Constant structure
A scale is said to have constant structure (CS) if its interval classes are distinct. That is, each interval size that occurs in the scale always subtends the same number of scale steps. This means that you never get something like an interval being counted as a fourth one place, and a fifth another place.
If a scale has constant structure, that scale can be mapped to an isomorphic keyboard or similar isomorphic instrument such that each chord with the same interval structure can be played using the same fingering shape.
The term "constant structure" was coined by Erv Wilson. In academic music theory, constant structure is called the partitioning property, but Erv got there first.
In terms of Rothenberg propriety, strictly proper scales have CS, and proper but not strictly proper scales do not. Improper scales generally do. However the 22edo scale C D E vF# G ^Ab B C (4-4-3-2-2-6-1) has both ambiguity (C-vF# 4th equals vF#-C 5th) and contradiction (^Ab-B 2nd exceeds E-G 3rd). The contradiction makes it improper and the ambiguity makes it not CS.
To determine if a scale has CS, all possible intervals between scale steps must be evaluated. An easy way to do this is with an interval matrix, in which each entry gives the interval spanning the number of scale steps indicated by the column, beginning with step indicated by the row. In a CS scale, each interval in the matrix must appear in only one column, corresponding to the “constant” number of steps for that interval.
Examples
Pentatonic scales
This common pentatonic scale is a constant structure: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1
Here is the interval matrix of this scale:
| 1 | 2 | 3 | 4 | 5 | (6) | |
|---|---|---|---|---|---|---|
| 1/1 | 1/1 | 9/8 | 5/4 | 3/2 | 5/3 | 2/1 |
| 9/8 | 1/1 | 10/9 | 4/3 | 40/27 | 16/9 | 2/1 |
| 5/4 | 1/1 | 6/5 | 4/3 | 8/5 | 9/5 | 2/1 |
| 3/2 | 1/1 | 10/9 | 4/3 | 3/2 | 5/3 | 2/1 |
| 5/3 | 1/1 | 6/5 | 27/20 | 3/2 | 9/5 | 2/1 |
Note that every interval always appears in the same position (column). For example, 3/2, which happens to appear three times, always spans four steps of this scale — never three or five.
In contrast, this pentatonic scale is not a constant structure: 1/1 - 25/24 - 6/5 - 3/2 - 5/3 - 2/1
Its interval matrix:
| 1 | 2 | 3 | 4 | 5 | (6) | |
|---|---|---|---|---|---|---|
| 1/1 | 1/1 | 25/24 | 6/5 | 3/2 | 5/3 | 2/1 |
| 25/24 | 1/1 | 144/125 | 36/25 | 8/5 | 48/25 | 2/1 |
| 6/5 | 1/1 | 5/4 | 25/18 | 5/3 | 125/72 | 2/1 |
| 3/2 | 1/1 | 10/9 | 4/3 | 25/18 | 8/5 | 2/1 |
| 5/3 | 1/1 | 6/5 | 5/4 | 36/25 | 9/5 | 2/1 |
Note the highlighted intervals that occur in more than one column. For example, 5/4 may occur as either two or three steps of the scale. Thus, this scale does not have constant structure.
Diatonic scales
Another example of a familiar scale that is not CS is the 12edo tuning of the 7-note diatonic scale.
Its interval matrix:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | (8) | |
|---|---|---|---|---|---|---|---|---|
| 0\12 | 0\12 | 2\12 | 4\12 | 5\12 | 7\12 | 9\12 | 11\12 | 12\12 |
| 2\12 | 0\12 | 2\12 | 3\12 | 5\12 | 7\12 | 9\12 | 10\12 | 12\12 |
| 4\12 | 0\12 | 1\12 | 3\12 | 5\12 | 7\12 | 8\12 | 10\12 | 12\12 |
| 5\12 | 0\12 | 2\12 | 4\12 | 6\12 | 7\12 | 9\12 | 11\12 | 12\12 |
| 7\12 | 0\12 | 2\12 | 4\12 | 5\12 | 7\12 | 9\12 | 10\12 | 12\12 |
| 9\12 | 0\12 | 2\12 | 3\12 | 5\12 | 7\12 | 8\12 | 10\12 | 12\12 |
| 11\12 | 0\12 | 1\12 | 3\12 | 5\12 | 6\12 | 8\12 | 10\12 | 12\12 |
The highlighted intervals, from F to B and from B to F, are the same size in 12edo: 6\12, or 600 cents. From F to B, this interval spans four steps of our diatonic scale (an “augmented fourth”); but from B to F it spans five (a “diminished fifth”). Since the same interval spans different numbers of scale steps at different points in the scale, this scale is not a constant structure.
