Constant structure: Difference between revisions
Wikispaces>genewardsmith **Imported revision 511014500 - Original comment: ** |
Wikispaces>Sarzadoce **Imported revision 553711800 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2015-06-11 15:55:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>553711800</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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The term "constant structure" seems to have been first used by [[Erv Wilson]]. | 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. | 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= | ||
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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.) | ||
=See also= | =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]] | [[Scale properties simplified]] | ||
[[epimorphic]] | [[epimorphic]] | ||
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The term &quot;constant structure&quot; seems to have been first used by <a class="wiki_link" href="/Erv%20Wilson">Erv Wilson</a>.<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 /> | <br /> | ||
To determine if a scale is CS, all possible intervals between scale steps must be evaluated. | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:1 -->Examples</h1> | <!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:1 -->Examples</h1> | ||
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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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="See also"></a><!-- ws:end:WikiTextHeadingRule: | <!-- 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> | ||
<a class="wiki_link" href="/Scale%20properties%20simplified">Scale properties simplified</a><br /> | <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" 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://tonalsoft.com/enc/c/constant-structure.aspx" rel="nofollow">Constant structure</a> (Tonalsoft Encyclopedia)<br /> | ||