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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-05-24 15:39:28 UTC</tt>.<br>
: The original revision id was <tt>511009828</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A [[scale]] is said to have //constant structure// if its generic interval classes are distinct. That is, each interval occurs always subtended by 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.


=Examples=  
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 (<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 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
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 is always the "fourth" of this scale.


Here is the interval matrix of this scale:
{| class="wikitable center-all"
!
! 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:
{| class="wikitable center-all"
!
! 1
! 2
! 3
! 4
! 5
! (6)
|-
! 1/1
| 1/1
| 25/24
| <span style="background-color: #ffcc44;">6/5</span>
| 3/2
| <span style="background-color: #ffcc44;">5/3</span>
| 2/1
|-
! 25/24
| 1/1
| 144/125
| 36/25
| <span style="background-color: #ffcc44;">8/5</span>
| 48/25
| 2/1
|-
! 6/5
| 1/1
| <span style="background-color: #ffcc44;">5/4</span>
| 25/18
| <span style="background-color: #ffcc44;">5/3</span>
| 125/72
| 2/1
|-
! 3/2
| 1/1
| 10/9
| 4/3
| 25/18
| <span style="background-color: #ffcc44;">8/5</span>
| 2/1
|-
! 5/3
| 1/1
| <span style="background-color: #ffcc44;">6/5</span>
| <span style="background-color: #ffcc44;">5/4</span>
| 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]].


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:
||~  ||~ 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 that 5/4 occurs as both the "second" and "third".


[[media type="custom" key="26024358"]]</pre></div>
{| class="wikitable center-all"
<h4>Original HTML content:</h4>
!
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;constant structure&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;a class="wiki_link" href="/scale"&gt;scale&lt;/a&gt; is said to have &lt;em&gt;constant structure&lt;/em&gt; if its generic interval classes are distinct. That is, each interval occurs always subtended by 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.&lt;br /&gt;
! 1
&lt;br /&gt;
! 2
&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;Examples&lt;/h1&gt;
! 3
&lt;br /&gt;
! 4
This common pentatonic scale is a constant structure: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1&lt;br /&gt;
! 5
Here is the &lt;a class="wiki_link" href="/interval%20matrix"&gt;interval matrix&lt;/a&gt; of this scale:&lt;br /&gt;
! 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
| <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
! 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
|}
 
