COMBINATORIAL MUSIC THEORY

Journal of the Audio Engineering Society, vol. 39, pp. 427-448. (1991 June).
© 1991, Audio Engineering Society and Andrew Duncan. All rights reserved.



Andrew Duncan
aduncan@cs.ucsb.edu
71035.1100@compuserve.com



[Graphs | Scales and Chords | The Fingerboard | Symmetries]

15   SYMMETRIES

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

x  ~ y  if and only if T(x) ~ T(y).

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.



Fig. 29


The graph shown on the left has four vertices, and the operation T1 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.



Fig. 30


By contrast, the operation T1 is a symmetry of the graph C12: 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 T0 through T11 are symmetries of C12. This set of twelve symmetry operations has a conventional name: we shall be calling it Z12.

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.


16   SYMMETRY GROUPS

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:

Ta (Tb (x)) = (Ta Tb ) (x).

The left-hand side of the equation describes a well-defined value: the number we get when we apply Tb to x, and then apply Ta to the result. On the right, the expression Ta Tb means the composition of the two operators: a third operator Tc whose action on all x is the same as the action produced by successive application of Tb and Ta. (Note the reversed order: Ta Tb means do Tb first and then Ta.) The fact that we can always find such a Tc is described by saying that Z12 is closed: in combining two members of the set, the result never strays outside the set. The closure of the symmetry operations is of primary importance if we are to make of them some sort of algebra. In fact, to do any useful work, we must really have several more properties. In particular, we need the existence of an identity element (here played by T0) and of inverses (here the inverse of Tk is evidently T12-k ). Additionally, symmetries combine in a way that satisfies the associative property: order of grouping makes no difference, so that (Ta Tb )Tc = Ta (Tb Tc).

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 C12 is a twelve-element group. For the non- directed graph C12, there is more to say.

Any symmetry of the directed graph C12 will also work for the non-directed graph C12, so the symmetry group of the undirected graph will contain Z12. 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 C12 are the elements of the dihedral group of order 24, D12. These elements are the 12 cyclic rotations of Z12, 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).



Fig. 31


Any element in this group may be expressed as the composition of a cyclic rotation Tn, 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.



Fig. 32


It is now apparent why we wished to consider the scales and chords to be 2-colorings of the directed graph C12: 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.



Fig. 33


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.


17   SYMMETRIES OF THE GRAPH H12

As we saw earlier, the symmetries of the graph C12 were the twelve rotations Z12 = {T0 ... T11}. When we add a second (directed) cycle to get H12, 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


Fig. 34 shows the action of M on the graph H12. 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 C12, 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 H12. 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.



Fig. 35


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 Z12 into something larger. Now, in addition to the operations {T0 .. T11} we have M, and all its products with the elements of Z12: {M (=T0M), T1M, T2M, .. T11M}. This group of twenty-four elements is the symmetry group of the directed, uncolored graph H12. If we now allow reflections, we get a group of forty-eight members, including each of the twenty-four symmetries of the H12, and the product of each of those symmetries with the reflection I. We will call this group H12, as it is the symmetry group of the non-directed graph H12. 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 T1M is given by 0-1-6-7, 2-11- 8-5, 3-4-9-10. This means that T1M(0) = 1, T1M(1) = 6, T1M(6) = 7, T1M(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                        inverse

T0    <1,0>    0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11     self
T1    <1,1>    0-1-2-3-4-5-6-7-8-9-10-11                T11
T2    <1,2>    0-2-4-6-8-10, 1-3-5-7-9-11               T10
T3    <1,3>    0-3-6-9, 1-4-7-10, 2-5-8-11              T9
T4    <1,4>    0-4-8, 1-5-9, 2-6-10, 3-7-11             T8
T5    <1,5>    0-5-10-3-8-1-6-11-4-9-2-7                T7
T6    <1,6>    0-6, 1-7, 2-8, 3-9, 4-10, 5-11           self
T7    <1,7>    0-7-2-9-4-11-6-1-8-3-10-5                T5
T8    <1,8>    0-8-4, 1-9-5, 2-10-6, 3-11-7             T4
T9    <1,9>    0-9-6-3, 1-10-7-4, 2-11-8-5              T3
T10   <1,10>   0-10-8-6-4-2, 1-11-9-7-5-3               T2
T11   <1,11>   0-11-10-9-8-7-6-5-4-3-2-1                T1

