Constant structure

A scale is said to be a constant structure (CS) if its interval classes are distinct. That is, each interval size that occurs in the scale always spans 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 is a 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 are constant structures, and proper but not strictly proper scales are not. Improper scales generally are. 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 a CS.

To determine if a scale is a 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 is not a 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

Novel terminology

An interval that occurs in a scale is CS-consistent^{[idiosyncratic term]} if it always subtends the same number of scale steps. A scale is thus CS if and only if all its intervals are CS-consistent. This term could be useful because someone might only care about certain primes in a subgroup being CS-consistent.

See also

• Gallery of CS Scales
• Glossary of scale properties
• epimorphic
• Constant structure (Tonalsoft Encyclopedia)
• Introduction to Erv Wilson's Moments of Symmetry