© 1991, Audio Engineering Society and Andrew Duncan. All rights reserved.

andrewzboard at gmail dot com

I have posted a video lecture covering roughly the first half of this paper. There is also a Web page with an interactive graphic illustrating the graph *H*_{12} discussed below. In addition, I have been working on a MIDI controller that exploits some of the patterns described in this paper.

Musical patterns may be investigated with the mathematical tools more commonly applied in science and engineering. For example, the cyclic autocorrelation of a musical scale describes its interval content. Fingering patterns on string instruments are embedded in a space with an unusual topology. Ideas from crystallography may be applied to the description of structure-preserving transformations of melodies. These phenomena are explored for the particularly common case of the twelve-note equally-tempered scale.

“Nothing can be farther from the working musician’s mind than counting, nothing farther from the working mathematician’s mind than singing, and yet there is something common to both.” [1]

The purpose of this paper is to examine some unique properties of commonly used musical patterns. It assumes some familiarity with musical ideas: octaves and intervals, the major scale and the concept of key. It also assumes some knowledge of abstract algebra: elementary number theory, groups, and graphs. In order to address a readership of wide-ranging backgrounds, I present these ideas with the mathematical and musical aspects evolving in parallel.

We will see that there is a natural way to describe the internal structure of a
musical scale that is closely related to the process of *autocorrelation*
used in digital signal theory. Attempting to pull musical patterns into a second
dimension will reveal that these patterns may be thought of as embedded in a
12-point space with peculiar connectivity. The *automorphisms*, or
structure-preserving self-mappings of this space, will correspond to precisely
those musical transformations that preserve melodic and harmonic relations
between notes, or interchange those relations. These mappings are analogous to
the symmetry groups of tilings or crystals.

Where a term is used in a more restrictive sense than is common, or already has a
well-defined or circumscribed meaning, it appears (where defined) in **bold**.
In a case where I would stress or accent (or shout) a word when describing these
ideas verbally, the word is in *italics*. I hope this does not make for a
bouncy ride.

Sounds are vibrations in the air: variations in air pressure about a mean. The
average ambient air pressure is roughly 100 Newtons of force per square meter of
area, and the variations caused by ordinary conversation are about seven orders
of magnitude (powers of ten) smaller. We will consider a sound to be represented
by a real function of time describing this variation. In most cases of interest,
this function can be broken up into a sum of pure sine waves of different
frequencies. These sine waves are the **components** of the sound. The term
**frequency** refers to the number of times per second that the oscillation
goes through a full cycle. Frequency is measured in cycles per second, or
**Hertz** (Hz). It is conventional to describe the range of the human ear as
20 Hz - 20,000 Hz (20 kHz), although most people’s hearing falls rapidly after 15
kHz or so. For comparison, the notes on a piano range from 27 Hz (low A) to 4.2
kHz (highest C).

In certain cases of particular interest, such as sounds produced by vibrating
strings, columns of air, or reeds, the frequencies of the sound’s components are
related in a particularly simple way: they are all integer multiplies of a base
frequency. The sinusoidal component with this lowest frequency is called the
**fundamental** of the sound. When we refer to the frequency of a sound, we
mean the frequency of the fundamental. For example, although the fundamental
frequency of the piano’s middle C is 261 Hz, there are also components at higher
frequencies that give “body” to the note.

We use the measurement of frequency to give an *ordering* to the set of all
sinusoids, and to a large number of sounds, via their fundamental component. This
ordering corresponds closely to our intuitive notion of **pitch**:
“low-pitched” sounds have low frequencies, and so with high. Perception is
somewhat more complicated, but the ordering that frequency brings is of
fundamental importance to music theory.

The “universe” in which we work can be defined in a multi-step process. First,
we define the most fundamental musical relationship: the **octave**. Two
frequencies are said to be separated by an octave when their ratio is 2:1. We say
that 2 Hz is an octave above 1 Hz. To the ear, notes separated by any number of
octaves sound somehow “the same”. (There are well-known physical reasons for
this, which we will take as given.) Fig. 1 shows the frequency axis. (Note that
this figure is a fractal: it looks “the same” at all levels of magnification.)

Any point on this line corresponds to some frequency, some pitch. In this way,
frequency is used as a *coordinate* to locate pitches in an absolute way. On
the axis are marked all the notes related by octaves to the frequency of 1 Hz.
These frequencies are all powers of two: those frequencies greater than 1 Hz have
positive exponents; those less than one have negative exponents. These pitches
are our first landmarks on the frequency axis. Note that they are certainly not
the *only* available frequencies. For the moment *all* frequencies are
equally accessible.

We next proceed to deform the frequency axis and its labels in various convenient ways. In Fig. 2a, we have taken to using the exponents (of two) to represent the frequencies, rather than the frequencies themselves. This has the effect of converting multiplication into addition: the movement of an octave is now the addition or subtraction of 1. This is really a very fundamental change. We do this to conform to the ear’s feeling that movement of one octave constitutes a particular size “step” or jump, which is always the same “size” wherever it occurs. Algebraically, we perceive (or learn to perceive) pitch logarithmically. Acknowledging this, we stretch and squeeze the axis until these octaves are the same distance apart, in Fig. 2b.

We will have repeated occasion to divide the octave into equal parts. For various
reasons, it is very fruitful to divide the octave into twelve equal steps. To
avoid fractions, we multiply everything by 12, as shown in Fig. 3. Our
frequency axis is now essentially completed. It relates directly to conventional
musical ideas: for example, the frequencies of the keys of a piano are
*integer* points on this axis. However, we still are considering the axis to
be continuous. Observe that two notes that are an octave apart now have numbers
that differ by 12.

The next step quite overshadows the previous two: we twist the axis into a circle
(Fig. 4). We define two notes to be equivalent if we get the same remainder
when dividing either by 12. Thus 12 becomes 0, 13 becomes 1, and so forth. Doing this
splits up the set of all frequencies into classes, each containing precisely
those frequencies related by octaves. For example, one such class is {... -11, 1,
13, ...}. For simplicity of notation, we would designate this class by (1). We
still have an (uncountably) infinite number of these classes, one for every point
on the circle. In music theory, such a class is often referred to as a
**pitch-class** (abbreviated PC). In the following, we will use the term note
to refer specifically to a set of equivalent pitches. For instance, we refer to
the **note** C, we do not have in mind a *particular* C (middle C, or any
other), but the “idea of ‘C-ness’”. (More below on the letter names for notes.)
The circumference of this circle is one octave: we let this octave stand for
*all* octaves.

Finally, we decide to restrict ourselves to the *integer* notes on the
circle. Thus we have divided the octave into twelve notes, separated by twelve
equal steps. This is the **12-tone equally tempered scale**. Fig. 5 shows
the notes arranged in a circle. (We now consider there to be nothing between
them.) In the further interest of notational simplicity, we remove the brackets
denoting an equivalence class, and refer to the notes with bare numbers. These
numbers still represent a *coordinate system* for the 12 notes. Later we
will find uses for a *coordinate-free* representation.

It is certainly easy to envision dividing the octave up into a different number
of notes. The number twelve is a justly popular choice. We refer to the count of
notes in an octave as the **order** of a musical system (The word “size” will
be used differently later on.)