However, in other tunings of the diatonic scale, the F–B and B–F intervals may have distinct sizes. For example, 31edo (meantone) tunes F–B and B–F to 15\31 (581¢) and 16\31 (619¢) respectively:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | (8) | |
|---|---|---|---|---|---|---|---|---|
| 0\31 | 0\31 | 5\31 | 10\31 | 13\31 | 18\31 | 23\31 | 28\31 | 31\31 |
| 5\31 | 0\31 | 5\31 | 8\31 | 13\31 | 18\31 | 23\31 | 26\31 | 31\31 |
| 10\31 | 0\31 | 3\31 | 8\31 | 13\31 | 18\31 | 21\31 | 26\31 | 31\31 |
| 13\31 | 0\31 | 5\31 | 10\31 | 15\31 | 18\31 | 23\31 | 28\31 | 31\31 |
| 18\31 | 0\31 | 5\31 | 10\31 | 13\31 | 18\31 | 23\31 | 26\31 | 31\31 |
| 23\31 | 0\31 | 5\31 | 8\31 | 13\31 | 18\31 | 21\31 | 26\31 | 31\31 |
| 28\31 | 0\31 | 3\31 | 8\31 | 13\31 | 16\31 | 21\31 | 26\31 | 31\31 |
Since each interval in the 31edo table appears in a consistent column, the 31edo tuning of the diatonic scale is a constant structure.
Similarly, the 22edo diatonic scale, which tunes F–B wider than B–F, is also a constant structure. Even though it has a four-scale-step interval that is larger than a five-step interval (making it “improper”), each distinct interval size still appears in only one column:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | (8) | |
|---|---|---|---|---|---|---|---|---|
| 0\22 | 0\22 | 4\22 | 8\22 | 9\22 | 13\22 | 17\22 | 21\22 | 22\22 |
| 4\22 | 0\22 | 4\22 | 5\22 | 9\22 | 13\22 | 17\22 | 18\22 | 22\22 |
| 8\22 | 0\22 | 1\22 | 5\22 | 9\22 | 13\22 | 14\22 | 18\22 | 22\22 |
| 9\22 | 0\22 | 4\22 | 8\22 | 12\22 | 13\22 | 17\22 | 21\22 | 22\22 |
| 13\22 | 0\22 | 4\22 | 8\22 | 9\22 | 13\22 | 17\22 | 18\22 | 22\22 |
| 17\22 | 0\22 | 4\22 | 5\22 | 9\22 | 13\22 | 14\22 | 18\22 | 22\22 |
| 21\22 | 0\22 | 1\22 | 5\22 | 9\22 | 10\22 | 14\22 | 18\22 | 22\22 |
Density of CS scales in EDOs
| EDO | Number of CS Scales | Percent of Scales CS | Corresponding Fraction |
|---|---|---|---|
| 1 | 1 | 100.0% | 1/1 |
| 2 | 1 | 100.0% | 1/1 |
| 3 | 2 | 100.0% | 1/1 |
| 4 | 2 | 66.7% | 2/3 |
| 5 | 5 | 83.3% | 5/6 |
| 6 | 4 | 44.4% | 4/9 |
| 7 | 11 | 61.1% | 11/18 |
| 8 | 11 | 36.7% | 11/30 |
| 9 | 22 | 39.3% | 11/28 |
| 10 | 20 | 20.2% | 20/99 |
| 11 | 45 | 24.2% | 15/62 |
| 12 | 47 | 14.0% | 47/335 |
| 13 | 85 | 13.5% | 17/126 |
| 14 | 88 | 7.6% | 88/1161 |
| 15 | 163 | 7.5% | 163/2182 |
| 16 | 165 | 4.0% | 11/272 |
| 17 | 294 | 3.8% | 49/1285 |
| 18 | 313 | 2.2% | 313/14532 |
| 19 | 534 | 1.9% | 89/4599 |
| 20 | 541 | 1.0% | 541/52377 |