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 ==


&lt;table class="wiki_table"&gt;
{| class="wikitable right-all"
    &lt;tr&gt;
! EDO
        &lt;th&gt;&lt;br /&gt;
! Number of CS Scales
&lt;/th&gt;
! Percent of Scales CS
        &lt;th&gt;1&lt;br /&gt;
! Corresponding Fraction
&lt;/th&gt;
|-
        &lt;th&gt;2&lt;br /&gt;
| 1
&lt;/th&gt;
| 1
        &lt;th&gt;3&lt;br /&gt;
| 100.0%
&lt;/th&gt;
| 1/1
        &lt;th&gt;4&lt;br /&gt;
|-
&lt;/th&gt;
| 2
        &lt;th&gt;5&lt;br /&gt;
| 1
&lt;/th&gt;
| 100.0%
        &lt;th&gt;(6)&lt;br /&gt;
| 1/1
&lt;/th&gt;
|-
    &lt;/tr&gt;
| 3
    &lt;tr&gt;
| 2
        &lt;th&gt;1/1&lt;br /&gt;
| 100.0%
&lt;/th&gt;
| 1/1
        &lt;td&gt;1/1&lt;br /&gt;
|-
&lt;/td&gt;
| 4
        &lt;td&gt;9/8&lt;br /&gt;
| 2
&lt;/td&gt;
| 66.7%
        &lt;td&gt;5/4&lt;br /&gt;
| 2/3
&lt;/td&gt;
|-
        &lt;td&gt;3/2&lt;br /&gt;
| 5
&lt;/td&gt;
| 5
        &lt;td&gt;5/3&lt;br /&gt;
| 83.3%
&lt;/td&gt;
| 5/6
        &lt;td&gt;2/1&lt;br /&gt;
|-
&lt;/td&gt;
| 6
    &lt;/tr&gt;
| 4
    &lt;tr&gt;
| 44.4%
        &lt;th&gt;9/8&lt;br /&gt;
| 4/9
&lt;/th&gt;
|-
        &lt;td&gt;1/1&lt;br /&gt;
| 7
&lt;/td&gt;
| 11
        &lt;td&gt;10/9&lt;br /&gt;
| 61.1%
&lt;/td&gt;
| 11/18
        &lt;td&gt;4/3&lt;br /&gt;
|-
&lt;/td&gt;
| 8
        &lt;td&gt;40/27&lt;br /&gt;
| 11
&lt;/td&gt;
| 36.7%
        &lt;td&gt;16/9&lt;br /&gt;
| 11/30
&lt;/td&gt;
|-
        &lt;td&gt;2/1&lt;br /&gt;
| 9
&lt;/td&gt;
| 22
    &lt;/tr&gt;
| 39.3%
    &lt;tr&gt;
| 11/28
        &lt;th&gt;5/4&lt;br /&gt;
|-
&lt;/th&gt;
| 10
        &lt;td&gt;1/1&lt;br /&gt;
| 20
&lt;/td&gt;
| 20.2%
        &lt;td&gt;6/5&lt;br /&gt;
| 20/99
&lt;/td&gt;
|-
        &lt;td&gt;4/3&lt;br /&gt;
| 11
&lt;/td&gt;
| 45
        &lt;td&gt;8/5&lt;br /&gt;
| 24.2%
&lt;/td&gt;
| 15/62
        &lt;td&gt;9/5&lt;br /&gt;
|-
&lt;/td&gt;
| 12
        &lt;td&gt;2/1&lt;br /&gt;
| 47
&lt;/td&gt;
| 14.0%
    &lt;/tr&gt;
| 47/335
    &lt;tr&gt;
|-
        &lt;th&gt;3/2&lt;br /&gt;
| 13
&lt;/th&gt;
| 85
        &lt;td&gt;1/1&lt;br /&gt;
| 13.5%
&lt;/td&gt;
| 17/126
        &lt;td&gt;10/9&lt;br /&gt;
|-
&lt;/td&gt;
| 14
        &lt;td&gt;4/3&lt;br /&gt;
| 88
&lt;/td&gt;
| 7.6%
        &lt;td&gt;3/2&lt;br /&gt;
| 88/1161
&lt;/td&gt;
|-
        &lt;td&gt;5/3&lt;br /&gt;
| 15
&lt;/td&gt;
| 163
        &lt;td&gt;2/1&lt;br /&gt;
| 7.5%
&lt;/td&gt;
| 163/2182
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 16
        &lt;th&gt;5/3&lt;br /&gt;
| 165
&lt;/th&gt;
| 4.0%
        &lt;td&gt;1/1&lt;br /&gt;
| 11/272
&lt;/td&gt;
|-
        &lt;td&gt;6/5&lt;br /&gt;
| 17
&lt;/td&gt;
| 294
        &lt;td&gt;27/20&lt;br /&gt;
| 3.8%
&lt;/td&gt;
| 49/1285
        &lt;td&gt;3/2&lt;br /&gt;
|-
&lt;/td&gt;
| 18
        &lt;td&gt;9/5&lt;br /&gt;
| 313
&lt;/td&gt;
| 2.2%
        &lt;td&gt;2/1&lt;br /&gt;
| 313/14532
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 19
&lt;/table&gt;
| 534
| 1.9%
| 89/4599
|-
| 20
| 541
| 1.0%
| 541/52377
|}


Note that every interval always appears in the same position (column). For example, 3/2 is always the &amp;quot;fourth&amp;quot; of this scale.&lt;br /&gt;
== Novel terminology ==
&lt;br /&gt;
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.
&lt;br /&gt;
This pentatonic scale is not a constant structure: 1/1 - 25/24 - 6/5 - 3/2 - 5/3 - 2/1&lt;br /&gt;
Its interval matrix:&lt;br /&gt;


== See also ==


&lt;table class="wiki_table"&gt;
* [[Gallery of CS Scales]]
    &lt;tr&gt;
* [[Glossary of scale properties]]
        &lt;th&gt;&lt;br /&gt;
* [[epimorphic]]
&lt;/th&gt;
* [http://tonalsoft.com/enc/c/constant-structure.aspx Constant structure] (Tonalsoft Encyclopedia)
        &lt;th&gt;1&lt;br /&gt;
* [http://anaphoria.com/wilsonintroMOS.html#cs Introduction to Erv Wilson's Moments of Symmetry]
&lt;/th&gt;
        &lt;th&gt;2&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;3&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;4&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;5&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;(6)&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;1/1&lt;br /&gt;
&lt;/th&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25/24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;25/24&lt;br /&gt;
&lt;/th&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;144/125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;36/25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;48/25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;6/5&lt;br /&gt;
&lt;/th&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25/18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;125/72&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;3/2&lt;br /&gt;
&lt;/th&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25/18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;5/3&lt;br /&gt;
&lt;/th&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;36/25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


Note that 5/4 occurs as both the &amp;quot;second&amp;quot; and &amp;quot;third&amp;quot;.&lt;br /&gt;
[[Category:Scale]]
&lt;br /&gt;
[[Category:Terms]]
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