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Constant structure: Difference between revisions

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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 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=
== Examples ==
 
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


Here is the interval matrix of this scale:
Here is the interval matrix of this scale:


{| class="wikitable"
{| class="wikitable center-all"
|-
!
| |
! 1
| | '''1'''
! 2
| | '''2'''
! 3
| | '''3'''
! 4
| | '''4'''
! 5
| | '''5'''
! (6)
| | '''(6)'''
|-
|-
| | '''1/1'''
! 1/1
| | 1/1
| 1/1
| | 9/8
| 9/8
| | 5/4
| 5/4
| | 3/2
| 3/2
| | 5/3
| 5/3
| | 2/1
| 2/1
|-
|-
| | '''9/8'''
! 9/8
| | 1/1
| 1/1
| | 10/9
| 10/9
| | 4/3
| 4/3
| | 40/27
| 40/27
| | 16/9
| 16/9
| | 2/1
| 2/1
|-
|-
| | '''5/4'''
! 5/4
| | 1/1
| 1/1
| | 6/5
| 6/5
| | 4/3
| 4/3
| | 8/5
| 8/5
| | 9/5
| 9/5
| | 2/1
| 2/1
|-
|-
| | '''3/2'''
! 3/2
| | 1/1
| 1/1
| | 10/9
| 10/9
| | 4/3
| 4/3
| | 3/2
| 3/2
| | 5/3
| 5/3
| | 2/1
| 2/1
|-
|-
| | '''5/3'''
! 5/3
| | 1/1
| 1/1
| | 6/5
| 6/5
| | 27/20
| 27/20
| | 3/2
| 3/2
| | 9/5
| 9/5
| | 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, is always the "fourth" of this scale - never the "third" or "fifth".
Line 66: Line 66:
Its interval matrix:
Its interval matrix:


{| class="wikitable"
{| class="wikitable center-all"
|-
!
| |
! 1
| | '''1'''
! 2
| | '''2'''
! 3
| | '''3'''
! 4
| | '''4'''
! 5
| | '''5'''
! (6)
| | '''(6)'''
|-
|-
| | '''1/1'''
! 1/1
| | 1/1
| 1/1
| | 25/24
| 25/24
| | <span style="background-color: #ffcc44;">6/5</span>
| <span style="background-color: #ffcc44;">6/5</span>
| | 3/2
| 3/2
| | <span style="background-color: #ffcc44;">5/3</span>
| <span style="background-color: #ffcc44;">5/3</span>
| | 2/1
| 2/1
|-
|-
| | '''25/24'''
! 25/24
| | 1/1
| 1/1
| | 144/125
| 144/125
| | 36/25
| 36/25
| | <span style="background-color: #ffcc44;">8/5</span>
| <span style="background-color: #ffcc44;">8/5</span>
| | 48/25
| 48/25
| | 2/1
| 2/1
|-
|-
| | '''6/5'''
! 6/5
| | 1/1
| 1/1
| | <span style="background-color: #ffcc44;">5/4</span>
| <span style="background-color: #ffcc44;">5/4</span>
| | 25/18
| 25/18
| | <span style="background-color: #ffcc44;">5/3</span>
| <span style="background-color: #ffcc44;">5/3</span>
| | 125/72
| 125/72
| | 2/1
| 2/1
|-
|-
| | '''3/2'''
! 3/2
| | 1/1
| 1/1
| | 10/9
| 10/9
| | 4/3
| 4/3
| | 25/18
| 25/18
| | <span style="background-color: #ffcc44;">8/5</span>
| <span style="background-color: #ffcc44;">8/5</span>
| | 2/1
| 2/1
|-
|-
| | '''5/3'''
! 5/3
| | 1/1
| 1/1
| | <span style="background-color: #ffcc44;">6/5</span>
| <span style="background-color: #ffcc44;">6/5</span>
| | <span style="background-color: #ffcc44;">5/4</span>
| <span style="background-color: #ffcc44;">5/4</span>
| | 36/25
| 36/25
| | 9/5
| 9/5
| | 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 both the "second" and "third" steps of the scale. Thus, this scale does not have constant structure.
Line 122: Line 121:
Interval matrix as steps of 12edo:
Interval matrix as steps of 12edo:


{| class="wikitable"
{| class="wikitable center-all"
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! (8)
|-
|-
| |
! 0
| | '''1'''
| 0
| | '''2'''
| 2
| | '''3'''
| 4
| | '''4'''
| 5
| | '''5'''
| 7
| | '''6'''
| 9
| | '''7'''
| 11
| | '''(8)'''
| 12
|-
|-
| | 0
! 2
| | 0
| 0
| | 2
| 2
| | 4
| 3
| | 5
| 5
| | 7
| 7
| | 9
| 9
| | 11
| 10
| | 12
| 12
|-
|-
| | '''2'''
! 4
| | 0
| 0
| | 2
| 1
| | 3
| 3
| | 5
| 5
| | 7
| 7
| | 9
| 8
| | 10
| 10
| | 12
| 12
|-
|-
| | '''4'''
! 7
| | 0
| 0
| | 1
| 2
| | 3
| 4
| | 5
| <span style="background-color: #ffcc44;">6</span>
| | 7
| 7
| | 8
| 9
| | 10
| 11
| | 12
| 12
|-
|-
| | '''7'''
! 9
| | 0
| 0
| | 2
| 2
| | 4
| 4
| | <span style="background-color: #ffcc44;">6</span>
| 5
| | 7
| 7
| | 9
| 9
| | 11
| 10
| | 12
| 12
|-
|-
| | '''9'''
! 11
| | 0
| 0
| | 2
| 2
| | 4
| 3
| | 5
| 5
| | 7
| 7
| | 9
| 8
| | 10
| 10
| | 12
| 12
|-
|-
| | '''11'''
! 12
| | 0
| 0
| | 2
| 1
| | 3
| 3
| | 5
| 5
| | 7
| <span style="background-color: #ffcc44;">6</span>
| | 8
| 8
| | 10
| 10
| | 12
| 12
|-
| | '''12'''
| | 0
| | 1
| | 3
| | 5
| | <span style="background-color: #ffcc44;">6</span>
| | 8
| | 10
| | 12
|}
|}


Interval matrix as note names:
Interval matrix as note names:


{| class="wikitable"
{| class="wikitable center-all"
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! (8)
|-
|-
| |
! C
| | '''1'''
| C
| | '''2'''
| D
| | '''3'''
| E
| | '''4'''
| F
| | '''5'''
| G
| | '''6'''
| A
| | '''7'''
| B
| | '''(8)'''
| C
|-
|-
| | '''C'''
! D
| | C
| C
| | D
| D
| | E
| Eb
| | F
| F
| | G
| G
| | A
| A
| | B
| Bb
| | C
| C
|-
|-
| | '''D'''
! E
| | C
| C
| | D
| Db
| | Eb
| Eb
| | F
| F
| | G
| G
| | A
| Ab
| | Bb
| Bb
| | C
| C
|-
|-
| | '''E'''
! F
| | C
| C
| | Db
| D
| | Eb
| E
| | F
| <span style="background-color: #ffcc44;">F#</span>
| | G
| G
| | Ab
| A
| | Bb
| B
| | C
| C
|-
|-
| | '''F'''
! G
| | C
| C
| | D
| D
| | E
| E
| | <span style="background-color: #ffcc44;">F#</span>
| F
| | G
| G
| | A
| A
| | B
| Bb
| | C
| C
|-
|-
| | '''G'''
! A
| | C
| C
| | D
| D
| | E
| Eb
| | F
| F
| | G
| G
| | A
| Ab
| | Bb
| Bb
| | C
| C
|-
|-
| | '''A'''
! B
| | C
| C
| | D
| Db
| | Eb
| Eb
| | F
| F
| | G
| <span style="background-color: #ffcc44;">Gb</span>
| | Ab
| Ab
| | Bb
| Bb
| | C
| C
|-
| | '''B'''
| | C
| | Db
| | Eb
| | F
| | <span style="background-color: #ffcc44;">Gb</span>
| | 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# and Gb are distinguished, would have constant structure.)
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# and Gb are distinguished, would have constant structure.)