Historically, there have been letter names also associated with each note. One
such convention, called the **scientific tuning**, gives the name C to the
note 0. (Recall that this note represents the frequencies {... 1/2, 1, 2, 4,
...}.) Using this convention, middle C would have a frequency of 256 Hz. As noted
above, middle C is customarily put at (approximately) 261 Hz. This is another
convention called **A 440** tuning. Still another convention, **concert
tuning**, puts middle C somewhere else.

For convenience, we introduce the convention of naming the top (zero) note “C”. Ascending through the melodic circle, the names for the notes are as follows: C, C#/D♭, D, D#/E♭, E, F, F#/G♭, G, G#/A♭, A, A#/B♭, B, (and back to C again). (It is a matter of musical context whether, for example, one refers to the second note as C# or D♭.) Fig. 5 shows the cyclic twelve-note scale with the letter names for each note. Observe that we have assigned the note name C to represent the note number 0. This is a common choice, but is not a requirement.

Why do we use these names instead of, for instance, naming them A through L? The
notes whose names contain no **accidentals** - sharps or flats - form a
particularly interesting subset of the twelve notes: the **diatonic scale**.
More about this scale in the following.

We may define two notes to be **adjacent** if their difference is 1. This
*relation* between notes can be represented by a **graph**,
conventionally known as *C*_{12}, as shown here in Fig. 6.

We often blur the distinction
between notes and the vertex that represents them. For example, we start to think
of the top vertex as *being* the “note” 0. What is the difference between
this graph, and for example Fig. 5? A graph is a picture of a relationship: the
round dots, or **vertices**, represent objects of some sort (for us, notes)
and the lines, or **edges**, connect any two objects that have the given
relationship (in our case musical adjacency). Thus we have abstracted away all
but the essential elements.

Note that since we have numbered the vertices, the graph is at least implicity
*directed*: we distinguish between clockwise and counterclockwise motion
around the graph. One edge in Fig. 6 has an arrow along it defining our
“forward” direction. This distinction affects our later considerations of
symmetry. In particular, the mirror image of a directed graph can not properly be
superimposed on the original, as the direction of traversal along the edges will
be reversed. Where the distinction is important, we will denote directed graphs
by bold italic symbols (e.g. *C*_{12}), in the same way vectors are
sometimes denoted.

We now examine the question “How many ‘essentially different’ subscales of the 12-note scale are there?” This question is really the same as “How many different 12-bead necklaces are there with some beads colored black and the rest white?”

To clarify these remarks, we illustrate in Fig. 7a a scale of particular
interest: the **diatonic** or **major/minor scale**. Here, notes that are
in the scale correspond to vertices that are colored black, and notes not in the
scale to white vertices. (Observe that this is the *reverse* of their
coloring on the keys of a piano.) The connection between the two illustrations is
that Fig. 7b is obtained by *rotating* Fig. 7a. If the top vertex
represents the note C, then Fig. 7a represents the C major scale, whereas
Fig. 7b represents the D major scale. We feel that both of these scales are the
same *sort* of scale, and so this leads us to a more explicit definition of
what we mean by “essentially different”, and to a more restrictive definition of
the word “scale”.

We define a ** k-species** to be a set of

In order to formalize this idea we use again the idea of an equivalence relation.
We define two species to be *equivalent* if their graphs are related by the
operation of rotation (which is musical *transposition*; note that this word
appears with a different meaning in some mathematical contexts). This means we
can build up a class of equivalent species from a single “seed” by rotating it,
and collecting the results. Each class of equivalent species is referred to as a
**scale** or **chord**. Musically, the difference between a scale and a
chord is that a scale refers to a set of notes played sequentially; in a chord
they are played simultaneously. The distinction is irrelevant here, and the terms
will be used interchangeably. Using this terminology, the graphs of Fig. 7
represent two different species or representatives of the same scale.

This process of moving to equivalence classes takes us *away* from our
coordinate-based conception of the 12-note system into a more coordinate-free
perspective. Now the absolute location of a pattern is not so important as the
relative disposition of its constituent notes.

It is evident that there are 2^{12} = 4096 different species in the
twelve-note musical system. The definition of equivalence given above sorts these
species into classes of equivalent scales. This sorting can be written down
explicitly as shown in Table 1. (The entire listing, printed in 9-point type,
runs more than 90 pages!)

4095 111111111111 # 1 of 1 in class 1 of 1 for size 12 4094 111111111110 # 1 of 12 in class 1 of 1 for size 11 4093 111111111101 # 2 of 12 in class 1 of 1 for size 11 4092 111111111100 # 1 of 12 in class 1 of 6 for size 10 4091 111111111011 # 3 of 12 in class 1 of 1 for size 11 4090 111111111010 # 1 of 12 in class 2 of 6 for size 10 ... ... 1162 010010001010 # 7 of 12 in class 39 of 43 for size 4 1161 010010001001 # 7 of 12 in class 40 of 43 for size 4 1160 010010001000 # 4 of 12 in class 17 of 19 for size 3 1159 010010000111 # 7 of 12 in class 10 of 66 for size 5 1158 010010000110 # 6 of 12 in class 18 of 43 for size 4 ... ... 7 000000000111 #12 of 12 in class 1 of 19 for size 3 6 000000000110 #11 of 12 in class 1 of 6 for size 2 5 000000000101 #12 of 12 in class 2 of 6 for size 2 4 000000000100 #10 of 12 in class 1 of 1 for size 1 3 000000000011 #12 of 12 in class 1 of 6 for size 2 2 000000000010 #11 of 12 in class 1 of 1 for size 1 1 000000000001 #12 of 12 in class 1 of 1 for size 1 0 000000000000 # 1 of 1 in class 1 of 1 for size 0

A (very large) diagram of all the 4096 scales, colored by equivalence class, is available here. More explanation in this video.

What this table means may be illustrated by a particular example. Entry 1160
in Table 1 tells us about the species 010010001000, consisting of the notes
{1, 4, 8} or {C#, E, G#}. For the purpose of explanation, we will read the line
from the right. The number on the far right informs us that the **size** of
this species is 3 (the number of notes contained in it). The next number to the
left tells us that using the equivalence relation of rotation the collection of
all 3-note species breaks up into 19 equivalence classes. In other words, there
are basically only 19 different three-note chords or scales. As we scan down the
list, this class is the 17th one we come to. This class consists of all rotations
(musical transpositions) of the given notes. For example, {2, 5, 9} = {D, F, A}
and {4, 7, 11} = {E, G, B} are also in this class. In music theory, this
particular equivalence class is referred to as the **minor triad**, and it is
one of the 19 possible three-note chords. Finally, this class has 12 members (all
rotations of the pattern are distinct), and this species is the 4th member of the
class to appear in the list.

We may interpret the above counting as saying that there are 19 *essentially
different* ways to play a group of three notes. In this sense, all major
triads are considered to be the same sort of triad, as are all minor triads. This
kind of sorting is the first step to describing the terrain of the 12-note
musical system.

