58edo: Difference between revisions

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The //58 equal temperament//, often abbreviated 58-tET, 58-EDO, or 58-ET, is the scale derived by dividing the [[octave]] into 58 equally-sized steps. Each step represents a frequency ratio of 20.69 cents. It tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the [[11-limit|11]], [[13-limit|13]] and [[17-limit]]s. It is the smallest [[edo|equal temperament]] which is [[consistent]] through the 17-limit, and is also the first et to map the entire 11-limit [[tonality diamond]] to distinct scale steps, and hence the first et which can define a version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]]. It supports [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[Hemifamity temperaments#Mystery|mystery]], [[Hemifamity temperaments#Buzzard|buzzard]] and [[Starling temperaments#Thuja|thuja]] [[Regular Temperaments|temperament]]s, and supplies the [[optimal patent val]] for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank three temperaments [[Starling family#Thrush|thrush]], [[Starling family#Thrush-Bluebird|bluebird]], [[Starling family#Aplonis|aplonis]] and [[Breed family#Jove,%20aka%20Wonder-Jofur|jofur]].

While the 17th harmonic is a cent and a half cent flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. 58 = 2*29, and 58 shares the same excellent fifth with [[29edo]].

=Scales= 
[[hemif7]]
[[hemif10]]
[[hemif17]]

==Intervals== 
|| degree of 58edo || cents value || ratios ||
|| 0 || 0.00 || 1/1 ||
|| 1 || 20.69 || 56/55, 64/63, 81/80, 128/125 ||
|| 2 || 41.38 || 36/35, 49/48, 50/49, 55/54 ||
|| 3 || 62.07 || 25/24, 26/25, 27/26, 28/27, 33/32 ||
|| 4 || 82.76 || 21/20, 22/21 ||
|| 5 || 103.45 || 16/15, 17/16, 18/17 ||
|| 6 || 124.14 || 14/13, 15/14, 27/25 ||
|| 7 || 144.83 || 12/11, 13/12 ||
|| 8 || 165.52 || 11/10 ||
|| 9 || 186.21 || 10/9 ||
|| 10 || 206.9 || 9/8, 17/15 ||
|| 11 || 227.59 || 8/7 ||
|| 12 || 248.28 || 15/13 ||
|| 13 || 268.97 || 7/6 ||
|| 14 || 289.66 || 13/11, 20/17 ||
|| 15 || 310.34 || 6/5 ||
|| 16 || 331.03 || 17/14 ||
|| 17 || 351.72 || 11/9, 16/13 ||
|| 18 || 372.41 || 21/17 ||
|| 19 || 393.1 || 5/4 ||
|| 20 || 413.79 || 14/11 ||
|| 21 || 434.48 || 9/7 ||
|| 22 || 455.17 || 13/10, 17/13, 22/17 ||
|| 23 || 475.86 || 21/16 ||
|| 24 || 496.55 || 4/3 ||
|| 25 || 517.24 || 27/20 ||
|| 26 || 537.93 || 15/11 ||
|| 27 || 558.62 || 11/8, 18/13 ||
|| 28 || 579.31 || 7/5 ||
|| 29 || 600 || 17/12, 24/17 ||
|| 30 || 620.69 || 10/7 ||
|| 31 || 641.38 || 13/9, 16/11 ||
|| 32 || 662.07 || 22/15 ||
|| 33 || 682.76 || 40/27 ||
|| 34 || 703.45 || 3/2 ||
|| 35 || 724.14 || 32/21 ||
|| 36 || 744.83 || 20/13, 26/17, 17/11 ||
|| 37 || 765.52 || 14/9 ||
|| 38 || 786.21 || 11/7 ||
|| 39 || 806.9 || 8/5 ||
|| 40 || 827.59 || 34/21 ||
|| 41 || 848.28 || 13/8, 18/11 ||
|| 42 || 868.97 || 28/17 ||
|| 43 || 889.66 || 5/3 ||
|| 44 || 910.34 || 22/13, 17/10 ||
|| 45 || 931.03 || 12/7 ||
|| 46 || 951.72 || 26/15 ||
|| 47 || 972.41 || 7/4 ||
|| 48 || 993.1 || 16/9 ||
|| 49 || 1013.79 || 9/5 ||
|| 50 || 1034.48 || 20/11 ||
|| 51 || 1055.17 || 11/6, 24/13 ||
|| 52 || 1075.86 || 13/7, 28/15 ||
|| 53 || 1096.55 || 15/8, 32/17, 17/9 ||
|| 54 || 1117.24 || 40/21, 21/11 ||
|| 55 || 1137.93 ||   ||
|| 56 || 1158.62 ||   ||
|| 57 || 1179.31 ||   ||
==Rank two temperaments== 
||~ Period ||~ Generator ||~ Name ||
|| 1\1 || 1\58 ||   ||
||   || 3\58 ||   ||
||   || 5\58 ||   ||
||   || 7\58 ||   ||
||   || 9\58 ||   ||
||   || 11\58 || Gorgik ||
||   || 13\58 ||   ||
||   || 15\58 || Myna ||
||   || 17\58 || Hemififths ||
||   || 19\58 ||   ||
||   || 21\58 ||   ||
||   || 23\58 || Buzzard ||
||   || 25\58 ||   ||
||   || 27\58 || Thuja ||
|| 1\2 || 1\58 ||   ||
||   || 2\58 ||   ||
||   || 3\58 ||   ||
||   || 4\58 || Harry ||
||   || 5\58 || Srutal/Diaschismic ||
||   || 6\58 ||   ||
||   || 7\58 ||   ||
||   || 8\58 || Echidna, Supers ||
||   || 9\58 || Secant ||
||   || 10\58 ||   ||
||   || 11\58 ||   ||
||   || 12\58 || Sruti ||
||   || 13\58 ||   ||
||   || 14\58 ||   ||
|| 1\29 || 1\58 || Mystery ||

