Enumeration in Music Theory

James Barr

Western New England University

Department of Mathematics

May 10, 2014

Contents

1 Introduction 2

1.1 Intervals, Chords, Scales, and Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Enumeration of k-note lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Differentiating k-note lines 5

2.1 Genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Enumeration of genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Circle of fifths 10

3.1 Proof of Theorem 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Enumeration of chords 12

5 Conclusion 14

1

1 Introduction

Mathematics can be used to understand the complexity of music by defining musical terms

and aspects such as tones, intervals, scales and rhythm. The mathematical aspects of music

have a remarkable collection of number properties simply from the fact that nature in itself

holds and can be represented by mathematics. To attempt to structure new ways of com-

posing, hearing and understanding music, links between different math fields such as graph

theory, group and set theory, number theory and music have all been made which have led

to many musical applications.

Music is an art form that is all around us. Defined by its musical tones of composed

harmonies and melodies, music is an important part in our lives. One can really think of

music in two different ways, atonal music and tonal music. Atonal music can be thought

of as a lack of a key, specifically a key signature. It does not take into account and avoids

the use of major and minor chords, scales, keys, rhythm, repetition, etc. It focuses more on

individual sounds and subjectivity; this type of music is not as commonly used by composers.

This is different from tonal music which conforms to major and minor keys. It uses chords

and scales where relationships between chords and scales are most abundant.

1.1 Intervals, Chords, Scales, and Lines

2

A tone is an individual note, looking at the above figure [5], each individual note on the

piano is a tone. A sequence of two notes is called an interval ; for example C − F is an

interval. The chromatic set is the set of tones corresponding to both the black and white

keys on a piano which is illustrated in the figure above and listed below.

{C, C#/Db, D, D#/Eb, E, F, F#/Gb, G, G#/Ab, A, A#/Bb, B}

The diatonic set is the set of all white keys on a piano {C,D,E, F,G,A,B}. The distance

between two consecutive elements of the chromatic set is a semitone. The figure above

illustrates the chromatic set and the diatonic set. Notice each black key on a piano has

two different names. Consider Eb; it is one semitone down from E. However, it can also be

thought of as being one semitone above D, also known as D#. For an interval, the diatonic

length is the number of steps needed in the diatonic scale to move from the first note of the

interval to the second. Similarly, the chromatic length is the numbers of steps needed in

the chromatic scale to move from the first note to the second. The direction of movement

from the first note to the second is always up the scale, wrapping around to the beginning, if

needed. For example, the interval (C−F ) has a diatonic length of 3 and a chromatic length

of 5.

A chord is an unordered nonempty subset of the chromatic set. For example, {B,C,E}

is a chord. A scale is like a chord, except it is an ordered nonempty sequence of notes of

the chromatic set. For example, (B,C,E) and (E,B,C) are two different three note scales,

but they are the same three note chord. A scale of k distinct notes is called a k-note line;

so we can think of a line and scale as being the same thing. Because they seem the same

at face value, why do we distinguish the two apart? We distinguish a line and a scale apart

3

because lines can be manipulated and represented more by a mathematical standpoint. For

the main purposes of this paper, we shall restrict ourselves to the usage of k-note lines.

We can now extend the notion of intervals to k-note lines. We can represent the chromatic

and diatonic lengths of a k-note line with length arrays. Diatonic and chromatic length arrays

measure the diatonic and chromatic lengths between each note of the k-note line respectively.

So given a k-note line with notes (n1, n2..., nk), the diatonic length array is calculated

< d1, d2, ..., dk >D=< n2 − n1, n3 − n2, ..., nk − nk−1, n1 − nk >D

where all subtraction is performed modulo 7, or di = (di+1 − d1) mod 7

If all notes are in increasing order, thenk∑

i=1

d1 = 7. For example, consider the three note

line (C − E − F ), it would have a diatonic length array of < 2, 1, 4 >D, and a chromatic

length array of < 4, 1, 7 >C .

1.2 Enumeration of k-note lines

How many different k-note lines are there in the musical universe?

Theorem 1.2.1: For the standard diatonic set {C,D,E, F,G,A,B}, there are a total of

13, 699 different k-note lines, 1 ≤ k ≤ 7.

Proof: A simple counting argument can be applied to find all of these different k-note lines.