As we noted above, there are 4096 different species in the twelve-note musical
system. For any *k*, the number of *k*-species is the same as the
number of ways of choosing *k* distinct elements from a set of 12 elements,
a well-known quantity:

# of *k*-species = 2! / (k! * (12-k)!).

Oviously if we add up these numbers for all *k* we must get 4096. The total
number of *scales* (or chords) is 352; finding this number is not nearly so
easy. The theoretical approach [2] requires using Burnside’s Theorem (which is
elegant), and the direct approach (listing them all) is tedious. For now
will observe some of the results of the direct approach.

For example, there are 12 different 1-species: 12 different 1-note subsets of the
full 12-note set. Fig. 8 shows graphs of these species, along with the binary
numbers associated with the species. All these species are members of the
*same* scale or chord: they are all equivalent. Thus there is exactly one
1-scale or 1-chord.

The previous example was somewhat degenerate, and we first encounter some non-
trivial features when we examine the case *k* = 2. For example, the
equivalent species 110000000000 = 011000000000 = . . . all represent the same
2-scale: a scale built up of two adjacent notes. However, no amount of rotating
will ever turn this scale into 100000100000. This number represents a different
2-scale. Fig. 9 shows the 6 species of the latter scale.

This is the only 2-scale which does not have twelve species in its equivalence
class. The reason is that once we have rotated it six steps, we have the original
species again. For this reason it occurs as a special case later on. In Fig. 10
we see graphs for each of the *different* 2-scales.

Which of the equivalent species was chosen to represent a particular scale? The
first to appear in Table 1, the one with the largest binary number associated
with it. We will refer to this species as the **canonical species** of the
scale.

The 2-scales (or equivalently 2-chords) have a special name: the
**intervals**. We see that there are six of them. Matching them up with their
conventional music-theoretical names, we have the results of Table 2.

110000000000 m2/M7 101000000000 M2/m7 100100000000 m3/M6 100010000000 M3/m6 100001000000 P4/P5 100000100000 +4/-5

We note that there is no way to distinguish a major third, for example, from a
minor sixth: C to E is a major third, after all, but then E up to C is a minor
sixth. Or, equivalently, from C *down to* E is a minor sixth. Thus, although
in general we distinguish between moving up and down, with intervals we cannot. A
mathematical way of saying this is that all intervals exhibit **dihedral**
symmetry.

The diatonic scale mentioned above is a particular 7-scale, one with many
interesting properties. We will investigate further 5-scales and 7-scales, as
they are of particular theoretic interest. Note that every 5-scale defines a
unique 7-scale (its complement). For this reason, a discussion of certain
properties of 5-scales may sum up the properties of 7-scales as well. In a way,
the *most* interesting 5-scale is the **pentatonic scale**: {0, 2, 4, 7,
9}, or 101010010100, or {C, D, E, G, A}. (See Fig. 11. Strictly speaking, we
have just given one species of this scale.)

This scale forms the backbone for nearly all musical traditions: Western,
Eastern, and all other quarters of the compass. It is the source of countless
melodies in both classical and folk music. It is also the starting point of
musical improvisation. The pattern of the pentatonic scale on the fingerboard is
*the* most fundamental of patterns, the most important habit for the fingers
to develop. In addition, it is the *complement* of the diatonic scale. That
the two most fundamental patterns in the 12-note system should be related like
this is quite remarkable.

Where does the pentatonic scale stand among its peers, the 5-scales? Table 3 lists all the 66 different 5-scales, listed in decreasing binary order, with each scale represented by the largest (canonical) binary number in its equivalence class.

1: 111110000000 21: 111000100100 41: 110100100100 61: 110001010010 2: 111101000000 22: 111000100010 42: 110100100010 62: 110001001010 3: 111100100000 23: 111000011000 43: 110100011000 63: 110000101010 4: 111100010000 24: 111000010100 44: 110100010100 64: 101010101000 5: 111100001000 25: 111000010010 45: 110100010010 65: 101010100100 6: 111100000100 26: 111000001100 46: 110100001100 66: 101010010100 7: 111100000010 27: 111000001010 47: 110100001010 8: 111011000000 28: 111000000110 48: 110011001000 9: 111010100000 29: 110110100000 49: 110011000100 10: 111010010000 30: 110110010000 50: 110011000010 11: 111010001000 31: 110110001000 51: 110010110000 12: 111010000100 32: 110110000100 52: 110010101000 13: 111010000010 33: 110110000010 53: 110010100100 14: 111001100000 34: 110101100000 54: 110010100010 15: 111001010000 35: 110101010000 55: 110010011000 16: 111001001000 36: 110101001000 56: 110010010100 17: 111001000100 37: 110101000100 57: 110010010010 18: 111001000010 38: 110101000010 58: 110010001010 19: 111000110000 39: 110100110000 59: 110001100010 20: 111000101000 40: 110100101000 60: 110001010100

The “percolation” to the right of the 1s in Table 3 follows an interesting pattern. We may think of the 1s as little people, for example explorers. These people are moving around on a circular world with only 12 positions at which to stand. The fellow on the right moves out, perhaps scouting for snarks or woozles. When he reaches the eleventh position (scale #7) he sees that one more step will bring him adjacent to somebody he sees “up ahead”. (He doesn’t realize that he is looking at the back of the first explorer! See Fig. 12.)

Alarmed, he runs back to the group and gets a friend. Together, they advance
one step (scale #8), and then the same procedure repeats. As we proceed, the
scales become progressively less “dense”. Intuitively, it also seems that the
scales become more “useful”. For example, the first scale, 111110000000, has
little internal variety. We might think of this scale as having minimum entropy.
One feels it is not very fruitful ground for melodic ideas. When we get to the
*last* scale in this ordering, we discover that it is the pentatonic scale!
This is grounds for considering the pentatonic scale to be a particularly special
5-scale. A similar phenomenon occurs with 7-scales: the diatonic scale is the
last entry.

How does this correspond to our idea of entropy in the physical sciences? In the
context of a gas, for example, minimum entropy would occur when all the molecules
were crowded up into a corner of the room, and maximum entropy when they were
uniformly diffused throughout. This is approximately what happens in our scales.
In the case of 6-scales, the canonical listing starts with 111111000000 and ends
with 101010101010, a diffusion so uniform as to be again structured. Thus
paradoxically the scale that has maximum “entropy” comes out highly patterned.
With k = 5 or 7, we can’t quite approach this uniformity of diffusion, as neither
5 nor 7 have any common factors with 12. But there is a sense in which these
scales are the *most* rich in content.

(As noted above, one might prefer to assign bit 0 to note 0 in the binary representation of a species. In addition, one might sort the species from low to high, both inside the equivalence classes to determine the canonical representative, and in the ordering of those representatives. Thus there are many ways to generate the tables of scales & chords here excerpted. However, the phenomena do not vary in essence.)

“No single number and no single tone is what it is without the others.” [1]

One way of characterizing the “richness” of a particular scale is by its
**interval spectrum**. By this we mean the number of jumps of size 1, of size
2, 3, ... contained in the scale. For example, the five note scale 101010101000
contains four jumps of size 2, four jumps of size 4, and two jumps of size 6.
Note that any two equivalent species will have the same interval content: the
same inventory of jumps. That is, the interval spectrum is a well-defined
function on chords/scales. We might describe the interval spectrum of the scale
just mentioned by the string of numbers 504040204040: it contains five jumps of
size 0, none of size 1, four of size 2, . . ., two of size 6, . . ., four of size
eight (which is really the same as size four), etc. Note that the symmetry of
the spectrum is a consequence of the above mentioned dihedral symmetry of
intervals. This is why we may properly refer to the spectrum as describing the
interval content of a scale. A list of the interval content of all the 5-scales
is excerpted in Table 4.

