Constant structure: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Internal link, misc. edits
Bcmills (talk | contribs)
811 edits
clarified and expanded diatonic scale example and interval matrix discussion
Line 1: Line 1:
A [[scale]] is said to have '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always [[subtend]]s 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 have '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always [[subtend]]s 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.


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.
Line 5: Line 5:
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 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.


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 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 ==
== 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 64:
| 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 119:
| 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 does not have 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 138:
! (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
! 2\12
| 0
| 0\12
| 2
| 2\12
| 3
| 3\12
| 5
| 5\12
| 7
| 7\12
| 9
| 9\12
| 10
| 10\12
| 12
| 12\12
|-
|-
! 4
! 4\12
| 0
| 0\12
| 1
| 1\12
| 3
| 3\12
| 5
| 5\12
| 7
| 7\12
| 8
| 8\12
| 10
| 10\12
| 12
| 12\12
|-
|-
! 5
! 5\12
| 0
| 0\12
| 2
| 2\12
| 4
| 4\12
| <span style="background-color: #ffcc44;">6</span>
| <span style="background-color: #ffcc44;">6\12</span>
| 7
| 7\12
| 9
| 9\12
| 11
| 11\12
| 12
| 12\12
|-
|-
! 7
! 7\12
| 0
| 0\12
| 2
| 2\12
| 4
| 4\12
| 5
| 5\12
| 7
| 7\12
| 9
| 9\12
| 10
| 10\12
| 12
| 12\12
|-
|-
! 9
! 9\12
| 0
| 0\12
| 2
| 2\12
| 3
| 3\12
| 5
| 5\12
| 7
| 7\12
| 8
| 8\12
| 10
| 10\12
| 12
| 12\12
|-
|-
! 11
! 11\12
| 0
| 0\12
| 1
| 1\12
| 3
| 3\12
| 5
| 5\12
| <span style="background-color: #ffcc44;">6</span>
| <span style="background-color: #ffcc44;">6\12</span>
| 8
| 8\12
| 10
| 10\12
| 12
| 12\12
|}
|}


Interval matrix as note names:
The corresponding note names:


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 229: Line 233:
|-
|-
! D
! D
| C
| D
| D
| Eb
| E
| F
| F
| G
| G
| A
| A
| Bb
| B
| C
| C
| D
|-
|-
! E
! E
| C
| E
| Db
| Eb
| F
| F
| G
| G
| Ab
| A
| Bb
| B
| C
| C
| D
| E
|-
|-
! F
! F
| F
| G
| A
| <span style="background-color: #ffcc44;">B</span>
| C
| C
| D
| D
| E
| E
| <span style="background-color: #ffcc44;">F#</span>
| F
|-
! G
| G
| G
| A
| A
| B
| B
| C
|-
! G
| C
| C
| D
| D
Line 264: Line 271:
| F
| F
| G
| G
| A
| Bb
| C
|-
|-
! A
! A
| A
| B
| C
| C
| D
| D
| Eb
| E
| F
| F
| G
| G
| Ab
| A
| Bb
| C
|-
|-
! B
! B
| B
| C
| C
| Db
| D
| Eb
| E
| F
| <span style="background-color: #ffcc44;">F</span>
| <span style="background-color: #ffcc44;">Gb</span>
| G
| Ab
| A
| Bb
| B
| C
|}
 
In 12edo, the intervals from F to B and from B to F are the same size: 6\12, or 600 cents. From F to B, this interval spans four steps of our diatonic scale; but from B to F it spans five. Since the same 6\12 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
! 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
| <span style="background-color: #ffcc44;">15\31</span>
| 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
| <span style="background-color: #ffcc44;">16\31</span>
| 21\31
| 26\31
| 31\31
|}
|}


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.
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"
!
! 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
| <span style="background-color: #ffcc44;">12\22</span>
| 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
| <span style="background-color: #ffcc44;">10\22</span>
| 14\22
| 18\22
| 22\22
|}


== Density of CS scales in EDOs ==
== Density of CS scales in EDOs ==

Revision as of 15:03, 3 May 2024

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.

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 corresponding note names:

1 2 3 4 5 6 7 (8)
C C D E F G A B C
D D E F G A B C D
E E F G A B C D E
F F G A B C D E F
G G A B C D E F G
A A B C D E F G A
B B C D E F G A B

In 12edo, the intervals from F to B and from B to F are the same size: 6\12, or 600 cents. From F to B, this interval spans four steps of our diatonic scale; but from B to F it spans five. Since the same 6\12 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

See also

Retrieved from "https://en.xen.wiki/index.php?title=Constant_structure&oldid=142324"