Constant structure: Difference between revisions

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A [[scale]] is said to have //constant structure// (CS) 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.

The term "constant structure" seems to have been first used by [[Erv Wilson]].

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 || <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 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 || 11 || 12 ||
|| **4** || 0 || 1 || 3 || 5 || 7 || 8 || 10 || 12 ||
|| **7** || 0 || 2 || 4 || <span style="background-color: #ffcc44;">6</span> || 7 || 9 || 11 || 12 ||
|| **9** || 0 || 2 || 4 || 5 || 7 || 9 || 10 || 12 ||
|| **11** || 0 || 2 || 3 || 5 || 7 || 8 || 10 || 12 ||
|| **12** || 0 || 1 || 3 || 5 || <span style="background-color: #ffcc44;">6</span> || 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 || B || C ||
|| **E** || C || Db || Eb || F || G || Ab || B || C ||
|| **F** || C || D || E || <span style="background-color: #ffcc44;">F#</span> || 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 || <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.)

=Density of CS Scales in EDO's= 

|| **EDO** || **Percent of Scales CS** || **Corresponding Fraction** ||
|| 1 || 100.0% || 1/1 ||
|| 2 || 100.0% || 1/1 ||
|| 3 || 100.0% || 1/1 ||
|| 4 || 66.7% || 2/3 ||
|| 5 || 83.3% || 5/6 ||
|| 6 || 44.4% || 4/9 ||
|| 7 || 61.1% || 11/18 ||
|| 8 || 36.7% || 11/30 ||
|| 9 || 39.3% || 11/28 ||
|| 10 || 20.2% || 20/99 ||
|| 11 || 24.2% || 15/62 ||
|| 12 || 14.0% || 47/335 ||
|| 13 || 13.5% || 17/126 ||
|| 14 || 7.6% || 88/1161 ||
|| 15 || 7.5% || 163/2182 ||
|| 16 || 4.0% || 11/272 ||
|| 17 || 3.8% || 49/1285 ||
|| 18 || 2.2% || 313/14532 ||
|| 19 || 1.9% || 89/4599 ||
|| 20 || 1.0% || 541/52377 ||

=See also= 
[[Scale properties simplified]]
[[epimorphic]]
[[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]]

[[media type="custom" key="26024358"]]

Original HTML content:

<html><head><title>constant structure</title></head><body>A <a class="wiki_link" href="/scale">scale</a> is said to have <em>constant structure</em> (CS) 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.<br />
<br />
The term &quot;constant structure&quot; seems to have been first used by <a class="wiki_link" href="/Erv%20Wilson">Erv Wilson</a>.<br />
<br />
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 <a class="wiki_link" href="/interval%20matrix">interval matrix</a> (<a class="wiki_link" href="/Scala">Scala</a> 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).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:1 -->Examples</h1>
 <br />
This common pentatonic scale is a constant structure: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1<br />
Here is the interval matrix of this scale:<br />


<table class="wiki_table">
    <tr>
        <td><br />
</td>
        <td><strong>1</strong><br />
</td>
        <td><strong>2</strong><br />
</td>
        <td><strong>3</strong><br />
</td>
        <td><strong>4</strong><br />
</td>
        <td><strong>5</strong><br />
</td>
        <td><strong>(6)</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>1/1</strong><br />
</td>
        <td>1/1<br />
</td>
        <td>9/8<br />
</td>
        <td>5/4<br />
</td>
        <td>3/2<br />
</td>
        <td>5/3<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td><strong>9/8</strong><br />
</td>
        <td>1/1<br />
</td>
        <td>10/9<br />
</td>
        <td>4/3<br />
</td>
        <td>40/27<br />
</td>
        <td>16/9<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td><strong>5/4</strong><br />
</td>
        <td>1/1<br />
</td>
        <td>6/5<br />
</td>
        <td>4/3<br />
</td>
        <td>8/5<br />
</td>
        <td>9/5<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td><strong>3/2</strong><br />
</td>
        <td>1/1<br />
</td>
        <td>10/9<br />
</td>
        <td>4/3<br />
</td>
        <td>3/2<br />
</td>
        <td>5/3<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td><strong>5/3</strong><br />
</td>
        <td>1/1<br />
</td>
        <td>6/5<br />
</td>
        <td>27/20<br />
</td>
        <td>3/2<br />
</td>
        <td>9/5<br />
</td>
        <td>2/1<br />
</td>
    </tr>
</table>

Note that every interval always appears in the same position (column). For example, 3/2, which happens to appear three times, is always the &quot;fourth&quot; of this scale - never the &quot;third&quot; or &quot;fifth&quot;.<br />
<br />
<br />
This pentatonic scale is not a constant structure: 1/1 - 25/24 - 6/5 - 3/2 - 5/3 - 2/1<br />
Its interval matrix:<br />