111110000000: 543210001234 111101000000: 533211011233 111100100000: 532211111223 111100010000: 532112121123 111100001000: 532112121123 111100000100: 532211111223 . . 110001010010: 512213131221 110001001010: 512223032221 110000101010: 513122122131 101010101000: 504040204040 101010100100: 503222122230 101010010100: 503214041230

In *this* list, the last entry, the pentatonic scale, is seen to contain
*more* jumps of size 5 or 7 (perfect 4ths/5ths) than any other scale. It
contains four such intervals, between 0 & 7, 2 & 7, 2 & 9, and 4 & 9, or with
note names, C & G, D & G, D & A, and E & A. Similarly, the interval spectrum list
for the 7-scales ends as shown in Table 5.

111111100000: 765432123456 111111010000: 755433133455 111111001000: 754443134445 111111000100: 754443134445 111111000010: 755433133455 111110110000: 754433233445 . . 110110110100: 733633333633 110110110010: 733633333633 110110101100: 733544244533 110110101010: 725444244452 110110011010: 733544244533 110101101010: 725436163452

In this list, the last entry, the diatonic scale, is also seen to contain more 4ths/5ths than any other seven-note scale: six. In addition, it is the only scale that has a unique content for each interval: for example, it contains three major 3rd/minor 6th intervals; every other interval is contained either more or less times. The interval spectra of the pentatonic and diatonic scales are further evidence of the unique value of these patterns in the 12-note musical system.

The interval spectrum of a scale or chord may be found by a process which resembles an operation carried out in other fields of math and engineering. For example, to find out how many jumps of size 2 (major seconds) are contained in the pentatonic scale, we line up two copies of the scale, one shifted (cyclically) from the other by just such a jump:

101010010100 001010100101

We count how many times the 1s line up together: three times in all. A little thought will show that this is also the number of major seconds contained in the scale. Fig. 13 shows how this looks using our cyclic graphs. The pentatonic scale is shown lined up with another copy, shifted zero places. All notes in the scale line up with each other.

In Fig. 14, the scale in front has been shifted two places, and now there are three locations where a black vertex in one graph lines up with a black vertex in the other.

This approach adds another perspective to our observations about the interval content of the pentatonic and diatonic scales. Suppose we seek a five-note scale which has the following property: when shifted by 5 or 7 steps (a perfect 4th/5th) it still has four members in common with the original scale. Our discovery is that the pentatonic is the only such scale! Similarly, the only 7- scale that has six members in common with its neighbors a perfect 4th or 5th away is the diatonic.

In the context of digital signal processing, we correlate two sequences of
numbers by lining them up, *multiplying* adjacent numbers, and then adding
the products. Different relative shifts of the sequences yield different sums,
and the collection of sums from all different shifts is called the
**cross-correlation** of the sequences. If one correlates a sequence to
itself, the result is called the **autocorrelation**. Our process of finding
the interval spectrum is remarkably similar to this operation: the multiplication
yields a 1 only when both entries are 1, and the sum then tells us how many times
this happens. In fact, it appears that our method is *exactly* the same as
cyclic autocorrelation, but there is one subtle difference. For example, we would
agree that the scale 100000100000 contains precisely one jump of size 6 - in fact
that is all it contains. But following the shift, multiply, and add procedure
gives an answer of two. This is because in the special case when the shift is six
(or in general, half the order of the scale), we count each matching twice: when
note x matches up with note y, then y also matches up with x on the other side of
the cycle. So we must divide this count by two. Thus our interval spectrum
differs slightly from the conventional cyclic autocorrelation. More generally,
when decomposing scales into subscales, we may divide the count by the
**index** of the scale, defined by

index = order of scale / size of scale’s equivalence class.

As we noted above, considering scales as equivalence classes moves us towards a coordinate-free perspective of patterns -- there is no absolute location for notes. Just as finding the correlation of two signals removes all information about absolute time or phase, examining the interval content of a scale removes all information about absolute pitch or key.

Mendeleyev’s periodic table of the elements was successful because of the added dimension it introduced. Rather than a simple list of known elements, they were arranged with a second axis, so that elements adjacent vertically shared common properties. For the same reasons it is very useful to arrange musical notes as patterns on the plane: the added dimension allows us to corral some correlations.

Fig. 15 shows the fingerboard of a fretted string instrument. We may think of
the notes played on this instrument as particular points on the surface of the
fingerboard, which is a section of a plane. (Here we are using the term note in
its weak sense: 0 is different from 12, etc.) The relation of adjacency we have
defined corresponds to physical adjacency moving *along* the neck of the
instrument. For example, placing a finger on the string at the position marked
*b* in Fig. 15, and plucking the lower string, will produce a note that is
the successor to the note produced by placing the finger at *a*, one fret
behind.

To place the discussion in a more general setting, we now introduce the concept
of the **normalized infinite fingerboard**. By normalized, we mean that the
frets are all equally spaced. This is to say, they do not correspond to a
physical instrument, but to a stylized one. Further, we identify the location of
a particular note at the junction of the fret and the string, rather than between
frets. Finally, we let the fingerboard extend indefinitely in all directions.
Fig. 16 shows a section of the normalized infinite fingerboard.

In this fingerboard the relations of adjacency and succession have two
*directions*: as illustrated in Fig. 17, a note will have four adjacents,
a pair both vertically and horizontally, and two successors. The fingerboard thus
appears as a sort of discrete space: one in which there are only four points
adjacent to any other. The global topology of this space is of particular
interest.

We decide on the convention that the “forward” directions are to the right and upward, as indicated in the figure. We would also like the horizontal direction to embody the relation of adjacency we previously discussed. This is to say, horizontally, note 0 is adjacent to note -1on the left and note 1 on the right. To which notes is it adjacent above and below? On many string instruments, such as the violin, viola, cello, bass, guitar, and Chapman Stick, motion across the neck corresponds to jumping by 5 or 7 steps. Our fingerboard convention will be that movement up one string constitutes a jump of five steps (the interval of a musical fourth). Fig. 18 shows a section of the fingerboard, with vertices numbered according to this convention.

Now we move to the strict definition of note: we identify notes separated by
octaves. Mathematically, we reduce the fingerboard mod 12, replacing 12, 24, 36,
etc. by 0; 13, 25, etc. by 1; and so forth, as illustrated in Fig. 19.

What does this mean for the fingerboard? Well, the fingerboard is periodic in at least one direction. Moving horizontally, every twelve steps brings us back to the same note. In fact, we have periodicity in the vertical direction, too. One may easily verify that twelve (and no fewer than twelve) repeated jumps by five notes will return to the same note. (This works because 12 and 5 have no common factors.) Thus a finite section of the fingerboard will serve to represent the whole. A 12 x 12 square of fingerboard is shown in Fig. 20.

The top and bottom of the square are considered to be identical, as are the left and right sides. We put the note 0 in all four corners, and color it dark wherever it appears. Since the top and bottom sides are really the same, we draw one of them with a solid line, and one with a dashed line. Fig. 21 shows a section of the fingerboard with letter names for the notes.

