58edo
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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:<h1> --><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:<h2> --><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:<h2> --><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>