There are 7 notes of the diatonic set {C,D,E, F,G,A,B}. Finding all possible k-note lines

is the same as finding all k-permutations of the set {C,D, ..., A,B} for all k, 1 ≤ k ≤ 7. For

example, for a 1-note line, there are a total of 7 different notes to choose from. For a 2-note

line, the first note can be any of the 7 notes but, since the second note must be different

4

from the first, there are now only 6 choices for the second note. Extending this now for

1 ≤ k ≤ 7, we can generalize this with the following formula

7∑k=1

P (7, k)

=7∑

k=1

7!(7−k)!

= 7 + 7 ∗ 6 + 7 ∗ 6 ∗ 5 + ...+ 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1

= 13, 699

QED

This type of argument could be generalized to any other musical universe.

Theorem 1.2.2: For a diatonic set with d notes, there are a total ofd∑

k=1

P (d, k) different

k-note lines, 1 ≤ k ≤ d.

2 Differentiating k-note lines

2.1 Genus

So, is there a way to classify all 13, 699 k-note lines? We can, with the help of the concept

of genus and species. Two k -note lines are in the same genus if they have the same diatonic

length array. For example, the genus of (C − E − F ) is the set

(C − E − F,D − F −G,E −G− A,F − A−B,G−B − C,A− C −D,B −D − E)

where this diatonic array is fixed under rotations in the diatonic set. This can be shown

pictorally; consider the figure below [7]. We fix the length array of the k-note line by drawing

5

lines going from each note to the next. Then, rotating about the circle, all other lines of the

same genus can be found.

Observe, that we can associate the notes of the diatonic set {C,D,E, F,G,A,B} with

the numbers {0, 1, 2, 3, 4, 5, 6} respectively. Thus we can calculate the diatonic length of an

interval using modular subtraction in Z7. For example, the length of (C − F ) would be

(5 − 2) mod 7 = 3 while the length of (F − C) would be (2 − 5) mod 7 = −3 mod 7 = 4.

Similarly, for the chromatic length of an interval, we use modular subtraction in Z12.

2.2 Enumeration of genus

How many different genera are there in the musical universe? To answer this, we shall use

a helpful counting theorem.

Theorem 2.2.1: The number of positive integral solutions to the equation

x1 + x2 + ...+ xk = n is(n−1k−1

).

Theorem 2.2.2: For the standard diatonic set {C,D,E, F,G,A,B,C} there are 1, 957

different genera.

6

The counting argument for finding these genera proves more difficult than k-note lines. To

be in the same genus, two k-note lines must have the same diatonic length array. Thus,

we can equivalently count all possible diatonic length arrays. Assuming the notes are in

increasing order,k∑

i=1

di = 7. With all of this in mind, finding all the different genera becomes

easier. Consider the different integer partitions of 7, that is, solving the equation

x1 + x2 + x3 + ...+ xk = 7 where xi ≥ 1

Each k-note line with notes in increasing order corresponds to a diatonic length array

< d1, d2, ..., dk > such thatk∑

i=1

di = 7, or equivalenty, to an integer partition of 7 into k positive

summands. Now lets extend this for the different values of k where 1 ≤ k ≤ 7 of k-note lines.

Without loss of generality, to find the arrays that are out of order, fix the first note in the

k-note line and then find the different permutations of the remaining notes to find the k-note

lines that are out of order. For example, look at < 1, 1, 5 > an array of increasing order.

Reordering the elements of this interval array gives us < 1, 1, 5 >,< 1, 5, 1 >,< 5, 1, 1 >.

Then we fix the first note, and permute the rest.

Proof:

k = 1: There is only one genus, namely < 7 >

< 7 >

= 1 genus

k = 2: There are 6 genera, the ones in order are < 1, 6 >,< 2, 5 >,< 3, 4 >, and the ones

out of order are < 6, 1 >,< 5, 2 >,< 4, 3 >.

< 1, 6 >,< 2, 5 >,< 3, 4 >,< 6, 1 >,< 5, 2 >,< 4, 3 >

7

= 6 genera

k = 3: Once we get to k = 3 there are 30 genera. The 4 integer partitions with summands

listed in increasing order are < 1, 1, 5 >,< 1, 2, 4 >,< 1, 3, 3 >,< 2, 2, 3 >. Look at the set

< 1, 1, 5 >, this corresponds to the diatonic array < 1, 1, 5 >D which is the three note line

(A,B,G). These notes are in clockwise order. Now we find all other 3-note lines using these

3 notes by fixing the note, A, as the first note and then permuting the remaining 2 notes,

which have 2 possible reorderings, (A,B,G) = < 1, 1, 5 >, and (A,G,B) = < 6, 2, 6 >. Again,

reordering the elements of the interval array gives the first value of the product, then fixing

the first note and permuting the remaining notes gives the second value of the product.