Musicians will note there is something fundamental about the relation of notes that are adjacent vertically. It is this added dimension that makes fingerboard patterns so revealing in structure.

For example, Fig. 22 shows a 12 x 12 section of the fingerboard with the notes of the pentatonic scale marked with dots. We can immediately see that this scale is composed of five notes all separated by jumps of size five: a perfect fourth. The autocorrelation properties may now be found by sliding the whole pattern to the right or left the required number of steps and seeing how many notes match up. We also see that the pentatonic scale may be played exclusively by moves of one step vertically or two steps horizontally, as the dots can be broken up into “bands” that proceed upward and to the left at a 45° angle.

A curious thing happens if we translate this pattern back into the binary numbers we used to represent the different species of the pentatonic scale. Table 6 shows the result: the black dots of Fig. 22 have been turned into 1s and the excluded notes into 0s.

101010010100 = 2708 base 10 101001010100 = 2644 101001010010 . 100101010010 . 100101001010 . 010101001010 . 010100101010 . 010100101001 . 010010101001 . 010010100101 . 001010100101 001010010101 = 661

When we convert these numbers into their decimal form, we discover that they are
completely sorted from highest to lowest. So we have this unexpected connection
between the physical layout of the pentatonic scale on the fingerboard and the
pattern that forms on the printed page when the equivalent species of that scale
are listed in numerical order. This occurs because pentatonic species that are
consecutive in magnitude are related by cyclic shifts of five places. This is the
*only* 5-scale that has this property. Similarly, when the diatonic species
are sorted by magnitude, consecutive entries are related by cyclic shifts to the
right of seven places. The diatonic is also the *only* such 7- scale.

We notice some interesting patterns in Fig. 22 or Table 6. For example, if we consider the top “edge” to be connected to the bottom, the vertical columns consist of strings of five consecutive ones and seven consecutive zeroes. If we also consider the “sides” to be connected, then each column is derived from the column to its immediate left by the operation of shifting downwards by five places. In addition, each row is shifted five places to the right from the row immediately above it. These are all consequences of considering notes separated by an octave to be identical. Patterns that have a linear periodicity in a one-dimensional space (such as the piano keyboard) acquire something of the character of tilings. In fact, a little consideration will reveal that any scale or chord on the fingerboard will map perfectly onto itself with a translation of a units to the right and b units up precisely when a + 5b = 0 (mod 12). Now we wish to examine a single “tile” of the fingerboard.

As we observed in Fig. 4 with the frequency axis, identifying notes separated by octaves has a “curling” or “looping” effect. In two dimensions, the periodicity along the neck has the effect of turning the infinite fingerboard into a cylinder, as shown in Fig. 23. The periodicity across the neck curls the cylinder upon itself, forming a torus, as in Fig. 24.

We are still not through paring down the fingerboard. As Fig. 20 suggests, there are several copies of the note 0 contained in it. In fact, we may consider the 12 x 12 torus to be tiled with smaller tori -- to wit, twelve of them, each containing one copy of each distinct note.

Fig. 25 shows one of these smaller tori embedded in the larger 12 x 12 version.
This scrap of fingerboard is topologically a torus in the same sense that the
larger one is -- its top-right and bottom-left sides, for example, are considered
to be the same. In Fig. 25, the sides are drawn with dashed lines, with the
“duplicate” notes dotted lines. (We do not want to use solid lines, because in
this case, the *sides* of the torus are not *edges* of the graph.) Thus
what appear to be two copies of vertices 0, 8, and 4 are seen to be single
copies.

If we want to get a more concrete idea of what this small torus looks like, we may carry out the following construction. First its top-right side is to be bent around and “glued” to its bottom-left. When we do that, we get an object like that depicted in Fig. 26. Next, we join the ends, to produce the structure shown in Fig. 27 (compare with [3, Fig. 5]).

This graph has 12 notes (vertices), and each note is connected by an edge to its
four adjacents. The different color (gray vs solid) edges represent the
perpendicular directions on the fingerboard. This graph, which I refer to as
**H**_{12}, represents the discrete topology, or
*connectivity* of the fingerboard. As the vertices are numbered, the
ascending direction is distinct from the descending, hence the graph is at least
implicitly directed. This graph has many interesting properties. For example, we
first note that it consists of two loops, each of which goes through each note
exactly once. (Such a loop is referred to in graph theory as a **Hamilton
cycle**.) Each loop is the edge of a Mûbius band with 1 1/2 twists, the bands
for the two loops being of opposite handedness. Each loop also constitutes a
**trefoil knot**: a fundamental way of knotting a loop in three dimensions.
One may think of the graph as having its notes divided into six pairs, a pair
consisting of any vertex and that vertex which is six steps away (by either
cycle). For example, the notes 0 and 6 are close to each other in the figure, but
traveling along the edges of the graph it takes six steps to go from one to the
other. As the graph *H*_{12} is cyclic along both loops, this is the
farthest distance separating any two vertices. As drawn in Fig. 27, the notes
farthest apart topologically are closest physically. The graph can, however, be
drawn so that both types of distance are maxima.

We illustrate this observation by the following process. Going back to our graph
of *C*_{12}, we add new edges connecting vertices whose numbers
differ (mod 12) by 5. Doing this corresponds exactly to adding the vertical
dimension of the fingerboard. Doing this creates a starlike pattern inside the
circle (Fig. 28). Again, as 5 and 12 are relatively prime, the star visits
every note once before returning to its origin. This graph is *also*
*H*_{12}. Embedding it in the plane forces us to draw edges as
crossing when they do not really meet. The reader may verify that any two notes
that are farthest apart within the graph are now similarly disposed on the
page.

If we travel around the outside (horizontal) loop of *H*_{12}, we are ascending
sequentially through the 12-note scale. In musical terminology, we are ascending
in a **melodic** or **chromatic** circle. If one travels instead in the
vertical direction (the inside loop), the sequence of notes is 0, 5, 10, 3, 8, 1,
6, 11, 4, 9, 2, 7, 0, or C, F, B♭, E♭, A♭ (or G#), C#, F#, B, E, A, D, G, C. In
musical terminology, we are moving around a **harmonic** cycle, the **circle
of fourths**. If we move in the harmonic cycle in the opposite direction, it is
called the **circle of fifths**. (The words “fourth” and “fifth” refer to the
number of notes it takes in the diatonic scale to span these intervals.) The
combination of melodic and harmonic motions is the life and breath of music. The
arraying of patterns on the fingerboard so that both their chromatic and harmonic
content is readily apparent is what makes this perspective so useful.

