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]

11   THE FINGERBOARD

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


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.



Fig. 16


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.



Fig. 17


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.



Fig. 18


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.



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.



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.



Fig. 21


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.



Fig. 22


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

Table 6

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.


12   THE GRAPH H12

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.



Fig. 23


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.



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


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.



Fig. 26


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]).

Fig. 27


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 H12, 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 H12 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.



Fig. 28


We illustrate this observation by the following process. Going back to our graph of C12, 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 H12. 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.


13   THE CIRCLES OF FOURTHS AND FIFTHS

If we travel around the outside (horizontal) loop of H12, 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, Bf, Ef, Af (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.


14   "NEARBY" FINGERBOARDS

The graph H12 may be thought of as a member of a family of graphs {Hn}, 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 H8, jumping every third vertex takes us to all vertices. Jumping every other vertex will not. All Hn 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, H8 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 Hn 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 H14, 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 H12.

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