{k}{2}) + (\frac{n−2}{2} ; \frac{k}{2}))) + \frac{1}{2n}(\binom{n}{k}k! + 2^{\frac{k}{2}} (\frac{k}{2} )!(\frac{n}{2} ; \frac{k}{2}))),

if n ≡ 0 mod 2 and k ≡ 0 mod 2. For integers k, v, v ≥ 0 the expression (k)_{v} is definied as:

(k)_{v}: = k · (k − 1) · . . . · (k − (v − 1)).

2.

\frac{1}{2}(\frac{1}{4} · 2 · 2^{\frac{k−1}{2}} (\frac{n−2}{2} ; \frac{k−1}{2})(\frac{k − 1}{2})! + \frac{1}{2n}\binom{n}{k}k!),

if n ≡ 0 mod 2 and k ≡ 1 mod 2. 3.

\frac{1}{2}(\frac{1}{2n}\binom{n}{k}k! + \frac{1}{2}2^{\frac{k}{2}} (\frac{n−1}{2} ; \frac{k}{2})(\frac{k}{2} )!),

if n ≡ 1 mod 2 and k ≡ 0 mod 2. 4.

\frac{1}{2}(\frac{1}{2n}\binom{n}{k}k! + \frac{1}{2}2^{\frac{k−1}{2}} (\frac{n−1}{2} ; \frac{k−1}{2})(\frac{k − 1}{2})!),

if n ≡ 1 mod 2 and k ≡ 1 mod 2. In the case n = 12 the number of patterns of k-rows is in table 3.

k  2 3 4  5  6  7 ; # of patterns 6 30 275 2 000 14 060 83 280
k  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.

The special case of Theorem 18 for k = n is Theorem 19 (Number of patterns of Tone-Rows) Let n ≥ 3. The number of patterns of tone-rows in n-tone music is

\frac{1}{4}((n − 1)! + (n − 1)!!)  if n ≡ 1 mod 2
\frac{1}{4}((n − 1)! + (n − 2)!!(\frac{n}{2} + 1)) if n ≡ 0 mod 2.

If n is in N then

n!! = \{ \begin{matrix}n · (n − 2) · . . . · 2 & if n ≡ 0 mod 2 \\ n · (n − 2) · . . . · 1 & if n ≡ 1 mod 2. \end{matrix}

Especially there are 9 985 920 patterns of tone-rows in 12-tone music.

2.3 Patterns of All-Interval-Rows

Let A and B be two finite sets. The set of all injective functions f: A → B will be denoted by Inj(A, B). For that reason the set of all tone-rows is Inj({0, 1, …, n − 1}, Z_{n}). In this chapter let n ≥ 3.

Definition 20 (All-Interval-Rows) Let us define a mapping

α: Inj({0, 1, …, n − 1}, Z_{n}) → {g|g: {1, 2, …, n − 1} → Int(n)}
f ↦ α(f)

and α(f)(i): = f(i)−f(i−1) for i = 1, 2, …, n−1. This is subtraction in Z_{n}. The function α(f) is called all-interval-row, iff α(f) is injective, that means α(f) ∈ Inj({1, 2, …, n − 1}, Int(n)). In other words a tone-row induces an all-interval-row, iff all possible intervals occur as differences between two successive tones of the tone-row. The set of all allinterval-rows will be denoted as Allint(n).

Let’s define some mappings:

1.

β: Inj({1, 2, …, n − 1}, Int(n)) → {g|g: {0, 1, …, n − 1} → Z_{n}}
f ↦ β(f)

β(f)(0): = 0 and β(f)(i): = β(f)(i − 1) + f(i) mod n for i = 1, 2, …, n − 1. You can easily derive that for i = 0, 1, …, n − 1

β(f)(i) ≡ ∑_{j=1}^{i} f(j) mod n.

2. Let l ∈ Z_{n}.

β̃: Inj({1, 2, …, n − 1}, Int(n)) → {g|g: {0, 1, …, n − 1} → Z_{n}}

f ↦ β̃(f),  β̃(f)(i) ≡ ∑_{j=1}^{i} f(j) + l mod n.

Theorem 21 Let f be a mapping f: {1, 2, …, n − 1} → Int(n). The following statements are equivalent:

1. f is an all-interval-row.