The graph *H*_{12} may be thought of as a member of a family of
graphs {*H*_{n}}, where n refers to the number of vertices.
(The letter *H* stands for the words “Hamiltonian” and “harmony”.) The way we form
these graphs is simple: we draw all those “stars” which go through every vertex
exactly once. In this context, the outside, circular loop also counts as a star.
For example, in *H*_{8}, jumping every third vertex takes us to all
vertices. Jumping every other vertex will not. All *H*_{n}
will be graphs composed only of Hamilton cycles, and the number of such cycles
will be *phi*(*n*)/2, where *phi* is Euler’s function. The values
of *n* for which we get two cycles are 5, 8, 10, and 12. Each of these
graphs represents the connectivity of a fingerboard of some (possibly
non-existent) string instrument. For example, *H*_{8} represents an
instrument that plays in a musical system of 8 notes to the octave (order = 8),
with the jump between strings being 3 notes These fingerboards are all “nearby”
the 12-tone one we will be examining most closely. Note that 12 appears here as
the largest number of notes in a musical system for which there is a unique
“circle of fourths/fifths.” In the upward direction, for n > 12, all
*H*_{n} have more than two cycles, and hence cannot represent
a two-dimensional fingerboard. However, if we just pick two of these cycles, the
pair will represent some fingerboard. For example, in *H*_{14},
choose the outside cycle, and the cycle jumping every fifth vertex. This graph
represents an instrument that plays a scale of 14 notes, with strings being
separated by 5 notes. This fingerboard is also near to *H*_{12}.

We are particularly interested in operations that preserve the adjacency and/or the successor relation between notes. Letting the symbol ~ stand for either relation, what we require is that

An operation that satisfies this rule is called a **symmetry** or
**automorphism** of the associated graph. An example of an operation where
this fails is shown in Fig. 29.

The graph shown on the left has four vertices, and the operation
*T*_{1} is the addition (mod 4) of 1 to the vertex number.
Performing this operation yields the graph in the middle. The operation is not a
symmetry of the graph because, for example, 1 is adjacent to 2 in (a), but not in
(b).

To make the comparison easier, we may wish to rotate the graph so that the
position of vertex 1 on the printed page is as it was before. This is considered
no change to the graph, since a graph’s disposition on the page is not important
to its identity: only the adjacency of vertices matters. In situations such as
this, we may consider the action of *T* to be the process that goes from
Fig. 29a to 29c: the *rotation* of the graph by one “step”, *without*
changing the position of the numbers. Thus we are now associating these
operations with physical manipulation of the graph on the printed page.

By contrast, the operation *T*_{1} is a symmetry of the graph
**C**_{12}: the rotation of the graph by 1/12 of a revolution does
preserve the adjacency relationship. (See Fig. 30.) In fact, all twelve
rotation operations *T*_{0} through *T*_{11} are
symmetries of **C**_{12}. This set of twelve symmetry operations has a
conventional name: we shall be calling it Z_{12}.

Using this terminology, we can describe the process we used to find the distinct scales and chords in a more general way: the scales correspond to colorings of a graph, which are to be considered equivalent if one may be mapped into the other by a symmetry of the graph.

There is a natural way of combining a pair of operations to yield a third, called composition. What this allows us to do is to say the following sort of thing:

The left-hand side of the equation describes a well-defined value: the number we
get when we apply *T _{b}* to

Any set with a rule for combining its elements that satisfies the above-named
conditions of closure, identity element, inverses, and associativity, is called a
group. The set of symmetries of any graph (or, indeed, almost any object,
physical or conceptual) will form a group. As we have seen, the symmetry group of
the directed graph *C*_{12} is a twelve-element group. For the non-
directed graph *C*_{12}, there is more to say.

Any symmetry of the directed graph **C**_{12} will also work
for the non-directed graph *C*_{12}, so the symmetry group of the
undirected graph will contain Z_{12}. But a non-directed graph will have
more symmetries: reflections across an axis through the center of the graph.
These reflections will reverse the directions of arrows in a directed graph,
hence altering the successor relation, although preserving adjacency. In fact,
the symmetries of *C*_{12} are the elements of the dihedral group of
order 24, D_{12}. These elements are the 12 cyclic rotations of
Z_{12}, and the twelve mirror-reflections across a line through the
center of the graph, and passing through two vertices, or precisely in between
two vertices (Fig. 31).

Any element in this group may be expressed as the
composition of a cyclic rotation *T _{n}*, and the zero inversion, or
mirror-reflection across the axis passing through vertices 0 and 6. (Note that
the term “inversion” is also used, differently, in a musical context.) We denote
the operation of reflection across this axis by the symbol I. For example, Fig.
32 shows how the reflection across the axis through vertices 1 and 7 can be
expressed as an inversion followed by a rotation of two steps.

It is now apparent why we wished to consider the scales and chords to be
2-colorings of the directed graph **C**_{12}: this excludes the
inversion symmetries from the process of finding equivalence classes. For
example, Fig. 33 shows how the inversion I takes a major chord to a minor
chord.

Using the inversions leads to a smaller number of distinct chords, with each class of equivalent chords growing to accept new species we might think functionally different. However, outside the context of defining chords and scales, these additional symmetries are of considerable interest.

As we saw earlier, the symmetries of the graph **C**_{12} were
the twelve rotations Z_{12} = {*T*_{0} ...
*T*_{11}}. When we add a second (directed) cycle to get
**H**_{12}, we will have more symmetries. Namely, there is a
symmetry operation that interchanges the harmonic and melodic cycles. The twelve
rotations could be expressed as operations that added (mod 12) a number to the
vertex numbers; this new interchange operation (which we will call *M*) can
be thought of as the process that multiplies the vertex numbers by 5. For
example, this operation takes 1 to 5 and 4 to 8 (which is the same as 20, mod
12).

Fig. 34 shows the action of *M* on the graph
**H**_{12}. Recall that to be a symmetry, an operation has to
preserve adjacency: if vertices (notes) *x* and *y* are adjacent
before the operation, they must remain adjacent after. When we restricted our
attention to **C**_{12}, this meant the operation did not turn
a stepwise melody into one containing leaps. The only symmetries we had seen were
the cyclic rotations, or musical transpositions. The new symmetries convert
melodic steps into harmonic ones. For example, to find the class of species that
make up the pentatonic scale, we took a particular species of that scale, and
looked at its images under the operations of rotation, which are symmetries of
*H*_{12}. The reader may verify that taking the species {0, 1, 2, 3,
4} = {C, C#, D, Ef, E} = 111110000000, and applying *M* yields {0, 5, 10, 3,
8} = {C, F, Bf, Ef, Af} = 100101001010, which is a species of the pentatonic
scale.

The observation that M interchanges the two cycles leads us to conclude that on the fingerboard, its action is to exchange the vertical and horizontal directions. More precisely, applying M reflects the fingerboard across an axis going throught the notes 3 and 9, as shown in Fig. 35. Interestingly, reflecting across the axis through the notes 0 and 6 has the same effect, so that either of these operations may be thought of as the physical interpretation of the operation M.

