Constant structure
A scale is said to have constant structure (CS) if its generic interval classes are distinct. That is, each interval size that occurs always spans the same number of steps. This means that you never get something like an interval being counted as a fourth one place, and a fifth another place.
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 is CS, all possible intervals between scale steps must be evaluated. An easy way to do this is with an interval matrix (Scala can do this for you). A CS scale will never have the same interval appear in multiple columns of the matrix (columns correspond to generic interval classes).
Examples
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, is always the "fourth" of this scale - never the "third" or "fifth".
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 both the "second" and "third" steps of the scale. Thus, this scale does not have constant structure.
Another example of a familiar scale that is not CS is the 7-note diatonic scale in 12edo.
Interval matrix as steps of 12edo:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | (8) | |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 2 | 4 | 5 | 7 | 9 | 11 | 12 |
| 2 | 0 | 2 | 3 | 5 | 7 | 9 | 10 | 12 |
| 4 | 0 | 1 | 3 | 5 | 7 | 8 | 10 | 12 |
| 5 | 0 | 2 | 4 | 6 | 7 | 9 | 11 | 12 |
| 7 | 0 | 2 | 4 | 5 | 7 | 9 | 10 | 12 |
| 9 | 0 | 2 | 3 | 5 | 7 | 8 | 10 | 12 |
| 11 | 0 | 1 | 3 | 5 | 6 | 8 | 10 | 12 |
Interval matrix as note names:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | (8) | |
|---|---|---|---|---|---|---|---|---|
| C | C | D | E | F | G | A | B | C |
| D | C | D | Eb | F | G | A | Bb | C |
| E | C | Db | Eb | F | G | Ab | Bb | C |
| F | C | D | E | F# | G | A | B | C |
| G | C | D | E | F | G | A | Bb | C |
| A | C | D | Eb | F | G | Ab | Bb | C |
| B | C | Db | Eb | F | Gb | Ab | Bb | C |
F# and Gb are the same pitch (600 cents) in 12edo, and this interval occurs as both an (augmented) fourth and a (diminished) fifth - so not constant structure. However, a meantone tuning of this scale, in which F# is narrower than Gb, would have constant structure. As would a pythagorean tuning or superpyth tuning such as 22edo, in which F# is wider than Gb.
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 |