2. f ∈ Inj({1, 2, …, n − 1}, Int(n)) and β(f) ∈ Inj({0, 1, …, n − 1}, Z_{n}).

3. f ∈ Inj({1, 2, …, n − 1}, Int(n)) and β̃(f) ∈ Inj({0, 1, …, n − 1}, Z_{n}).

The proof is omitted. You can easily prove the following results:

1. If n ≡ 1 mod 2, there are no all-interval-rows.

2. If n ≡ 0 mod 2 the function f defined as

f(i): = \{ \begin{matrix}i & if i ≡ 1 mod 2 \\ −i & if i ≡ 0 mod 2 \end{matrix}

is an all-interval-row. For the rest of this chapter let n ≥ 4 and n ≡ 0 mod 2.

3. f ∈ Allint(n) implies β(f)(n − 1) = \frac{n}{2}.

4. f ∈ Allint(n) implies f(1) ≠ \frac{n}{2} and f(n − 1) ≠ \frac{n}{2}.

Remark 22 1. On Int(n) we have the following permutations:

I: Int(n) → Int(n), j ↦ I(j): = n − j.

I stands for inversion. I is of the type (1, \frac{n}{2} − 1, 0, …). In the case n = 12 there is a further permutation called

Q: Int(n) → Int(n), j ↦ Q(j): ≡ 5 · j mod 12.

Q stands for quartcircle symmetry. Since gcd(5, 12) = 1, Q is a permutation on Z_{n}, and since 5 · 0 = 0, Q is a permutation on Int(n). Q is of the type (3, 4, 0, …, 0). You can easily prove that (I ∘ Q)(j) = (Q ∘ I)(j) = 7 · j mod 12 and that it is of the type (5, 3, 0, …, 0). I ∘ Q is called quintcircle symmetry.

2. On the set {1, 2, …, n − 1} retrogradation R is a permutation, defined as

R: = (1, n − 1) ∘ (2, n − 2) ∘ … ∘ (\frac{n}{2} − 1, \frac{n}{2} + 1) ∘ (\frac{n}{2}).

3. If f ∈ Allint(n), then I ∘ f, f ∘ R are in Allint(n). Furthermore if n = 12 then Q ∘ f ∈ Allint(12).

4. For that reason we can define the following permutations on Allint(n).

φ_{I}, φ_{R}, φ_{Q}: Allint(n) → Allint(n)
f ↦ φ_{I}(f): = I ∘ f, f ↦ φ_{R}(f): = f ∘ R, f ↦ φ_{Q}(f): = Q ∘ f.

For φ_{Q} we need the assumption that n = 12.

5. It is easy to prove that these permutations commute in pairs and that φ_{I}^{2} = φ_{R}^{2} = φ_{Q}^{2} = id.

6. In [4] there is a further permutation E called exchange at \frac{n}{2}. It is defined as

E: Allint(n) → Allint(n), f ↦ E(f)

and

E(f)(i): = \{ \begin{matrix}f(f^{−1}(\frac{n}{2}) + i) & if i < n − f^{−1}(\frac{n}{2}) \\ \frac{n}{2} & if i = n − f^{−1}(\frac{n}{2}) \\ f(i − n + f^{−1}(\frac{n}{2})) & if i > n − f^{−1}(\frac{n}{2}). \end{matrix}

I have already mentioned, that f(1) ≠ \frac{n}{2} and f(n − 1) ≠ \frac{n}{2}. Since f ∈ Allint(n) is bijective, there exists exactly one j, such that 1 < j < n−1 and f(j) = \frac{n}{2}. The values of the function E(f)(i) for i = 1, 2, …, n−1 are f(j+1), f(j+2), …, f(n−1), f(j) = \frac{n}{2}, f(1), f(2), …, f(j − 1). The permutation E is defined for n ≥ 4, but in the case n = 4 we have E = φ_{R}.

7. The following formulas hold: E ∘ φ_{I} = φ_{I} ∘ E, E ∘ φ_{Q} = φ_{Q} ∘ E, E ∘ φ_{R} = φ_{R} ∘ E and E^{2} = id.

8. Let us define three permutation groups on Allint(n).

G_{1}: = ⟨φ_{I}, φ_{R}⟩, G_{2}: = ⟨φ_{I}, φ_{R}, E⟩ und G_{3}: = ⟨φ_{I}, φ_{R}, E, φ_{Q}⟩. For G_{2} we must assume n ≥ 6, and for G_{3} we must assume n = 12. We calculate that |G_{1}| = 4, |G_{2}| = 8, |G_{3}| = 16.