Including the operation *M* expands the group Z_{12} into something
larger. Now, in addition to the operations {*T*_{0} ..
*T*_{11}} we have *M*, and all its products with the elements
of Z_{12}: {*M* (=*T*_{0}*M*),
*T*_{1}*M*, *T*_{2}*M*, ..
*T*_{11}*M*}. This group of twenty-four elements is the
symmetry group of the directed, uncolored graph *H*_{12}. If we now
allow reflections, we get a group of forty-eight members, including each of the
twenty-four symmetries of the *H*_{12}, and the product of each of
those symmetries with the reflection *I*. We will call this group
H_{12}, as it is the symmetry group of the non-directed graph
*H*_{12}. We will examine the action of these symmetries on patterns
in the fingerboard. To facilitate this investigation, we present in Table 7 a
complete list (from [4]) of these symmetries, their action on the numbers 0-11,
and their inverses. The action of each symmetry is given in terms of its cycle
structure. For example, the action of *T*_{1}*M* is given by
0-1-6-7, 2-11- 8-5, 3-4-9-10. This means that *T*_{1}*M*(0) =
1, *T*_{1}*M*(1) = 6, *T*_{1}*M*(6) = 7,
*T*_{1}*M*(7) = 0, and so forth. In addition, an alternate
notation is given for each element, consisting of two numbers surrounded by
<angled brackets>.

operation cycles inverseT_{0}<1,0> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 selfT_{1}<1,1> 0-1-2-3-4-5-6-7-8-9-10-11T_{11}T_{2}<1,2> 0-2-4-6-8-10, 1-3-5-7-9-11T_{10}T_{3}<1,3> 0-3-6-9, 1-4-7-10, 2-5-8-11T_{9}T_{4}<1,4> 0-4-8, 1-5-9, 2-6-10, 3-7-11T_{8}T_{5}<1,5> 0-5-10-3-8-1-6-11-4-9-2-7T_{7}T_{6}<1,6> 0-6, 1-7, 2-8, 3-9, 4-10, 5-11 selfT_{7}<1,7> 0-7-2-9-4-11-6-1-8-3-10-5T_{5}T_{8}<1,8> 0-8-4, 1-9-5, 2-10-6, 3-11-7T_{4}T_{9}<1,9> 0-9-6-3, 1-10-7-4, 2-11-8-5T_{3}T_{10}<1,10> 0-10-8-6-4-2, 1-11-9-7-5-3T_{2}T_{11}<1,11> 0-11-10-9-8-7-6-5-4-3-2-1T_{1}M_{ }<5,0> 0, 1-5, 2-10, 3, 4-8, 6, 7-11, 9 selfT_{1}M<5,1> 0-1-6-7, 2-11-8-5, 3-4-9-10T_{7}MT_{2}M<5,2> 0-2, 1-7, 3-5, 4-10, 6-8, 9-11 selfT_{3}M<5,3> 0-3-6-9, 1-8-7-2, 4-11-10-5T_{9}MT_{4}M<5,4> 0-4, 1-9, 2, 3-7, 5, 6-10, 8, 11 selfT_{5}M<5,5> 0-5-6-11, 1-10-7-4, 2-3-8-9T_{11}MT_{6}M<5,6> 0-6, 1-11, 2-4, 3-9, 5-7, 8-10 selfT_{7}M<5,7> 0-7-6-1, 2-5-8-11, 3-10-9-4T_{1}MT_{8}M<5,8> 0-8, 1, 2-6, 3-11, 4, 5-9, 10, 7 selfT_{9}M<5,9> 0-9-6-3, 1-2-7-8, 4-5-10-11T_{3}MT_{10}M<5,10> 0-10, 1-3, 2-8, 4-6, 5-11, 7-9 selfT_{11}M<5,11> 0-11-6-5, 1-4-7-10, 2-9-8-3T_{5}MMI_{ }<7,0> 0, 1-7, 2, 3-9, 4, 5-11, 6, 8, 10 selfT_{1}MI<7,1> 0-1-8-9-4-5, 2-3-10-11-6-7T_{5}MIT_{2}MI<7,2> 0-2-4-6-8-10, 1-9-5, 3-11-7T_{10}MIT_{3}MI<7,3> 0-3, 1-10, 2-5, 4-7, 6-9, 8-11 selfT_{4}MI<7,4> 0-4-8, 1-11-9-7-5-3, 2-6-10T_{8}MIT_{5}MI<7,5> 0-5-4-9-8-1, 2-7-6-11-10-3T_{1}MIT_{6}MI<7,6> 0-6, 1, 2-8, 3, 4-10, 5, 7, 9, 11 selfT_{7}MI<7,7> 0-7-8-3-4-11, 1-2-9-10-5-6T_{11}MIT_{8}MI<7,8> 0-8-4, 1-3-5-7-9-11, 2-10-6T_{4}MIT_{9}MI<7,9> 0-9, 1-4, 2-11, 3-6, 5-8, 7-10 selfT_{10}MI<7,10> 0-10-8-6-4-2, 1-5-9, 3-7-11T_{2}MIT_{11}MI<7,11> 0-11-4-3-8-7, 1-6-5-10-9-2T_{7}MII_{ }<11,0> 0, 1-11, 2-10, 3-9, 4-8, 5-7, 6 selfT_{1}I<11,1> 0-1, 2-11, 3-10, 4-9, 5-8, 6-7 selfT_{2}I<11,2> 0-2, 1, 3-11, 4-10, 5-9, 6-8, 7 selfT_{3}I<11,3> 0-3, 1-2, 4-11, 5-10, 6-9, 7-8 selfT_{4}I<11,4> 0-4, 1-3, 2, 5-11, 6-10, 7-9, 8 selfT_{5}I<11,5> 0-5, 1-4, 2-3, 6-11, 7-10, 8-9 selfT_{6}I<11,6> 0-6, 1-5, 2-4, 3, 7-11, 8-10, 9 selfT_{7}I<11,7> 0-7, 1-6, 2-5, 3-4, 8-11, 9-10 selfT_{8}I<11,8> 0-8, 1-7, 2-6, 3-5, 4, 9-11, 10 selfT_{9}I<11,9> 0-9, 1-8, 2-7, 3-6, 4-5, 10-11 selfT_{10}I<11,10> 0-10, 1-9, 2-8, 3-7, 4-6, 5, 11 selfT_{11}I<11,11> 0-11, 1-10, 2-9, 3-8, 4-7, 5-6 self

The angle bracket notation may be explained as follows: if the operator *T*
has associated with it the symbol <a,b>, it means that *T*(*x*) =
a*x* + *b* (mod 12). Why this works is easy to see by looking at the
second block of operations. For example, we have already remarked that the
symmetry *M* corresponds to multiplying the vertex numbers by five, so its
bracket symbol is <5,0>. Similarly, the symmetry *T*_{1}*M*
corresponds to performing *M* (multiplication by five), followed by the
operation *T*_{1} (addition of one), so its symbol will be
<5,1>. A similar argument holds for all the symmetries in H_{12}. In
extending this argument, we note that multiplication by any of the integers
having no common factors with twelve corresponds to some symmetry:
*T*_{0} is multiplication by one, *M* is multiplication by
five, *MI* is multiplication by seven, and *I* is multiplication by
eleven.

The above listing gives a complete account of the members of H_{12} and
their actions, but it is not graphically illuminating. For example, Figures 34
and 35 illustrate the action of the operator *M* at least as well as the
listing of the cycle structure. In Fig. 34, the arrows indicate that the
operation exchanges 1 & 5, 2 & 10, etc., and leaves 0, 3, 6, and 9 fixed. Fig.
35 indicates that on the fingerboard, reflection across the given axis has the
same effect.

Fig. 36 shows the action of all of the forty-eight members of H_{12} on
the graph *H*_{12} drawn as a circular array of vertices and as a
section of the fingerboard. Some comments on the general patterns involved
follow.