M     <5,0>    0, 1-5, 2-10, 3, 4-8, 6, 7-11, 9         self
T1M   <5,1>    0-1-6-7, 2-11-8-5, 3-4-9-10              T7M
T2M   <5,2>    0-2, 1-7, 3-5, 4-10, 6-8, 9-11           self
T3M   <5,3>    0-3-6-9, 1-8-7-2, 4-11-10-5              T9M
T4M   <5,4>    0-4, 1-9, 2, 3-7, 5, 6-10, 8, 11         self
T5M   <5,5>    0-5-6-11, 1-10-7-4, 2-3-8-9              T11M
T6M   <5,6>    0-6, 1-11, 2-4, 3-9, 5-7, 8-10           self
T7M   <5,7>    0-7-6-1, 2-5-8-11, 3-10-9-4              T1M
T8M   <5,8>    0-8, 1, 2-6, 3-11, 4, 5-9, 10, 7         self
T9M   <5,9>    0-9-6-3, 1-2-7-8, 4-5-10-11              T3M	
T10M  <5,10>   0-10, 1-3, 2-8, 4-6, 5-11, 7-9           self	
T11M  <5,11>   0-11-6-5, 1-4-7-10, 2-9-8-3              T5M

MI    <7,0>    0, 1-7, 2, 3-9, 4, 5-11, 6, 8, 10        self
T1MI  <7,1>    0-1-8-9-4-5, 2-3-10-11-6-7               T5MI
T2MI  <7,2>    0-2-4-6-8-10, 1-9-5, 3-11-7              T10MI
T3MI  <7,3>    0-3, 1-10, 2-5, 4-7, 6-9, 8-11           self
T4MI  <7,4>    0-4-8, 1-11-9-7-5-3, 2-6-10              T8MI
T5MI  <7,5>    0-5-4-9-8-1, 2-7-6-11-10-3               T1MI
T6MI  <7,6>    0-6, 1, 2-8, 3, 4-10, 5, 7, 9, 11        self
T7MI  <7,7>    0-7-8-3-4-11, 1-2-9-10-5-6               T11MI
T8MI  <7,8>    0-8-4, 1-3-5-7-9-11, 2-10-6              T4MI
T9MI  <7,9>    0-9, 1-4, 2-11, 3-6, 5-8, 7-10           self
T10MI <7,10>   0-10-8-6-4-2, 1-5-9, 3-7-11              T2MI
T11MI <7,11>   0-11-4-3-8-7, 1-6-5-10-9-2               T7MI	

I     <11,0>   0, 1-11, 2-10, 3-9, 4-8, 5-7, 6          self
T1I   <11,1>   0-1, 2-11, 3-10, 4-9, 5-8, 6-7           self
T2I   <11,2>   0-2, 1, 3-11, 4-10, 5-9, 6-8, 7          self
T3I   <11,3>   0-3, 1-2, 4-11, 5-10, 6-9, 7-8           self
T4I   <11,4>   0-4, 1-3, 2, 5-11, 6-10, 7-9, 8          self
T5I   <11,5>   0-5, 1-4, 2-3, 6-11, 7-10, 8-9           self
T6I   <11,6>   0-6, 1-5, 2-4, 3, 7-11, 8-10, 9          self
T7I   <11,7>   0-7, 1-6, 2-5, 3-4, 8-11, 9-10           self
T8I   <11,8>   0-8, 1-7, 2-6, 3-5, 4, 9-11, 10          self
T9I   <11,9>   0-9, 1-8, 2-7, 3-6, 4-5, 10-11           self
T10I  <11,10>  0-10, 1-9, 2-8, 3-7, 4-6, 5, 11          self
T11I  <11,11>  0-11, 1-10, 2-9, 3-8, 4-7, 5-6           self

Table 7

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) = ax + 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 T1M corresponds to performing M (multiplication by five), followed by the operation T1 (addition of one), so its symbol will be <5,1>. A similar argument holds for all the symmetries in H12. In extending this argument, we note that multiplication by any of the integers having no common factors with twelve corresponds to some symmetry: T0 is multiplication by one, M is multiplication by five, MI is multiplication by seven, and I is multiplication by eleven.


18   SYMMETRIES OF THE FINGERBOARD

The above listing gives a complete account of the members of H12 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 H12 on the graph H12 drawn as a circular array of vertices and as a section of the fingerboard. Some comments on the general patterns involved follow.



Fig. 36a




Fig. 36b




Fig. 36c




Fig. 36d


Where H12 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 T3 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 T1M 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 {T0 . . T11}. As the subscript increases, the operation is seen on the circular graph to break down into a larger number of progressively smaller cycles, until T6 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 T2M) 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.


19   CONCLUSION

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 H12 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 H12 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 H12. 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 H12. These could be considered as evolving chords or scales, with the resulting motions determined by the local laws of evolution.

20   ACKNOWLEDGEMENTS

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 H12, 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.

REFERENCES

[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 T2MI and T10MI are in error.)


THE AUTHOR

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.

[Since the publication of this paper, Mr. Duncan worked at Philips Media, writing software for MPEG digital video and interactive multimedia. He is now at the University of California at Santa Barbara pursuing his Ph.D. in Computer Science, studying compilers and interpreters.]

[Graphs | Scales and Chords | The Fingerboard | Symmetries]