Remark 23 (Counting of All-Interval-Rows) Let

x_{1}, x_{2}, …, x_{n−1}, y_{1}, y_{2}, …, y_{n−1}, z_{1}, z_{2}, …, z_{n−1}

be indeterminates over Q. Furthermore let f be a mapping f: {1, 2, …, n − 1} → Int(n). We define R: = Q[x_{1}, x_{2}, …, x_{n−1}, z_{1}, z_{2}, …, z_{n−1}] and

W (f): = ∏_{i:=1}^{n−1} w_{i}(f(i)).

The functions w_{i} are defined as w_{i}: Int(n) → R, j ↦ w_{i}(j): = z_{j} ∏_{ν:=i}^{n−1} x_{ν}^{j}. After calculating W (f) you have to replace terms of the form x_{ν}^{j} by y_{j mod n}. Then you get W̃ (f) ∈ Q[y_{1}, y_{2}, …, y_{n−1}, z_{1}, z_{2}, …, z_{n−1}]. According to Theorem 21 f is an all-interval-row, if and only if, W̃ (f) = ∏_{i=1}^{n−1} y_{i}z_{i}. Consequently the number of all-interval-rows in n-tone music is the coefficient of ∏_{i=1}^{n−1} y_{i}z_{i} in

∏_{i=1}^{n−1}(∑_{j=1}^{n−1} z_{j} ∏_{k=i}^{n−1} x_{k}^{j})|_{x_{ν}^{j}=y_{j mod n}}.

Remark 24 For φ ∈ G_{1} or G_{2} or G_{3} we want to calculate

χ(φ): = |{f ∈ Allint(n)|φ(f) = f}|.

After some calculations we can derive that there are only 4 permutions φ such that χ(φ) ≠ 0. In Remark 23 we calculated χ(id). The value of χ(φ_{I} ∘φ_{R}) is the coefficient of ∏_{i=1}^{n−1} y_{i}z_{i} in

∏_{i=1}^{\frac{n}{2} −1}( ∑_{{j=1 \atop j≠ \frac{n}{2}}}^{n−1} z_{j}z_{n−j} ∏_{k=i}^{n−1} x_{k}^{j} ∏_{k=n−i}^{n−1} x_{k}^{n−j})z\frac{n}{2} ∏_{k= \frac{n}{2}}^{n−1} x_{k} ^{\frac{n}{2}} |_{x_{ν}^{j}=y_{j mod n}}.

Now let n ≥ 6. The value of χ(φ_{I} ∘ V) is the coefficient of ∏_{i=1}^{n−1} y_{i}z_{i} in

∏_{i=1}^{\frac{n}{2} −1}( ∑_{{j=1 \atop j≠ \frac{n}{2}}}^{n−1} z_{j}z_{n−j} ∏_{k=i}^{n−1} x_{k}^{j} ∏_{k=( \frac{n}{2} +i)}^{n−1} x_{k}^{n−j})z\frac{n}{2} ∏_{k= \frac{n}{2}}^{n−1} x_{k} ^{\frac{n}{2}} |_{x_{ν}^{j}=y_{j mod n}}.

Now let n = 12. In order to calculate χ(φ_{Q} ∘ V ∘ φ_{R}) you must compute

∑_{i=1}^{5}(z_{6} ∏_{j=2i}^{11} x_{j}^{6}z_{3}z_{9}(∏_{j=i}^{11} x_{j}^{3} ∏_{j=i+6}^{11} x_{j}^{9} + ∏_{j=i}^{11} x_{j}^{9} ∏_{j=i+6}^{11} x_{j}^{3})·

· ∏_{j=1}^{i−1}( ∑_{{k=1 \atop k∉\{3,6,9\}}}^{n−1} z_{k}z_{5k mod 12} ∏_{l=j}^{11} x_{l}^{k} ∏_{l=2i−j}^{11} x_{l}^{5k mod 12})·

· ∏_{j=2i+1}^{i+5}( ∑_{{k=1 \atop k∉\{3,6,9\}}}^{n−1} z_{k}z_{5k mod 12} ∏_{l=j}^{11} x_{l}^{k} ∏_{l=12+2i−j}^{11} x_{l}^{5k mod 12})).