Where *H*_{12} is drawn as a circle, the edges are omitted for
clarity. The note zero is understood to be at the top. The lines drawn between
vertices indicate the images of those vertices under the operation. For example,
the illustration for *T*_{3} shows that 0 -> 3 -> 6 -> 9 -> 0. There
are two other cycles for this operation. In the diagram, each appears as a square
connecting four vertices. The direction of each cycle is indicated by an arrow on
one of its edges. In the case when two vertices are exchanged by the operation,
as in the case of *M* exchanging 1 and 5, no arrow is drawn.

Where the graph is drawn as a 12-note section of the fingerboard (calling notes
identical that are situated opposite each other) each note or vertex in the graph
is shown as a solid black dot. The note zero is understood to be at the corners.
As with the circular graph, the edges are omitted. An axis of reflection is
indicated by a center line: alternate long and short segments. The operation of
**glide reflection** is indicated by a center line with half-arrows on
opposite sides of the line. For example, the operation
*T*_{1}*M* is a glide reflection, and may be described as
follows. Reflect the fingerboard across (either) axis. Now the half-arrows have
exchanged sides. Next slide the fingerboard in the direction of the arrows, until
the trailing arrow is superimposed with the leading arrow’s former position. If
the direction of glide is immaterial, the arrows are shown as bidirectional.

The operation of rotation through 180° is indicated by an open circle about the point of rotation. Finally, the operation of translation is shown by an arrow. The arrow is “anchored” to a particular note, and points to that note’s image, although every note is understood to move in a parallel motion.

The group breaks naturally into four subsets (the technical term is
**cosets**.) In the first, we have just the twelve rotations
{*T*_{0} . . *T*_{11}}. As the subscript increases, the
operation is seen on the circular graph to break down into a larger number of
progressively smaller cycles, until *T*_{6} is composed of six
cycles of length two. Then cycles reverse the process and their direction. On the
fingerboard, the operations are just translations. One might draw them as just
translations of length zero to eleven, but we choose to draw them as completely
embedded within the patch of fingerboard shown. Several equivalent translations
are drawn.

The next coset consists of the operation *M* and its products with the
rotations. In the circular format, these are seen as operations that have a
particularly “square” character. The lines connecting notes and their images
always intersect at right angles, and often form little rectangles. On the
fingerboard, there are three reflections and nine glide reflections. For each of
these there are two possible axes, and both these axes are parallel to the short
side of the fingerboard. For the glide reflections, different translation
distances occur. In addition, for three of these (for example, for
*T*_{2}*M*) the glide may be taken the same distance in either
direction, and so is indicated by double-headed half-arrows. For the other glide
reflections, one might also move opposite the arrows after reflection, but the
distance would be farther. We also note that the axes of reflection and glide
reflection move progressively down and to the right as the subscript
increases.

Next we examine the coset formed by the operation *MI* and its products with
the rotations. In the circular format, these operations have a hexagonal
character. On the fingerboard, we have again reflections and glide reflections,
this time across axes parallel to the long side of the fingerboard. As before,
these axes move across the fingerboard, this time upwards and to the right.

Finally, the coset consisting of *I* and its products with the rotations. In
the circular format these appear as pure reflections across an axis that
precesses clockwise. On the fingerboard, they are rotations about any one of four
points. These four points form a small rectangle, whose height and width are half
of the corresponding dimensions of the full patch of fingerboard. This rectangle
is seen to move slowly to the right, (with its corners disappearing off one end
of the fingerboard and reappearing on the other), until after twelve steps it
will return to its starting place.

The combinatorial analysis of scales, chords, and species yields valuable insight into the particularly distinguished nature of the standard pentatonic and diatonic scales within the 12-note system. It also provides a language in which to press further inquiries. I believe the treasures of the diatonic pattern are not yet mined out.

The recognition of *H*_{12} as a description of the connectivity of the fingerboard of a string
instrument helps us apply these combinatorial tools to the study of fingering patterns. The
equivalent representations of *H*_{12} as a circular graph, a planar periodic graph, or a graph
embedded in a torus give an attractive geometric interpretation to the elements of the previously
described group H_{12}. These automorphisms of the fingerboard are an exhaustive catalog of the
operations that may be performed on musical patterns without disturbing their interval content --
that is, without doing topological violence to the embedding space.
Future research will include investigating cellular automata (for example, Conway’s
ubiquitous game of “Life”) on such graphs as *H*_{12}. These could be considered as evolving chords
or scales, with the resulting motions determined by the local laws of evolution.

I would like to thank Professor Daniel Hitt for steering me toward the *Journal of
Music Theory* in the stacks at the UC Santa Cruz library, and Jonathan V. Post for
bringing Reiner’s paper to my attention. I am indebted to Professor Dragan
Marusic for his stimulating discussion about the symmetries of
*H*_{12}, and to Professor Gerhard Ringel for his advice and
support. Finally, I owe a great deal to Linda Wahler, whose careful and thorough
critique of this paper in its early stages shaped it considerably.

[1] Victor Zuckerkandl, *Man the Musician, Sound and Symbol*, vol. 2,
(Princeton University Press, Princeton NJ, 1973).

[2] David L. Reiner, “Enumeration in Music Theory”, *Am Math. Monthly*, Jan 1985.

[3] Roger N. Shepard, “Demonstrations of Circular Components of Pitch”, *J. Audio
Eng. Soc.*, vol. 31, pp. 641-649 (1983 Sep.).

[4] Daniel Starr, “Sets, Invariance and Partitions”, *J. Music Theory*,
vol. 22, (Spring 1978).
(Note that the entries in Starr’s table for *T*_{2}*MI* and
*T*_{10}*MI* are in error.)

Andrew Duncan was born in London, U.K., in 1960. He received a B.S. degree in Engineering and Applied Science from the California Institute of Technology in 1983. He taught high school physics for 2 years in Pasadena CA, while at Caltech, and for another year after graduation. He then worked for several years with Dr. Marshall Buck at Cerwin-Vega! Inc. In 1986 Mr. Duncan moved to Santa Cruz CA, where he worked as a consultant at E-mu Systems Inc., writing software for electronic synthesizers, and studying mathematics at the University of California.

In 1988 he published the paper “The Analytic Impulse”, *J. Audio Eng. Soc.*,
vol. 6, no. 5, pp 315-327 (1988 May) on the mathematics of energy-time curves,
for which he subsequently received an AES Publication Award. In 1989 he received
am M.A. in Pure Mathematics. Mr. Duncan is now [1991] working as a consultant at
the MAMA Foundation, a small nonprofit recording studio founded by Gene
Czerwinski, owner of Cerwin-Vega! He spends much of his time wiring XLR
connectors in ways not intended by any standards organization.

A member of the AES and AMS, Mr. Duncan’s interests center on music, mathematics, and their union and intersection. He plays the piano, guitar, electric bass, and Chapman Stick, studying the music of Scott Joplin, John Fahey, Phil Lesh and J.S. Bach. He is working with Harvey Starr of Starrswitch Inc. on a custom MIDI controller that will enable a musician to extend a two-handed tapping technique to a fully electronic fingerboard. His current academic interests include object-oriented programming techniques and tiling theory. For relaxation, he is a competitive swimmer.