31edo: Difference between revisions
Wikispaces>spt3125 **Imported revision 470340618 - Original comment: ** |
Wikispaces>Gedankenwelt **Imported revision 475413562 - 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:Gedankenwelt|Gedankenwelt]] and made on <tt>2013-12-07 01:25:25 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>475413562</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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<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">[[toc|flat]] | <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">[[toc|flat]] | ||
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//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).It also deals with the 11-limit fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and | //Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It also deals with the 11-limit fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit. | ||
31edo is the 11th [[prime numbers|prime]] edo, following [[29edo]] and coming before [[37edo]]. | 31edo is the 11th [[prime numbers|prime]] edo, following [[29edo]] and coming before [[37edo]]. | ||
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|| 3\31 || 116.13 || [[Miracle]] || | || 3\31 || 116.13 || [[Miracle]] || | ||
|| 4\31 || 154.84 || [[Nusecond]] || | || 4\31 || 154.84 || [[Nusecond]] || | ||
|| 5\31 || 193.55 || [[Luna]]/[[Hemithird]]/[[Hemiwürschmidt]] || | || 5\31 || 193.55 || [[Luna]]/[[Hemithird|Hemithirds]]/[[Hemiwürschmidt]] || | ||
|| 6\31 || 232.26 || [[ | || 6\31 || 232.26 || [[Mothra]]/[[Mosura]] || | ||
|| 7\31 || 270.97 || [[Orson]]/[[Orwell]]/[[Semicomma family#Orwell-Winston|Winston]] || | || 7\31 || 270.97 || [[Orson]]/[[Orwell]]/[[Semicomma family#Orwell-Winston|Winston]] || | ||
|| 8\31 || 309.68 || [[Myna]] || | || 8\31 || 309.68 || [[Myna]] || | ||
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|| 12\31 || 464.52 || [[Semisept]] || | || 12\31 || 464.52 || [[Semisept]] || | ||
|| 13\31 || 503.23 || [[Meantone]]/[[Meanpop]] || | || 13\31 || 503.23 || [[Meantone]]/[[Meanpop]] || | ||
|| 14\31 || 541.94 || [[Casablanca]]/[[ | || 14\31 || 541.94 || [[Casablanca]]/[[Cypress]]/[[Oracle]] || | ||
|| 15\31 || 580.65 || [[Tritonic]] || | || 15\31 || 580.65 || [[Tritonic]]/[[Tritoni]] || | ||
=Intervals= | =Intervals= | ||
===1\31 octave - approx. 38.71¢ - Diesis=== | ===1\31 octave - approx. 38.71¢ - Diesis=== | ||
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<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"><html><head><title>31edo</title></head><body><!-- ws:start:WikiTextTocRule:82:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --> | <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"><html><head><title>31edo</title></head><body><!-- ws:start:WikiTextTocRule:82:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --> | ||
<!-- ws:end:WikiTextTocRule:124 --><hr /> | <!-- ws:end:WikiTextTocRule:124 --><hr /> | ||
<em>Thirty-one tone equal temperament</em>, also called <em>31-tET</em>, <em>31-EDO</em>, <em>31-et</em>, or <em>tricesimoprimal meantone temperament</em>, is the scale derived by dividing the octave into 31 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by <a class="wiki_link" href="/Adriaan%20Fokker">Adriaan Fokker</a>. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 <a class="wiki_link" href="/cents">cents</a>. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).It also deals with the 11-limit fairly well, and is consistent through it, but is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and | <em>Thirty-one tone equal temperament</em>, also called <em>31-tET</em>, <em>31-EDO</em>, <em>31-et</em>, or <em>tricesimoprimal meantone temperament</em>, is the scale derived by dividing the octave into 31 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by <a class="wiki_link" href="/Adriaan%20Fokker">Adriaan Fokker</a>. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 <a class="wiki_link" href="/cents">cents</a>. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It also deals with the 11-limit fairly well, and is consistent through it, but is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.<br /> | ||
<br /> | <br /> | ||
31edo is the 11th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/29edo">29edo</a> and coming before <a class="wiki_link" href="/37edo">37edo</a>.<br /> | 31edo is the 11th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/29edo">29edo</a> and coming before <a class="wiki_link" href="/37edo">37edo</a>.<br /> | ||
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<td>193.55<br /> | <td>193.55<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Luna">Luna</a>/<a class="wiki_link" href="/Hemithird"> | <td><a class="wiki_link" href="/Luna">Luna</a>/<a class="wiki_link" href="/Hemithird">Hemithirds</a>/<a class="wiki_link" href="/Hemiw%C3%BCrschmidt">Hemiwürschmidt</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>232.26<br /> | <td>232.26<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/ | <td><a class="wiki_link" href="/Mothra">Mothra</a>/<a class="wiki_link" href="/Mosura">Mosura</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>541.94<br /> | <td>541.94<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Casablanca">Casablanca</a>/<a class="wiki_link" href="/ | <td><a class="wiki_link" href="/Casablanca">Casablanca</a>/<a class="wiki_link" href="/Cypress">Cypress</a>/<a class="wiki_link" href="/Oracle">Oracle</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>580.65<br /> | <td>580.65<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Tritonic">Tritonic</a><br /> | <td><a class="wiki_link" href="/Tritonic">Tritonic</a>/<a class="wiki_link" href="/Tritoni">Tritoni</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||