Then substitute y_{j mod 12} for x_{ν}^{j} and find the coefficient of ∏_{i=1}^{11} y_{i}z_{i}.

Theorem 25 (Number of Patterns of All-Interval-Rows) For i = 1, 2, 3 the number of patterns of all-interval-rows in regard to G_{i} is

1. \frac{1}{4}(χ(id) + χ(φ_{I} ∘ φ_{R})) for i = 1.

2. \frac{1}{8}(χ(id) + χ(φ_{I} ∘ φ_{R}) + χ(φ_{I} ∘ V)) for i = 2.

3. For i = 3 we calculate

\frac{1}{16}(χ(id) + χ(φ_{I} ∘ φ_{R}) + χ(φ_{I} ∘ V) + χ(φ_{Q} ∘ φ_{R} ∘ V)) =

= \frac{1}{16}(3 856 + 176 + 120 + 120) = 267.

This is an application of the Lemma of Bunside of [1].

2.4 Patterns of Rhythms

Definition 26 (n-Bar, Entry-time, k-Rhythm) A bar is an important contribution in a composition. Usually a lot of bars of the same form follow one another. If you know the smallest rhythmical subdivision of a bar, you can figure out how many entry-times (think of rhythmical accents played on a drum) a bar holds. If there are n entry-times in a bar, I call it an n-bar. In mathematical terms an n-bar is expressed as the cyclic group Z_{n}. We can define cyclic temporal shifting S as a permutation S: Z_{n} → Z_{n}, t ↦ S(t): = t + 1. Retrogradation R (temporal inversion) is defined as R: Z_{n} → Z_{n}, t ↦ R(t): = −t. The group ⟨S⟩ is ζ_{n}^{(E)} and ⟨S, R⟩ = ϑ_{n}^{(E)}. A k-rhythm in an n-bar is a subset of k elements of Z_{n}. The permutation groups ζ_{n}^{(E)} or ϑ_{n}^{(E)} 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 5.

2.5 Patterns of Motifs

Definition 27 (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 ∈ Z_{m}, y ∈ Z_{n}} = Z_{m} × Z_{n}

is the set of all possible combinations of entry-times in the m-bar Z_{m} and pitches in the n-scale Z_{n}. Furthermore let G be a permutation group on M. In Remark 29 we are going to study two special groups G. The group G defines an equivalence relation on M:

(x_{1}, y_{1}) ∼ (x_{2}, y_{2}): ⟺ ∃g ∈ G with (x_{2}, y_{2}) = g(x_{1}, y_{1}).

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.

Theorem 28 (Number of Patterns of k-Motifs) The number of patterns of k-motifs in an n-scale and in an m-bar is the coefficient of x^{k} in

CI(G; 1 + x, 1 + x^{2}, …, 1 + x^{m·n}).

This completely follows from Pólya’s Theorem of [1].

Remark 29 (Special Permutation Groups) Now I want to demonstrate two examples for group G.

1. In Definition 2 we had a permutation group G_{2} = ζ_{n}^{(E)} or G_{2} = ϑ_{n}^{(E)} acting on the n-scale Z_{n}. Moreover in Definition 26 there was a permutation group G_{1} = ζ_{m}^{(E)} or G_{1} = ϑ_{m}^{(E)} defined on the m-bar Z_{m}. For that reason, we define the group G as G: = G_{1} ⊗G_{2}. Two elements (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ M are called equivalent with respect to G, iff there exist φ ∈ G_{1} and ψ ∈ G_{2}, with

(x_{2}, y_{2}) = (φ, ψ)(x_{1}, y_{1}) = (φ(x_{1}), ψ(y_{1})).

Because of the fact that we know how to calculate the cycle index of G_{1} ⊗ G_{2}, we can compute the 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 G is defined as G: = ⟨T, S, φ_{A}|A ∈ Gl (2, Z_{n})⟩, with

T: M → M, ( \begin{matrix}x \\ y \end{matrix} ) ↦ T ( \begin{matrix}x \\ y \end{matrix} ): = ( \begin{matrix}x \\ y + 1 \end{matrix} )

S: M → M, ( \begin{matrix}x \\ y \end{matrix} ) ↦ S ( \begin{matrix}x \\ y \end{matrix} ): = ( \begin{matrix}x + 1 \\ y \end{matrix} )

φ_{A}: M → M, ( \begin{matrix}x \\ y \end{matrix} ) ↦ φ_{A} ( \begin{matrix}x \\ y \end{matrix} ): = A ( \begin{matrix}x \\ y \end{matrix} ) .