<table class="wiki_table">
    <tr>
        <td><br />
</td>
        <td><strong>1</strong><br />
</td>
        <td><strong>2</strong><br />
</td>
        <td><strong>3</strong><br />
</td>
        <td><strong>4</strong><br />
</td>
        <td><strong>5</strong><br />
</td>
        <td><strong>(6)</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>1/1</strong><br />
</td>
        <td>1/1<br />
</td>
        <td>25/24<br />
</td>
        <td><span style="background-color: #ffcc44;">6/5</span><br />
</td>
        <td>3/2<br />
</td>
        <td><span style="background-color: #ffcc44;">5/3</span><br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td><strong>25/24</strong><br />
</td>
        <td>1/1<br />
</td>
        <td>144/125<br />
</td>
        <td>36/25<br />
</td>
        <td><span style="background-color: #ffcc44;">8/5</span><br />
</td>
        <td>48/25<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td><strong>6/5</strong><br />
</td>
        <td>1/1<br />
</td>
        <td><span style="background-color: #ffcc44;">5/4</span><br />
</td>
        <td>25/18<br />
</td>
        <td><span style="background-color: #ffcc44;">5/3</span><br />
</td>
        <td>125/72<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td><strong>3/2</strong><br />
</td>
        <td>1/1<br />
</td>
        <td>10/9<br />
</td>
        <td>4/3<br />
</td>
        <td>25/18<br />
</td>
        <td><span style="background-color: #ffcc44;">8/5</span><br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td><strong>5/3</strong><br />
</td>
        <td>1/1<br />
</td>
        <td><span style="background-color: #ffcc44;">6/5</span><br />
</td>
        <td><span style="background-color: #ffcc44;">5/4</span><br />
</td>
        <td>36/25<br />
</td>
        <td>9/5<br />
</td>
        <td>2/1<br />
</td>
    </tr>
</table>

Note the highlighted intervals that occur in more than one column. For example, 5/4 may occur as both the &quot;second&quot; and &quot;third&quot; steps of the scale. Thus, this scale does not have constant structure.<br />
<br />
<br />
Another example of a familiar scale that is not CS is the 7-note diatonic scale in <a class="wiki_link" href="/12edo">12edo</a>.<br />
Interval matrix as steps of 12edo:<br />


<table class="wiki_table">
    <tr>
        <td><br />
</td>
        <td><strong>1</strong><br />
</td>
        <td><strong>2</strong><br />
</td>
        <td><strong>3</strong><br />
</td>
        <td><strong>4</strong><br />
</td>
        <td><strong>5</strong><br />
</td>
        <td><strong>6</strong><br />
</td>
        <td><strong>7</strong><br />
</td>
        <td><strong>(8)</strong><br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>2<br />
</td>
        <td>4<br />
</td>
        <td>5<br />
</td>
        <td>7<br />
</td>
        <td>9<br />
</td>
        <td>11<br />
</td>
        <td>12<br />
</td>
    </tr>
    <tr>
        <td><strong>2</strong><br />
</td>
        <td>0<br />
</td>
        <td>2<br />
</td>
        <td>3<br />
</td>
        <td>5<br />
</td>
        <td>7<br />
</td>
        <td>9<br />
</td>
        <td>11<br />
</td>
        <td>12<br />
</td>
    </tr>
    <tr>
        <td><strong>4</strong><br />
</td>
        <td>0<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td>5<br />
</td>
        <td>7<br />
</td>
        <td>8<br />
</td>
        <td>10<br />
</td>
        <td>12<br />
</td>
    </tr>
    <tr>
        <td><strong>7</strong><br />
</td>
        <td>0<br />
</td>
        <td>2<br />
</td>
        <td>4<br />
</td>
        <td><span style="background-color: #ffcc44;">6</span><br />
</td>
        <td>7<br />
</td>
        <td>9<br />
</td>
        <td>11<br />
</td>
        <td>12<br />
</td>
    </tr>
    <tr>
        <td><strong>9</strong><br />
</td>
        <td>0<br />
</td>
        <td>2<br />
</td>
        <td>4<br />
</td>
        <td>5<br />
</td>
        <td>7<br />
</td>
        <td>9<br />
</td>
        <td>10<br />
</td>
        <td>12<br />
</td>
    </tr>
    <tr>
        <td><strong>11</strong><br />
</td>
        <td>0<br />
</td>
        <td>2<br />
</td>
        <td>3<br />
</td>
        <td>5<br />
</td>
        <td>7<br />
</td>
        <td>8<br />
</td>
        <td>10<br />
</td>
        <td>12<br />
</td>
    </tr>
    <tr>
        <td><strong>12</strong><br />
</td>
        <td>0<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td>5<br />
</td>
        <td><span style="background-color: #ffcc44;">6</span><br />
</td>
        <td>8<br />
</td>
        <td>10<br />
</td>
        <td>12<br />
</td>
    </tr>
</table>