Original HTML content:

<html><head><title>58edo</title></head><body>The <em>58 equal temperament</em>, often abbreviated 58-tET, 58-EDO, or 58-ET, is the scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 58 equally-sized steps. Each step represents a frequency ratio of 20.69 cents. It tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the <a class="wiki_link" href="/11-limit">11</a>, <a class="wiki_link" href="/13-limit">13</a> and <a class="wiki_link" href="/17-limit">17-limit</a>s. It is the smallest <a class="wiki_link" href="/edo">equal temperament</a> which is <a class="wiki_link" href="/consistent">consistent</a> through the 17-limit, and is also the first et to map the entire 11-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a> to distinct scale steps, and hence the first et which can define a version of the famous 43-note <a class="wiki_link" href="/Harry%20Partch%20related%20scales">Genesis scale</a> of <a class="wiki_link" href="/Harry%20Partch">Harry Partch</a>. It supports <a class="wiki_link" href="/hemififths">hemififths</a>, <a class="wiki_link" href="/myna">myna</a>, <a class="wiki_link" href="/diaschismic">diaschismic</a>, <a class="wiki_link" href="/harry">harry</a>, <a class="wiki_link" href="/Hemifamity%20temperaments#Mystery">mystery</a>, <a class="wiki_link" href="/Hemifamity%20temperaments#Buzzard">buzzard</a> and <a class="wiki_link" href="/Starling%20temperaments#Thuja">thuja</a> <a class="wiki_link" href="/Regular%20Temperaments">temperament</a>s, and supplies the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank three temperaments <a class="wiki_link" href="/Starling%20family#Thrush">thrush</a>, <a class="wiki_link" href="/Starling%20family#Thrush-Bluebird">bluebird</a>, <a class="wiki_link" href="/Starling%20family#Aplonis">aplonis</a> and <a class="wiki_link" href="/Breed%20family#Jove,%20aka%20Wonder-Jofur">jofur</a>.<br />
<br />
While the 17th harmonic is a cent and a half cent flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. 58 = 2*29, and 58 shares the same excellent fifth with <a class="wiki_link" href="/29edo">29edo</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Scales</h1>
 <a class="wiki_link" href="/hemif7">hemif7</a><br />
<a class="wiki_link" href="/hemif10">hemif10</a><br />
<a class="wiki_link" href="/hemif17">hemif17</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Scales-Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h2>
 