The multiplication A · ( \begin{matrix}x \\ y \end{matrix} ) stands for matrix multiplication. The set Gl (2, Z_{n}) is

the group of all regular 2 × 2-matrices over Z_{n}.

You can easily derive the following results:

(a) T^{n} = S^{n} = id_{M} and T^{j} ≠ id_{M} and S^{j} ≠ id_{M} 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: T^{i} ∘ S^{j} ∉ ⟨φ_{A}|A ∈ Gl (2, Z_{n})⟩, iff i ≠ 0 or j ≠ 0.

(d) Let A: = ( \begin{matrix}a & b \\ c & d \end{matrix} ), then: φ_{A} ∘ T^{k} ∘ S^{l} = T^{(cl+dk)} ∘ S^{(al+bk)} ∘ φ_{A}.

(e) G is the group of all affine mappings Z_{n}^{2} → Z_{n}^{2}.

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

Example 30 Let us consider the case, that n = m = 12.

1. If G is defined as G: = ϑ_{n}^{(E)} ⊗ ϑ_{n}^{(E)}, then we derive

CI(G; x_{1}, x_{2}, …, x_{144}) =

= \frac{1}{576}(x_{1}^{144} +12x_{1}^{24}x_{2}^{60} +36x_{1}^{4}x_{2}^{70} +147x_{2}^{72} +8x_{3}^{48} +24x_{3}^{8}x_{6}^{20} +60x_{4}^{36} +96x_{6}^{24} +192x_{12}^{12}).

By applying Theorem 28, the number of patterns of k-motifs is the coefficient of x^{k} in 1 + x + 48x^{2} + 937x^{3} + 31 261x^{4} + 840 006x^{5}19 392 669x^{6} + 381 561 281x^{7} + 6 532 510 709x^{8} + 98 700 483 548x^{9} + 1 332 424 197 746x^{10} + . . ..

2. If G: = ⟨T, S, φ_{A}|A ∈ Gl (2, Z_{n})⟩, I computed the cycle index of G with a Turbo Pascal program as

CI(G; x_{1}, x_{2}, …, x_{144}) = \frac{1}{663 552}(x_{1}^{144} + 18x_{1}^{72}x_{2}^{36} + 36x_{1}^{48}x_{2}^{48} + …).

By applying Theorem 28, the number of patterns of k-motifs is the coefficient of x^{k} in 1 + x + 5x^{2} + 26x^{3} + 216x^{4} + 2 024x^{5} + 27 806x^{6} + 417 209x^{7} + 6 345 735x^{8} + 90 590 713x^{9} + 1 190 322 956x^{10} + . . .. For k = 1, 2, 3, 4 these numbers are the same as in [8]. In the case k = 5 however, it is stated that there exist 2 032 different patterns of 5-motifs, while here we get 2 024 of these patterns.

2.6 Patterns of Tropes

Definition 31 (Trope) 1. If you divide the set of 12 tones in 12-tone music into 2 disjointed sets, each containing 6 elements, and if you label these sets as a first and a second set, we will speak of a trope. This definition goes back to Josef Matthias Hauer. Two tropes are called equivalent, iff transposing, inversion, changing the labels of the two sets or arbitrary sequences of these operations transform one trope into the other.

2. For a mathematical definition let n ≥ 4 and n ≡ 0 mod 2. A trope in n-tone music is a function f: Z_{n} → F: = \{1, 2\} such that |f^{−1}(\{1\})| = |f^{−1}(\{2\})| = \frac{n}{2}. f(i) = k is translated into: The tone i lies in the set with label k. Furthermore T and I are permutations on Z_{n} as in Definition 2. The group ⟨T, I⟩ is ϑ_{n}^{(E)}. Two tropes f_{1}, f_{2} are called equivalent, if and only if, ∃π ∈ ϑ_{n}^{(E)} ∃φ ∈ S_{2} such that f_{2} = φ^{−1} ∘ f_{1} ∘ π.

