58edo: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-13 12:20:08 UTC</tt>.<br>

: The original revision id was <tt>353000566</tt>.<br>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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, aka Wonder-Jofur|jofur]].

|| degree of 58edo || cents value || ratios || associated temperament ||

|| 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 || Myna ||

|| 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 ||   ||  ||

|| 41 || 848.28 || 13/8, 18/11 ||  ||

|| 42 || 868.97 ||   ||  ||

|| 43 || 889.66 ||   ||  ||

|| 44 || 910.34 ||   ||  ||

|| 45 || 931.03 ||   ||  ||

|| 46 || 951.72 ||   ||  ||

|| 47 || 972.41 || 7/4 ||  ||

|| 48 || 993.1 || 16/9 ||  ||

|| 49 || 1013.79 || 9/5 ||  ||

|| 50 || 1034.48 ||  ||  ||

|| 51 || 1055.17 ||  ||  ||

|| 52 || 1075.86 ||  ||  ||

|| 53 || 1096.55 ||  ||  ||

|| 54 || 1117.24 ||  ||  ||

|| 55 || 1137.93 ||  ||  ||

|| 56 || 1158.62 ||  ||  ||

|| 57 || 1179.31 ||  ||  ||</pre></div>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;58edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;58 equal temperament&lt;/em&gt;, often abbreviated 58-tET, 58-EDO, or 58-ET, is the scale derived by dividing the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; 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 &lt;a class="wiki_link" href="/11-limit"&gt;11&lt;/a&gt;, &lt;a class="wiki_link" href="/13-limit"&gt;13&lt;/a&gt; and &lt;a class="wiki_link" href="/17-limit"&gt;17-limit&lt;/a&gt;s. It is the smallest &lt;a class="wiki_link" href="/edo"&gt;equal temperament&lt;/a&gt; which is &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; through the 17-limit, and is also the first et to map the entire 11-limit &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt; to distinct scale steps, and hence the first et which can define a version of the famous 43-note &lt;a class="wiki_link" href="/Harry%20Partch%20related%20scales"&gt;Genesis scale&lt;/a&gt; of &lt;a class="wiki_link" href="/Harry%20Partch"&gt;Harry Partch&lt;/a&gt;. It supports &lt;a class="wiki_link" href="/hemififths"&gt;hemififths&lt;/a&gt;, &lt;a class="wiki_link" href="/myna"&gt;myna&lt;/a&gt;, &lt;a class="wiki_link" href="/diaschismic"&gt;diaschismic&lt;/a&gt;, &lt;a class="wiki_link" href="/harry"&gt;harry&lt;/a&gt;, &lt;a class="wiki_link" href="/Hemifamity%20temperaments#Mystery"&gt;mystery&lt;/a&gt;, &lt;a class="wiki_link" href="/Hemifamity%20temperaments#Buzzard"&gt;buzzard&lt;/a&gt; and &lt;a class="wiki_link" href="/Starling%20temperaments#Thuja"&gt;thuja&lt;/a&gt; &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;temperament&lt;/a&gt;s, and supplies the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; 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 &lt;a class="wiki_link" href="/Starling%20family#Thrush"&gt;thrush&lt;/a&gt;, &lt;a class="wiki_link" href="/Starling%20family#Thrush-Bluebird"&gt;bluebird&lt;/a&gt;, &lt;a class="wiki_link" href="/Starling%20family#Aplonis"&gt;aplonis&lt;/a&gt; and &lt;a class="wiki_link" href="/Breed%20family#Jove, aka Wonder-Jofur"&gt;jofur&lt;/a&gt;.&lt;br /&gt;

         &lt;td&gt;&lt;br /&gt;

        &lt;td&gt;&lt;br /&gt;

         &lt;td&gt;43&lt;br /&gt;

         &lt;td&gt;889.66&lt;br /&gt;

         &lt;td&gt;44&lt;br /&gt;

         &lt;td&gt;910.34&lt;br /&gt;

         &lt;td&gt;45&lt;br /&gt;

         &lt;td&gt;931.03&lt;br /&gt;

         &lt;td&gt;46&lt;br /&gt;

         &lt;td&gt;951.72&lt;br /&gt;

         &lt;td&gt;47&lt;br /&gt;

         &lt;td&gt;972.41&lt;br /&gt;

         &lt;td&gt;7/4&lt;br /&gt;

         &lt;td&gt;48&lt;br /&gt;

         &lt;td&gt;993.1&lt;br /&gt;

         &lt;td&gt;16/9&lt;br /&gt;

         &lt;td&gt;49&lt;br /&gt;

         &lt;td&gt;1013.79&lt;br /&gt;

        &lt;td&gt;9/5&lt;br /&gt;

         &lt;td&gt;50&lt;br /&gt;

         &lt;td&gt;1034.48&lt;br /&gt;

         &lt;td&gt;51&lt;br /&gt;

         &lt;td&gt;1055.17&lt;br /&gt;

         &lt;td&gt;52&lt;br /&gt;

         &lt;td&gt;1075.86&lt;br /&gt;

         &lt;td&gt;53&lt;br /&gt;

         &lt;td&gt;1096.55&lt;br /&gt;

         &lt;td&gt;54&lt;br /&gt;

         &lt;td&gt;1117.24&lt;br /&gt;

         &lt;td&gt;55&lt;br /&gt;

         &lt;td&gt;1137.93&lt;br /&gt;

         &lt;td&gt;56&lt;br /&gt;

         &lt;td&gt;1158.62&lt;br /&gt;

         &lt;td&gt;57&lt;br /&gt;

         &lt;td&gt;1179.31&lt;br /&gt;