<table class="wiki_table">
    <tr>
        <td>degree of 58edo<br />
</td>
        <td>cents value<br />
</td>
        <td>ratios<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0.00<br />
</td>
        <td>1/1<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>20.69<br />
</td>
        <td>56/55, 64/63, 81/80, 128/125<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>41.38<br />
</td>
        <td>36/35, 49/48, 50/49, 55/54<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>62.07<br />
</td>
        <td>25/24, 26/25, 27/26, 28/27, 33/32<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>82.76<br />
</td>
        <td>21/20, 22/21<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>103.45<br />
</td>
        <td>16/15, 17/16, 18/17<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>124.14<br />
</td>
        <td>14/13, 15/14, 27/25<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>144.83<br />
</td>
        <td>12/11, 13/12<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>165.52<br />
</td>
        <td>11/10<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>186.21<br />
</td>
        <td>10/9<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>206.9<br />
</td>
        <td>9/8, 17/15<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>227.59<br />
</td>
        <td>8/7<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>248.28<br />
</td>
        <td>15/13<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>268.97<br />
</td>
        <td>7/6<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>289.66<br />
</td>
        <td>13/11, 20/17<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>310.34<br />
</td>
        <td>6/5<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>331.03<br />
</td>
        <td>17/14<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>351.72<br />
</td>
        <td>11/9, 16/13<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>372.41<br />
</td>
        <td>21/17<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>393.1<br />
</td>
        <td>5/4<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>413.79<br />
</td>
        <td>14/11<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>434.48<br />
</td>
        <td>9/7<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>455.17<br />
</td>
        <td>13/10, 17/13, 22/17<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>475.86<br />
</td>
        <td>21/16<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>496.55<br />
</td>
        <td>4/3<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>517.24<br />
</td>
        <td>27/20<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>537.93<br />
</td>
        <td>15/11<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>558.62<br />
</td>
        <td>11/8, 18/13<br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>579.31<br />
</td>
        <td>7/5<br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>600<br />
</td>
        <td>17/12, 24/17<br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>620.69<br />
</td>
        <td>10/7<br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>641.38<br />
</td>
        <td>13/9, 16/11<br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>662.07<br />
</td>
        <td>22/15<br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>682.76<br />
</td>
        <td>40/27<br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>703.45<br />
</td>
        <td>3/2<br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>724.14<br />
</td>
        <td>32/21<br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>744.83<br />
</td>
        <td>20/13, 26/17, 17/11<br />
</td>
    </tr>
    <tr>
        <td>37<br />
</td>
        <td>765.52<br />
</td>
        <td>14/9<br />
</td>
    </tr>
    <tr>
        <td>38<br />
</td>
        <td>786.21<br />
</td>
        <td>11/7<br />
</td>
    </tr>
    <tr>
        <td>39<br />
</td>
        <td>806.9<br />
</td>
        <td>8/5<br />
</td>
    </tr>
    <tr>
        <td>40<br />
</td>
        <td>827.59<br />
</td>
        <td>34/21<br />
</td>
    </tr>
    <tr>
        <td>41<br />
</td>
        <td>848.28<br />
</td>
        <td>13/8, 18/11<br />
</td>
    </tr>
    <tr>
        <td>42<br />
</td>
        <td>868.97<br />
</td>
        <td>28/17<br />
</td>
    </tr>
    <tr>
        <td>43<br />
</td>
        <td>889.66<br />
</td>
        <td>5/3<br />
</td>
    </tr>
    <tr>
        <td>44<br />
</td>
        <td>910.34<br />
</td>
        <td>22/13, 17/10<br />
</td>
    </tr>
    <tr>
        <td>45<br />
</td>
        <td>931.03<br />
</td>
        <td>12/7<br />
</td>
    </tr>
    <tr>
        <td>46<br />
</td>
        <td>951.72<br />
</td>
        <td>26/15<br />
</td>
    </tr>
    <tr>
        <td>47<br />
</td>
        <td>972.41<br />
</td>
        <td>7/4<br />
</td>
    </tr>
    <tr>
        <td>48<br />
</td>
        <td>993.1<br />
</td>
        <td>16/9<br />
</td>
    </tr>
    <tr>
        <td>49<br />
</td>
        <td>1013.79<br />
</td>
        <td>9/5<br />
</td>
    </tr>
    <tr>
        <td>50<br />
</td>
        <td>1034.48<br />
</td>
        <td>20/11<br />
</td>
    </tr>
    <tr>
        <td>51<br />
</td>
        <td>1055.17<br />
</td>
        <td>11/6, 24/13<br />
</td>
    </tr>
    <tr>
        <td>52<br />
</td>
        <td>1075.86<br />
</td>
        <td>13/7, 28/15<br />
</td>
    </tr>
    <tr>
        <td>53<br />
</td>
        <td>1096.55<br />
</td>
        <td>15/8, 32/17, 17/9<br />
</td>
    </tr>
    <tr>
        <td>54<br />
</td>
        <td>1117.24<br />
</td>
        <td>40/21, 21/11<br />
</td>
    </tr>
    <tr>
        <td>55<br />
</td>
        <td>1137.93<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>56<br />
</td>
        <td>1158.62<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>57<br />
</td>
        <td>1179.31<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Scales-Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank two temperaments</h2>
 