3. Let x and y be indeterminates over Q. Define a function w: F → Q[x, y] by w(1): = x and w(2): = y. For f ∈ F^{Z_{n}} the weight of f is defined as product weight

W (f): = ∏_{x∈Z_{n}} w(f(x)).

A function f: Z_{n} → F: = \{1, 2\} is a trope, iff W (f) = x^{\frac{n}{2}} y^{\frac{n}{2}} .

Theorem 32 (Patterns of Tropes) Let φ be Euler’s φ-function. The number of patterns of tropes in regard to ϑ_{n}^{(E)} is

\frac{1}{4}(\frac{1}{n}(∑_{t| \frac{n}{2}}φ(t)( \frac{n}{t} ; \frac{n}{2t}) + ∑_{{t|n \atop t≡0 mod 2}}φ(t)2^{\frac{n}{t}} ) + (\frac{n}{2} ; \frac{n}{4}) + 2^{\frac{n}{2} −1}) if n ≡ 0 mod 4

\frac{1}{4}(\frac{1}{n}(∑_{t| \frac{n}{2}}φ(t)( \frac{n}{t} ; \frac{n}{2t}) + ∑_{{t|n \atop t≡0 mod 2}}φ(t)2^{\frac{n}{t}} ) + (\frac{n−2}{2} ; \frac{n−2}{4}) + 2^{\frac{n}{2} −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 Z_{n} was the cyclic group ⟨T ⟩.

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

2.7 Special Remarks on 12-tone music

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

Remark 33 (Quartcircle Symmetry) The quartcircle symmetry Q is defined as

Q: Z_{12} → Z_{12}, x ↦ Q(x): = 5x.

Q is a permutation on Z_{12}, since gcd(5, 12) = 1. Furthermore Q ∉ ⟨I, T ⟩, Q ∘ T = T^{5} ∘ Q, Q^{2} = id_{Z_{12}} and Q ∘ I = I ∘ Q = 7x, which is called the quintcircle symmetry. Let G be G: = ⟨I, T, Q⟩. Each element φ ∈ G can be written as φ = T^{k} ∘ I^{j} ∘ Q^{l} such that k ∈ {0, 1, …, n − 1}, j ∈ {0, 1}, and l ∈ {0, 1}. The cycle index of G: = ⟨I, T, Q⟩ is

CI(G; x_{1}, x_{2}, …, x_{12}) =

= \frac{1}{48}(∑_{t|12} φ(t)x_{t}^{\frac{12}{t}} + 2x_{1}^{6}x_{2}^{3} + 3x_{1}^{4}x_{2}^{4} + 6x_{1}^{2}x_{2}^{5} + 11x_{2}^{6} + 4x_{3}^{2}x_{6} + 6x_{4}^{3} + 4x_{6}^{2}).

This group G is an other permutation group acting on Z_{12} with a musical background. The question arises, how to generalize the quartcircle symmetry of 12-tone music to n-tone music. Should we take any unit in Z_{n} or only those units e such that e^{2} = 1 ?

References

[1] N.G. De Bruijn. Pólya’s Theory of Counting. In E.F. Beckenbach, editor, Applied Combinatorial Mathematics, chapter 5, pages 144 – 184. Wiley, New York, 1964.

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

[3] N.G. De Bruijn. A Survey of Generalizations of Pólya’s Enumeration Theorem. Nieuw Archief voor Wiskunde (2), XIX:89 – 112, 1971.

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

[5] H. Florey. Analytische Bemerkungen zu Josef Matthias Hauers letztem Zwölftonspiel. Beilage zu einer Schallplatte herausgegeben von der Hochschule für Musik und darstellende Kunst in Graz, 1988.

[6] H. Fripertinger. Untersuchung über die Anzahl verschiedener Intervalle, Akkorde, Tonreihen und anderer musikalischer Objekte in n-Ton Musik. Master’s thesis, Hochschule für Musik und Darstellende Kunst, Graz, 1991.

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

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

[9] G. Pólya. Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Mathematica, 68:145 – 254, 1937.

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

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

Address of the author:

H. Fripertinger Universität Graz Institut für Mathematik Heinrichstr. 36 A–8010 Graz, Austria