<br />
Interval matrix as note names:<br />


<table class="wiki_table">
    <tr>
        <td><br />
</td>
        <td><strong>1</strong><br />
</td>
        <td><strong>2</strong><br />
</td>
        <td><strong>3</strong><br />
</td>
        <td><strong>4</strong><br />
</td>
        <td><strong>5</strong><br />
</td>
        <td><strong>6</strong><br />
</td>
        <td><strong>7</strong><br />
</td>
        <td><strong>(8)</strong><br />
</td>
    </tr>
    <tr>
        <td><strong>C</strong><br />
</td>
        <td>C<br />
</td>
        <td>D<br />
</td>
        <td>E<br />
</td>
        <td>F<br />
</td>
        <td>G<br />
</td>
        <td>A<br />
</td>
        <td>B<br />
</td>
        <td>C<br />
</td>
    </tr>
    <tr>
        <td><strong>D</strong><br />
</td>
        <td>C<br />
</td>
        <td>D<br />
</td>
        <td>Eb<br />
</td>
        <td>F<br />
</td>
        <td>G<br />
</td>
        <td>A<br />
</td>
        <td>B<br />
</td>
        <td>C<br />
</td>
    </tr>
    <tr>
        <td><strong>E</strong><br />
</td>
        <td>C<br />
</td>
        <td>Db<br />
</td>
        <td>Eb<br />
</td>
        <td>F<br />
</td>
        <td>G<br />
</td>
        <td>Ab<br />
</td>
        <td>B<br />
</td>
        <td>C<br />
</td>
    </tr>
    <tr>
        <td><strong>F</strong><br />
</td>
        <td>C<br />
</td>
        <td>D<br />
</td>
        <td>E<br />
</td>
        <td><span style="background-color: #ffcc44;">F#</span><br />
</td>
        <td>G<br />
</td>
        <td>A<br />
</td>
        <td>B<br />
</td>
        <td>C<br />
</td>
    </tr>
    <tr>
        <td><strong>G</strong><br />
</td>
        <td>C<br />
</td>
        <td>D<br />
</td>
        <td>E<br />
</td>
        <td>F<br />
</td>
        <td>G<br />
</td>
        <td>A<br />
</td>
        <td>Bb<br />
</td>
        <td>C<br />
</td>
    </tr>
    <tr>
        <td><strong>A</strong><br />
</td>
        <td>C<br />
</td>
        <td>D<br />
</td>
        <td>Eb<br />
</td>
        <td>F<br />
</td>
        <td>G<br />
</td>
        <td>Ab<br />
</td>
        <td>Bb<br />
</td>
        <td>C<br />
</td>
    </tr>
    <tr>
        <td><strong>B</strong><br />
</td>
        <td>C<br />
</td>
        <td>Db<br />
</td>
        <td>Eb<br />
</td>
        <td>F<br />
</td>
        <td><span style="background-color: #ffcc44;">Gb</span><br />
</td>
        <td>Ab<br />
</td>
        <td>Bb<br />
</td>
        <td>C<br />
</td>
    </tr>
</table>

<br />
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.)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Density of CS Scales in EDO's"></a><!-- ws:end:WikiTextHeadingRule:3 -->Density of CS Scales in EDO's</h1>
 <br />


<table class="wiki_table">
    <tr>
        <td><strong>EDO</strong><br />
</td>
        <td><strong>Percent of Scales CS</strong><br />
</td>
        <td><strong>Corresponding Fraction</strong><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>100.0%<br />
</td>
        <td>1/1<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>100.0%<br />
</td>
        <td>1/1<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>100.0%<br />
</td>
        <td>1/1<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>66.7%<br />
</td>
        <td>2/3<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>83.3%<br />
</td>
        <td>5/6<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>44.4%<br />
</td>
        <td>4/9<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>61.1%<br />
</td>
        <td>11/18<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>36.7%<br />
</td>
        <td>11/30<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>39.3%<br />
</td>
        <td>11/28<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>20.2%<br />
</td>
        <td>20/99<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>24.2%<br />
</td>
        <td>15/62<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>14.0%<br />
</td>
        <td>47/335<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>13.5%<br />
</td>
        <td>17/126<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>7.6%<br />
</td>
        <td>88/1161<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>7.5%<br />
</td>
        <td>163/2182<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>4.0%<br />
</td>
        <td>11/272<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>3.8%<br />
</td>
        <td>49/1285<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>2.2%<br />
</td>
        <td>313/14532<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>1.9%<br />
</td>
        <td>89/4599<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>1.0%<br />
</td>
        <td>541/52377<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc2"><a name="See also"></a><!-- ws:end:WikiTextHeadingRule:5 -->See also</h1>
 <a class="wiki_link" href="/Scale%20properties%20simplified">Scale properties simplified</a><br />
<a class="wiki_link" href="/epimorphic">epimorphic</a><br />
<a class="wiki_link_ext" href="http://tonalsoft.com/enc/c/constant-structure.aspx" rel="nofollow">Constant structure</a> (Tonalsoft Encyclopedia)<br />
<a class="wiki_link_ext" href="http://anaphoria.com/wilsonintroMOS.html#cs" rel="nofollow">Introduction to Erv Wilson's Moments of Symmetry</a><br />
<br />
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