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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.
A [[scale]] is said to be a '''constant structure''' ('''CS''') if its [[interval class]]es 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.
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 (<code>4-4-3-2-2-6-1</code>) 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.
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 (<code>4-4-3-2-2-6-1</code>) 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 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).
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 ==
== Examples ==
=== Pentatonic scales ===


This common pentatonic scale is a constant structure: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1
This common pentatonic scale is a constant structure: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1
Line 62: Line 66:
| 2/1
| 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".
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.


This pentatonic scale is not a constant structure: 1/1 - 25/24 - 6/5 - 3/2 - 5/3 - 2/1
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:
Its interval matrix:
Line 117: Line 121:
| 2/1
| 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.
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.


Another example of a familiar scale that is not CS is the 7-note diatonic scale in [[12edo]].
=== Diatonic scales ===


Interval matrix as steps of 12edo:
Another example of a familiar scale that is ''not'' CS is the [[12edo]] tuning of the 7-note [[diatonic scale]].
 
Its interval matrix:


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 134: Line 140:
! (8)
! (8)
|-
|-
! 0
! 0\12
| 0
| 0\12
| 2
| 2\12
| 4
| 4\12
| 5
| 5\12
| 7
| 7\12
| 9
| 9\12
| 11
| 11\12
| 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
| <span style="background-color: #ffcc44;">6\12</span>
| 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
| <span style="background-color: #ffcc44;">6\12</span>
| 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:
{| class="wikitable center-all"
!
! 1
! 2
! 2
| 0
! 3
| 2
! 4
| 3
! 5
| 5
! 6
| 7
! 7
| 9
! (8)
| 10
|-
| 12
! 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
|-
|-
! 4
! 10\31
| 0
| 0\31
| 1
| 3\31
| 3
| 8\31
| 5
| 13\31
| 7
| 18\31
| 8
| 21\31
| 10
| 26\31
| 12
| 31\31
|-
|-
! 5
! 13\31
| 0
| 0\31
| 2
| 5\31
| 4
| 10\31
| <span style="background-color: #ffcc44;">6</span>
| <span style="background-color: #ffcc44;">15\31</span>
| 7
| 18\31
| 9
| 23\31
| 11
| 28\31
| 12
| 31\31
|-
|-
! 7
! 18\31
| 0
| 0\31
| 2
| 5\31
| 4
| 10\31
| 5
| 13\31
| 7
| 18\31
| 9
| 23\31
| 10
| 26\31
| 12
| 31\31
|-
|-
! 9
! 23\31
| 0
| 0\31
| 2
| 5\31
| 3
| 8\31
| 5
| 13\31
| 7
| 18\31
| 8
| 21\31
| 10
| 26\31
| 12
| 31\31
|-
|-
! 11
! 28\31
| 0
| 0\31
| 1
| 3\31
| 3
| 8\31
| 5
| 13\31
| <span style="background-color: #ffcc44;">6</span>
| <span style="background-color: #ffcc44;">16\31</span>
| 8
| 21\31
| 10
| 26\31
| 12
| 31\31
|}
|}


Interval matrix as note names:
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:


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 218: Line 312:
! (8)
! (8)
|-
|-
! C
! 0\22
| C
| 0\22
| D
| 4\22
| E
| 8\22
| F
| 9\22
| G
| 13\22
| A
| 17\22
| B
| 21\22
| C
| 22\22
|-
|-
! D
! 4\22
| C
| 0\22
| D
| 4\22
| Eb
| 5\22
| F
| 9\22
| G
| 13\22
| A
| 17\22
| Bb
| 18\22
| C
| 22\22
|-
|-
! E
! 8\22
| C
| 0\22
| Db
| 1\22
| Eb
| 5\22
| F
| 9\22
| G
| 13\22
| Ab
| 14\22
| Bb
| 18\22
| C
| 22\22
|-
|-
! F
! 9\22
| C
| 0\22
| D
| 4\22
| E
| 8\22
| <span style="background-color: #ffcc44;">F#</span>
| <span style="background-color: #ffcc44;">12\22</span>
| G
| 13\22
| A
| 17\22
| B
| 21\22
| C
| 22\22
|-
|-
! G
! 13\22
| C
| 0\22
| D
| 4\22
| E
| 8\22
| F
| 9\22
| G
| 13\22
| A
| 17\22
| Bb
| 18\22
| C
| 22\22
|-
|-
! A
! 17\22
| C
| 0\22
| D
| 4\22
| Eb
| 5\22
| F
| 9\22
| G
| 13\22
| Ab
| 14\22
| Bb
| 18\22
| C
| 22\22
|-
|-
! B
! 21\22
| C
| 0\22
| Db
| 1\22
| Eb
| 5\22
| F
| 9\22
| <span style="background-color: #ffcc44;">Gb</span>
| <span style="background-color: #ffcc44;">10\22</span>
| Ab
| 14\22
| Bb
| 18\22
| C
| 22\22
|}
|}
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 ==
== Density of CS scales in EDOs ==
Line 399: Line 491:
| 541/52377
| 541/52377
|}
|}
== Novel terminology ==
An interval that occurs in a scale is ''CS-consistent''{{idiosyncratic}} 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 ==
== See also ==


* [[Gallery of CS Scales]]
* [[Gallery of CS Scales]]
* [[Scale properties simplified]]
* [[Glossary of scale properties]]
* [[epimorphic]]
* [[epimorphic]]
* [http://tonalsoft.com/enc/c/constant-structure.aspx Constant structure] (Tonalsoft Encyclopedia)
* [http://tonalsoft.com/enc/c/constant-structure.aspx Constant structure] (Tonalsoft Encyclopedia)
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