< 1, 1, 5 >,< 1, 2, 4 >,< 1, 3, 3 >,< 2, 2, 3 >

= 3 · 2 + 6 · 2 + 3 · 2 + 3 · 2 = 30 genera

A similar argument can now be made for the other summands and the overall argument

can be now extended for the remaining k values. The remaining different integer partitions

of 7 for the different k-note lines where 1 ≤ k ≤ 7 can be seen below.

k = 4:

< 1, 1, 1, 4 >,< 1, 1, 2, 3 >,< 1, 2, 2, 2 >

= 4 · 6 + 12 · 6 + 4 · 6 = 120 genera

k = 5:

< 1, 1, 1, 1, 3 >,< 1, 1, 1, 2, 2 >

= 5 · 4! +(52

)· 4! = 360 genera

k = 6:

8

< 1, 1, 1, 1, 1, 2 >

= 6 · 5! = 720 genera

k = 7:

< 1, 1, 1, 1, 1, 1, 1 >

= 1 · 6! = 720 genera

Adding up all the different genera for k-note lines where 1 ≤ k ≤ 7 gives the 1, 957 different

genera as required.

1 + 6 + 30 + 120 + 360 + 720 + 720 = 1, 957 genera

This type of counting argument where we find the diatonic arrays of consecutive order,

then reorder them and then find the remaining diatonic arrays by fixing the first note and

choosing the positions for the rest of the notes can be simplified. We can count all integer

partitions of 7, which corresponds to the following formula.

7∑k=1

(6

k−1

)(k − 1)! = 1, 957

QED

2.3 Species

We can now differentiate k-note lines of the same genus even further with species. Two

k -note lines are in the same species if they have the same chromatic length arrays.

For example, consider (C,B). It has two different species.

1: (B,A), (E,D), (A,G), (D,C), (G,F )

9

2: (C,B), (F,E)

Respectively, the 2-note lines in each of these species have a chromatic length of

< 10, 2 >C and < 11, 1 >C

This 2-note line contains two different species. Now consider (C,E, F ), which has three

different species,

1: (C − E − F ), (F − A−B)

2: (A− C −D), (B −D − E), (D − F −G), (E −G− A)

3: (F − A−B)

Respectively, the 3-note lines in each of these species have a chromatic length of

< 4, 1, 7 >C , < 3, 2, 7 >C , and < 4, 2, 6 >C

Notice, (C −B) is a 2-note line that has 2 different species. (C −E − F ) is a 3-note line

that contains 3 different species. This generalizes to nice theorem proven by Clough and

Myerson [1]

Theorem 2.3.1: Given any k, 1 < k < 7, and any k-note line, the genus containing that

line contains exactly k species.

3 Circle of fifths

The proof of the above theorem makes use of the circle of fifths, discovered by German math-

ematician Johann David Heinichen [6]. In music theory, the circle of fifths is a representation

of the relationships between the twelve tones of the chromatic scale, their key signatures and

10

major and minor keys. It is a sequence of pitches, represented in a circle, in which the next

pitch is found seven semitones higher than the previous. Seven semitones is the same as a

musical interval referred to as a fifth. A fifth is the interval between two notes on the circle,

and the distance between two consecutive notes is seven semitones.

3.1 Proof of Theorem 2.3.1

The proof of Theorem 2.3.1 can be illustrated using a generalized version of the circle of

fifths. Since we are interested only in diatonic notes, consider only the top portion of the

circle with an edge of chromatic length six added from B to F . Consider the line (C−E−F ),

with genus < 4, 1, 7 >. The other lines of the genus can be obtained by cycling clockwise

around the upper semicircle of the circle of fifths. We claim that there are three different

types of species of this 3-note line. The three species arise from the three different possible

locations of the interval B-F. The interval B-F is the only interval on the diatonic circle that

does not have chromatic length seven, thus it is the location of this interval within a line that

determines the line’s species. In general, there are, in fact, k different possible locations of

11

this interval and thus k different species. This argument assumes all notes are in clockwise

order and a complete proof can be found in [1]

