131edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2012-01-03 23:25:32 UTC</tt>.<br>
: The original revision id was <tt>289421963</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>
<h4>Original Wikitext content:</h4>
<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">=&lt;span style="color: #006985; font-family: "Times New Roman",Times,serif; font-size: 113%;"&gt;131 tone equal temperament&lt;/span&gt;=


**131-EDO**, or **131-tET**, divides the octave into 131 equal steps of approx. 9.160305 Cents, each one. 131edo is the next [[EDO]], after [[81edo]], on the "Golden Tone System" ([[Das Goldene Tonsystem]]) of Thorvald Kornerup, using the 131b val. The patent val has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out 81/80 it tempers out the immunity comma, 1638400/1594323. In the 7-limit it tempers out 3125/3087 and 245/243, so that it supports [[Sensamagic clan#Bohpier|bophier temperament]].
== Theory ==
131edo is in[[consistent]] to the [[5-odd-limit]] and the error of [[harmonic]] [[3/1|3]] is quite large. However, it is the next [[edo]] after [[81edo]] on the [[Golden meantone|Golden Tone System]] (''[[Das Goldene Tonsystem]]'') of Thorvald Kornerup, using the 131b [[val]]. The [[patent val]] has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out [[81/80]] it tempers out the [[immunity comma]], 1638400/1594323. In the 7-limit it tempers out [[3125/3087]] and [[245/243]], so that it [[support]]s [[bohpier]].


131edo is the 32nd [[prime numbers|prime]] EDO.
131edo is also notable for having a good approximation to [[natave|acoustic ''e'']], at 189\131, which is a [[semiconvergent]]. This number of steps, 189, is particularly well-factorizable, and logarithmic divisors of acoustic ''e'' form a sequence of rapidly converging approximations to small rationals. Among these are [[4/3]] (2\7[[EDN|edn]] = 54\131), [[5/4]] (2\9edn = 42\131), [[15/13]] (1\7edn = 27\131), [[19/17]] (1\9edn = 21\131), [[11/10]] (2\21edn = 18\131), [[14/13]] (2\27edn = 14\131), and [[32/31]] (2\63edn = 6\131), with accuracy increasing the smaller the fraction.


__**Some MOS Scales 'enjoyable' in 131-EDO:**__
=== Odd harmonics ===
{{Harmonics in equal|131|columns=15}}


|| 33 16 33 33 16 || [[Sub-diatonic Tuning]] (comparable with [[8edo]] and [[99edo]]) ||
=== Subsets and supersets ===
|| 23 23 8 23 23 23 8 || [[Pytagorean tuning]] (comparable with [[17edo]]) ||
131edo is the 32nd [[prime]] edo, following [[127edo]] and before [[137edo]].
|| 21 21 13 21 21 21 13 || [[Meantone tuning]] (comparable with [[50edo]]) ||
|| 19 12 19 19 12 19 19 12 || [[Father Tuning]] ([Octal Triatonic mold] comparable with [[55edo]]) ||
|| 18 18 18 18 18 18 18 5 || [[Porcupine Tuning]] (comparable with [[29edo]] and [[80edo]]) ||
|| 17 17 17 6 17 17 17 17 6 || [[Armodue-Hornbostel Tuning]] (comparable with [[23edo]]) ||
|| 13 13 9 13 13 13 9 13 13 13 9 || Improper [[Sensi-11 Tuning]] [Undecimal Triatonic mold] ||
|| 11 11 11 11 11 5 11 11 11 11 11 11 5 || De Vries 13-tone Tuning ||
|| 10 10 10 7 10 10 10 10 7 10 10 10 10 7 || [[11L 3s|[WITNOTS Tuning]]] [Tetradecimal Triatonic mold] ||
|| 21 17 21 17 17 21 17 || [[mohaha7]] ||
|| 4 17 17 17 4 17 17 4 17 17 || [[mohaha10]] ||