<table class="wiki_table">
    <tr>
        <th>Period<br />
</th>
        <th>Generator<br />
</th>
        <th>Name<br />
</th>
    </tr>
    <tr>
        <td>1\1<br />
</td>
        <td>1\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>5\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>7\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>9\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>11\58<br />
</td>
        <td>Gorgik<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>13\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>15\58<br />
</td>
        <td>Myna<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>17\58<br />
</td>
        <td>Hemififths<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>19\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>21\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>23\58<br />
</td>
        <td>Buzzard<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>25\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>27\58<br />
</td>
        <td>Thuja<br />
</td>
    </tr>
    <tr>
        <td>1\2<br />
</td>
        <td>1\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>2\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>3\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>4\58<br />
</td>
        <td>Harry<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>5\58<br />
</td>
        <td>Srutal/Diaschismic<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>6\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>7\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>8\58<br />
</td>
        <td>Echidna, Supers<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>9\58<br />
</td>
        <td>Secant<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>10\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>11\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>12\58<br />
</td>
        <td>Sruti<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>13\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>14\58<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1\29<br />
</td>
        <td>1\58<br />
</td>
        <td>Mystery<br />
</td>
    </tr>
</table>

</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:phylingual|phylingual]] and made on <tt>2012-07-13 13:17:27 UTC</tt>.<br>
+: This revision was by author [[User:manuphonic|manuphonic]] and made on <tt>2015-12-10 14:38:58 UTC</tt>.<br>
-: The original revision id was <tt>353008086</tt>.<br>
+: The original revision id was <tt>569765105</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>
@@ Line 82: / Line 82: @@
 ||   || 7\58 ||   ||
 ||   || 9\58 ||   ||
-||   || 11\58 ||   ||
+||   || 11\58 || Gorgik ||
 ||   || 13\58 ||   ||
 ||   || 15\58 || Myna ||
@@ Line 99: / Line 99: @@
 ||   || 7\58 ||   ||
 ||   || 8\58 || Echidna, Supers ||
-||   || 9\58 ||   ||
+||   || 9\58 || Secant ||
 ||   || 10\58 ||   ||
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          &lt;td&gt;11\58&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Gorgik&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 787: / Line 787: @@
          &lt;td&gt;9\58&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Secant&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;

58edo: Difference between revisions

Revision as of 14:38, 10 December 2015

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

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