4 Enumeration of chords

We have discussed counting k-note lines, which again, are the same as scales. Thus we

have discussed counting the scales of the musical universe. Next we ask how many d -note

diatonic pitch class sets, or chords, exist in a c-note chromatic universe? We start in the

c-note chromatic universe Zc, the group of integers mod c. We want to consider the d-

note pitch classes which are subsets of Zc with size d. Equivalently, we can count different

interval arrays withc∑

i=1

di. Without loss of generality, assume the notes are in increasing

order. Consider D2n, which represents the symmetries of a regular n-gon with n rotations

and n reflections (n+ n = 2n). Rotations and reflections can be thought of as traspositions

and inversions respectively. These operations permute groups or in this case, chords [2]. We

want to count the number of d-note pitch class sets by counting the number of equivalence

classes when D2n acts on the set of d-note pitch class sets. These sets can be represented

by interval arrays of the form < d1, d2, ..., dd >. So now the argument becomes counting the

number of equivalence classes when D2d acts of the set of interval arrays.

Theorem 4.1.1: The number of d -note pitch class sets in a c-note chromatic universe is

[4]

12dT (c, d) + 1

2I(c, d)

where T (c, d) =∑

j|gcd(c,d)Φ(j)

(c/jd/j

)12

and I(c,d) =

f(x) =

( c

2−1

⌊ d2⌋)

if c is even and d is odd(⌊ c2⌋

⌊ d2⌋)

otherwise

Where the Euler function Φ(j) is the number of positive integers that are less than or

equal to j that are also relatively prime to j [4].

The proof of this theorem uses famous results known as the Hockey Stick Theorem and

Burnside’s Lemma, tools used in combinatorics where here they help us count the number

of equivalence classes. We will state these theorems for use in the counting argument, the

complete proofs can be found in [4] [12]

Theorem 4.1.2 Burnside’s Lemma: Let G be a group acting on a set S. The number

of equivalence classes is

1|G|

∑g∈G

Fix(g)

where Fix(g) is the number of elements of S that are fixed by g. [10]

Theorem 4.1.3 Hockey Stick Theorem: If m and n are nonnegative integers, then

n∑k=0

(km

)=

(n+1m+1

)[11]

This theorem is well illustrated using Pascal’s Triangle [8]

13

In addition to these concepts, another theorem can be used to help rewrite some summa-

tions into neater form.

Theorem 4.1.4: Let j, k and n be integers such that 0 ≤ j ≤ k ≤ n. Then,

n−k+j∑m=j

(mj

)(n−mk−j

)=

(n+1k+1

)

5 Conclusion

There are more generalized theorems for scales than the usual chromatic and diatonic. We

have talked about the enumeration of scales and genus. The genus of a line and any of its

permutations have the same structure of species. Because of this we may talk about the

structure and enumeration of chords. Further details about enumeration of chords are left

to the reader.

14

References

[1] Clough, John and Myerson, Gerald. Musical Scales and the Generalized Circle of Fifths

The American Mathematical Monthly, Vol, 93, No.9, November 1986

[2] Hook, Julian. (Used this because of mention of translations and reflections; can take out)

Uniform Triadic Transformations Hook, 2002.

[3] Silverman, Danielle and Wiseman, Jim. Noting the difference: Musical scales and Per-

mutations

[4] Case, Joshua and Koban, Lori and LeGrand, Jordan. Counting Pitch Class Sets with

Burnside’s Lemma Department of Mathematics and Computer Science University of

Maine, UMF Math Department.

[5] Image. Half steps, whole steps and scale formulas.

<http://www.bandnotes.info/tidbits/scales/half-whl.htm>. (5 May 2014).

[6] Image. <http://www.sevenstring.org/forum/music-theory-lessons-techniques/197136-

circle-fifths.htm>. (5 May 2014).

[7] Image. <http://www.electricguitarsuk.com/images/clip image002.jpg>. (5 May 2014).

[8] Image. Pascal’s Triangle and its patterns.<http://britton.disted.camosun.bc.ca/pascal/p

1h.gif>. (5 May 2014).

[9] Home page. Circle of fifths. <http://www.circleoffifths.com/>. (5 May 2014).

[10] Contemporary Abstract Algebra Gallian, Joseph A. Houghton Mifflin Company Boston.

5th edition. 2002. (p. 487)

15

[11] Concrete Mathematics: A Foundation for Computer Science Addison-Wesley. Reading,

MA. 2nd edidtion. 1994. (p 169,174)

[12] An Application of Burnside’s Theorem to Music Theorem to Music Theory Graham,

Jeff. The UMAP Journal. Bedford, MA. Vol. 29, No. 1 2008

16