==!!!'!!!!'== </pre></div>
== Scales ==
<h4>Original HTML content:</h4>
=== Mos scales ===
<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;131edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x131 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #006985; font-family: "Times New Roman",Times,serif; font-size: 113%;"&gt;131 tone equal temperament&lt;/span&gt;&lt;/h1&gt;
{| class="wikitable"
&lt;br /&gt;
|-
&lt;strong&gt;131-EDO&lt;/strong&gt;, or &lt;strong&gt;131-tET&lt;/strong&gt;, divides the octave into 131 equal steps of approx. 9.160305 Cents, each one. 131edo is the next &lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt;, after &lt;a class="wiki_link" href="/81edo"&gt;81edo&lt;/a&gt;, on the &amp;quot;Golden Tone System&amp;quot; (&lt;a class="wiki_link" href="/Das%20Goldene%20Tonsystem"&gt;Das Goldene Tonsystem&lt;/a&gt;) of Thorvald Kornerup, using the 131b val. The patent val has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out 81/80 it tempers out the immunity comma, 1638400/1594323. In the 7-limit it tempers out 3125/3087 and 245/243, so that it supports &lt;a class="wiki_link" href="/Sensamagic%20clan#Bohpier"&gt;bophier temperament&lt;/a&gt;.&lt;br /&gt;
| 33 16 33 33 16
&lt;br /&gt;
| [[3L_2s|Pentatonic]] (comparable with [[8edo]] and [[99edo]])
131edo is the 32nd &lt;a class="wiki_link" href="/prime%20numbers"&gt;prime&lt;/a&gt; EDO.&lt;br /&gt;
|-
&lt;br /&gt;
| 23 23 8 23 23 23 8
&lt;u&gt;&lt;strong&gt;Some MOS Scales 'enjoyable' in 131-EDO:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
| [[5L_2s|Pythagorean tuning]] (comparable with [[17edo]])
&lt;br /&gt;
|-
| 21 21 13 21 21 21 13
| [[5L_2s|Meantone tuning]] (comparable with [[50edo]])
|-
| 19 12 19 19 12 19 19 12
| [[5L_3s|Oneirotonic tuning]] (comparable with [[55edo]])
|-
| 18 18 18 18 18 18 18 5
| [[7L_1s|Porcupine tuning]] (comparable with [[29edo]] and [[80edo]])
|-
| 17 17 17 6 17 17 17 17 6
| [[7L_2s|Superdiatonic tuning]] (comparable with [[23edo]])
|-
| 16 16 16 16 16 16 16 16 3
| [[8L 1s|Bohpier tuning]] (comparable with [[41edo]])
|-
| 13 13 9 13 13 13 9 13 13 13 9
| [[8L 3s|Sensi-11 Tuning]]
|-
| 11 11 11 11 11 5 11 11 11 11 11 11 5
| De Vries 13-tone Tuning
|-
| 10 10 10 7 10 10 10 10 7 10 10 10 10 7
| [[11L_3s|Ketradektriatoh Tuning]]
|-
| 21 17 21 17 17 21 17
| [[mohaha7]]
|-
| 4 17 17 17 4 17 17 4 17 17
| [[mohaha10]]
|}


 
[[Category:Bohpier]]
&lt;table class="wiki_table"&gt;
[[Category:Immunity]]
    &lt;tr&gt;
[[Category:Meantone]]
        &lt;td&gt;33 16 33 33 16&lt;br /&gt;
[[Category:Golden meantone]]
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Sub-diatonic%20Tuning"&gt;Sub-diatonic Tuning&lt;/a&gt; (comparable with &lt;a class="wiki_link" href="/8edo"&gt;8edo&lt;/a&gt; and &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt;)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;23 23 8 23 23 23 8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Pytagorean%20tuning"&gt;Pytagorean tuning&lt;/a&gt; (comparable with &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21 21 13 21 21 21 13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Meantone%20tuning"&gt;Meantone tuning&lt;/a&gt; (comparable with &lt;a class="wiki_link" href="/50edo"&gt;50edo&lt;/a&gt;)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19 12 19 19 12 19 19 12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Father%20Tuning"&gt;Father Tuning&lt;/a&gt; ([Octal Triatonic mold] comparable with &lt;a class="wiki_link" href="/55edo"&gt;55edo&lt;/a&gt;)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18 18 18 18 18 18 18 5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Porcupine%20Tuning"&gt;Porcupine Tuning&lt;/a&gt; (comparable with &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt; and &lt;a class="wiki_link" href="/80edo"&gt;80edo&lt;/a&gt;)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17 17 17 6 17 17 17 17 6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/Armodue-Hornbostel%20Tuning"&gt;Armodue-Hornbostel Tuning&lt;/a&gt; (comparable with &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt;)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13 13 9 13 13 13 9 13 13 13 9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Improper &lt;a class="wiki_link" href="/Sensi-11%20Tuning"&gt;Sensi-11 Tuning&lt;/a&gt; [Undecimal Triatonic mold]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11 11 11 11 11 5 11 11 11 11 11 11 5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;De Vries 13-tone Tuning&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10 10 10 7 10 10 10 10 7 10 10 10 10 7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/11L%203s"&gt;[WITNOTS Tuning&lt;/a&gt;] [Tetradecimal Triatonic mold]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21 17 21 17 17 21 17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/mohaha7"&gt;mohaha7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4 17 17 17 4 17 17 4 17 17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="/mohaha10"&gt;mohaha10&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x131 tone equal temperament-!!!'!!!!'"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;!!!'!!!!'&lt;/h2&gt;
&lt;/body&gt;&lt;/html&gt;</pre></div>