Miracle
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- This revision was by author keenanpepper and made on 2012-11-10 16:24:00 UTC.
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Original Wikitext content:
Miracle is an exceptionally efficient linear temperament which is a member of both the [[marvel temperaments]] and the [[gamelismic clan]]. It is quite accurate, with TOP error only 0.63 cents/octave, meaning intervals of the 11-limit tonality diamond are represented with only one or two cents of error. Yet it is also very "low complexity" (efficient), as evidenced by the high density of 11-odd-limit ratios (in bold) in the following table: || # of secors || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || || JI intervals represented || **1/1** || 16/15~15/14 || **8/7** || **11/9** || 21/16 || **7/5** || **3/2** || **8/5** || **12/7** || **11/6** || || # of secors || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || || JI intervals represented || 63/32~55/28~108/55~49/25~96/49~88/45 || 22/21~21/20 || **9/8** || **6/5** || **9/7** || **11/8** || 22/15 || **11/7** || 42/25 || **9/5** || || # of secors || 20 || 21 || 22 || || || || || || || || || JI intervals represented || 48/25 || || **11/10** || || || || || || || || Some temperaments have 11/9 as a "neutral third", meaning it's exactly half of a 3/2 (tempering out 243/242), and other temperaments (in the [[gamelismic clan]]) have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts. This is in fact the generator of miracle temperament, called a "secor", and it represents both 16/15 and 15/14. Miracle can also be thought of as a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 45/44~49/48~50/49~55/54~56/55~64/63 all tempered together. =Spectrum of Miracle Tunings by Eigenmonzos= ||~ Eigenmonzo ||~ Secor || || 8/7 || 115.587 || || 11/9 || 115.803 || || 5/4 || 116.241 || || 7/5 || 116.502 || || 6/5 || 116.588 (5 and 7 limit minimax) || || 11/10 || 116.591 || || 12/11 || 116.596 || || 14/11 || 116.617 || || 7/6 || 116.641 || || 10/9 || 116.716 (9 and 11 limit minimax, Secor's definition of secor) || || 11/8 || 116.755 || || 9/7 || 116.792 || || 4/3 || 116.993 ||
Original HTML content:
<html><head><title>Miracle</title></head><body>Miracle is an exceptionally efficient linear temperament which is a member of both the <a class="wiki_link" href="/marvel%20temperaments">marvel temperaments</a> and the <a class="wiki_link" href="/gamelismic%20clan">gamelismic clan</a>. It is quite accurate, with TOP error only 0.63 cents/octave, meaning intervals of the 11-limit tonality diamond are represented with only one or two cents of error. Yet it is also very "low complexity" (efficient), as evidenced by the high density of 11-odd-limit ratios (in bold) in the following table:<br />
<table class="wiki_table">
<tr>
<td><ol><li>of secors</li></ol></td>
<td>0<br />
</td>
<td>1<br />
</td>
<td>2<br />
</td>
<td>3<br />
</td>
<td>4<br />
</td>
<td>5<br />
</td>
<td>6<br />
</td>
<td>7<br />
</td>
<td>8<br />
</td>
<td>9<br />
</td>
</tr>
<tr>
<td>JI intervals represented<br />
</td>
<td><strong>1/1</strong><br />
</td>
<td>16/15~15/14<br />
</td>
<td><strong>8/7</strong><br />
</td>
<td><strong>11/9</strong><br />
</td>
<td>21/16<br />
</td>
<td><strong>7/5</strong><br />
</td>
<td><strong>3/2</strong><br />
</td>
<td><strong>8/5</strong><br />
</td>
<td><strong>12/7</strong><br />
</td>
<td><strong>11/6</strong><br />
</td>
</tr>
<tr>
<td><ol><li>of secors</li></ol></td>
<td>10<br />
</td>
<td>11<br />
</td>
<td>12<br />
</td>
<td>13<br />
</td>
<td>14<br />
</td>
<td>15<br />
</td>
<td>16<br />
</td>
<td>17<br />
</td>
<td>18<br />
</td>
<td>19<br />
</td>
</tr>
<tr>
<td>JI intervals represented<br />
</td>
<td>63/32~55/28~108/55~49/25~96/49~88/45<br />
</td>
<td>22/21~21/20<br />
</td>
<td><strong>9/8</strong><br />
</td>
<td><strong>6/5</strong><br />
</td>
<td><strong>9/7</strong><br />
</td>
<td><strong>11/8</strong><br />
</td>
<td>22/15<br />
</td>
<td><strong>11/7</strong><br />
</td>
<td>42/25<br />
</td>
<td><strong>9/5</strong><br />
</td>
</tr>
<tr>
<td><ol><li>of secors</li></ol></td>
<td>20<br />
</td>
<td>21<br />
</td>
<td>22<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>JI intervals represented<br />
</td>
<td>48/25<br />
</td>
<td><br />
</td>
<td><strong>11/10</strong><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
Some temperaments have 11/9 as a "neutral third", meaning it's exactly half of a 3/2 (tempering out 243/242), and other temperaments (in the <a class="wiki_link" href="/gamelismic%20clan">gamelismic clan</a>) have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts. This is in fact the generator of miracle temperament, called a "secor", and it represents both 16/15 and 15/14.<br />
<br />
Miracle can also be thought of as a <a class="wiki_link" href="/cluster%20temperament">cluster temperament</a> with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 45/44~49/48~50/49~55/54~56/55~64/63 all tempered together.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Spectrum of Miracle Tunings by Eigenmonzos"></a><!-- ws:end:WikiTextHeadingRule:0 -->Spectrum of Miracle Tunings by Eigenmonzos</h1>
<table class="wiki_table">
<tr>
<th>Eigenmonzo<br />
</th>
<th>Secor<br />
</th>
</tr>
<tr>
<td>8/7<br />
</td>
<td>115.587<br />
</td>
</tr>
<tr>
<td>11/9<br />
</td>
<td>115.803<br />
</td>
</tr>
<tr>
<td>5/4<br />
</td>
<td>116.241<br />
</td>
</tr>
<tr>
<td>7/5<br />
</td>
<td>116.502<br />
</td>
</tr>
<tr>
<td>6/5<br />
</td>
<td>116.588 (5 and 7 limit minimax)<br />
</td>
</tr>
<tr>
<td>11/10<br />
</td>
<td>116.591<br />
</td>
</tr>
<tr>
<td>12/11<br />
</td>
<td>116.596<br />
</td>
</tr>
<tr>
<td>14/11<br />
</td>
<td>116.617<br />
</td>
</tr>
<tr>
<td>7/6<br />
</td>
<td>116.641<br />
</td>
</tr>
<tr>
<td>10/9<br />
</td>
<td>116.716 (9 and 11 limit minimax, Secor's definition of secor)<br />
</td>
</tr>
<tr>
<td>11/8<br />
</td>
<td>116.755<br />
</td>
</tr>
<tr>
<td>9/7<br />
</td>
<td>116.792<br />
</td>
</tr>
<tr>
<td>4/3<br />
</td>
<td>116.993<br />
</td>
</tr>
</table>
</body></html>