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


{| class="wikitable"
{| class="wikitable right-all"
! EDO
! Number of CS Scales
! Percent of Scales CS
! Corresponding Fraction
|-
|-
| | '''EDO'''
| 1
| | '''Number of CS Scales'''
| 1
| | '''Percent of Scales CS'''
| 100.0%
| | '''Corresponding Fraction'''
| 1/1
|-
|-
| | 1
| 2
| | 1
| 1
| | 100.0%
| 100.0%
| | 1/1
| 1/1
|-
|-
| | 2
| 3
| | 1
| 2
| | 100.0%
| 100.0%
| | 1/1
| 1/1
|-
|-
| | 3
| 4
| | 2
| 2
| | 100.0%
| 66.7%
| | 1/1
| 2/3
|-
|-
| | 4
| 5
| | 2
| 5
| | 66.7%
| 83.3%
| | 2/3
| 5/6
|-
|-
| | 5
| 6
| | 5
| 4
| | 83.3%
| 44.4%
| | 5/6
| 4/9
|-
|-
| | 6
| 7
| | 4
| 11
| | 44.4%
| 61.1%
| | 4/9
| 11/18
|-
|-
| | 7
| 8
| | 11
| 11
| | 61.1%
| 36.7%
| | 11/18
| 11/30
|-
|-
| | 8
| 9
| | 11
| 22
| | 36.7%
| 39.3%
| | 11/30
| 11/28
|-
|-
| | 9
| 10
| | 22
| 20
| | 39.3%
| 20.2%
| | 11/28
| 20/99
|-
|-
| | 10
| 11
| | 20
| 45
| | 20.2%
| 24.2%
| | 20/99
| 15/62
|-
|-
| | 11
| 12
| | 45
| 47
| | 24.2%
| 14.0%
| | 15/62
| 47/335
|-
|-
| | 12
| 13
| | 47
| 85
| | 14.0%
| 13.5%
| | 47/335
| 17/126
|-
|-
| | 13
| 14
| | 85
| 88
| | 13.5%
| 7.6%
| | 17/126
| 88/1161
|-
|-
| | 14
| 15
| | 88
| 163
| | 7.6%
| 7.5%
| | 88/1161
| 163/2182
|-
|-
| | 15
| 16
| | 163
| 165
| | 7.5%
| 4.0%
| | 163/2182
| 11/272
|-
|-
| | 16
| 17
| | 165
| 294
| | 4.0%
| 3.8%
| | 11/272
| 49/1285
|-
|-
| | 17
| 18
| | 294
| 313
| | 3.8%
| 2.2%
| | 49/1285
| 313/14532
|-
|-
| | 18
| 19
| | 313
| 534
| | 2.2%
| 1.9%
| | 313/14532
| 89/4599
|-
|-
| | 19
| 20
| | 534
| 541
| | 1.9%
| 1.0%
| | 89/4599
| 541/52377
|-
| | 20
| | 541
| | 1.0%
| | 541/52377
|}
|}


=See also=
== See also ==
*[[Gallery of CS Scales]]
 
*[[Scale properties simplified]]
* [[Gallery of CS Scales]]
*[[epimorphic]]
* [[Scale properties simplified]]
*[http://tonalsoft.com/enc/c/constant-structure.aspx Constant structure] (Tonalsoft Encyclopedia)
* [[epimorphic]]
*[http://anaphoria.com/wilsonintroMOS.html#cs Introduction to Erv Wilson's Moments of Symmetry]
* [http://tonalsoft.com/enc/c/constant-structure.aspx Constant structure] (Tonalsoft Encyclopedia)
* [http://anaphoria.com/wilsonintroMOS.html#cs Introduction to Erv Wilson's Moments of Symmetry]


[[Category:Theory]]
[[Category:Term]]
[[Category:Constant structure]]
[[Category:Constant structure]]
[[Category:scales]]
[[Category:Scales]]
[[Category:term]]
[[Category:theory]]
Retrieved from "https://en.xen.wiki/w/Constant_structure"