Andrew Duncan
aduncan@cs.ucsb.edu
71035.1100@compuserve.com
We define a k-species to be a set of k notes out of our 12-note scale. We represent each species by a vertex 2-coloring of C12, with black vertices representing notes in the species and white vertices representing the notes left out. We may also represent each species by a binary number, with 1s representing notes in the species, and 0s for the notes left out. For ease of interpretation, the high-order digit in the binary number represents the top (zero) vertex. For example, the binary numbers representing the species of Fig. 7a and 7b are 101011010101 ( = 2773, in base 10) and 011010110101 ( = 1717) respectively. (Observe that this introduces a reversal: the note 0 is represented by the bit having place value 211 and the note 11 by the bit with value 20. The following analysis could proceed, without this reversal, essentially unaltered.)
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.
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
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:
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 very beautiful), and the direct approach (listing them all) is tedious. For now will observe some of the results of the direct approach.
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
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.)
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.
101010010100 001010100101
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
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.