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121edo: Difference between revisions

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121 isn't consistent in the 21-odd-limit, hence the table shows odd harmonics as per the template doc
 
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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:genewardsmith|genewardsmith]] and made on <tt>2014-01-26 13:37:06 UTC</tt>.<br>
: The original revision id was <tt>485361746</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">The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai clan|grendel temperament]]. It is [[consistent]] through to the 19 odd limit and uniquely consistent to the 15 odd limit.


Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.
== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].


=13-limit detempering of 121et=
Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.
[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]</pre></div>
 
<h4>Original HTML content:</h4>
=== Odd harmonics ===
<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;121edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for 13-limit &lt;a class="wiki_link" href="/Mirkwai%20clan"&gt;grendel temperament&lt;/a&gt;. It is &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; through to the 19 odd limit and uniquely consistent to the 15 odd limit.&lt;br /&gt;
{{Harmonics in equal|121}}
&lt;br /&gt;
 
Because it tempers out 540/539 it allows &lt;a class="wiki_link" href="/swetismic%20chords"&gt;swetismic chords&lt;/a&gt;, because it tempers out 325/324 it allows &lt;a class="wiki_link" href="/marveltwin%20triad"&gt;marveltwin chords&lt;/a&gt;, because it tempers out 640/637 it allows &lt;a class="wiki_link" href="/huntmic%20chords"&gt;huntmic chords&lt;/a&gt;, because it tempers out 352/351 it allows &lt;a class="wiki_link" href="/minthmic%20chords"&gt;minthmic chords&lt;/a&gt;, because it tempers out 364/363 it allows &lt;a class="wiki_link" href="/gentle%20chords"&gt;gentle chords&lt;/a&gt;, because it tempers out 676/675 it allows &lt;a class="wiki_link" href="/island%20tetrad"&gt;island chords&lt;/a&gt; and because it tempers out 1575/1573 it allows the &lt;a class="wiki_link" href="/nicolic%20tetrad"&gt;nicolic tetrad&lt;/a&gt;. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.&lt;br /&gt;
=== Subsets and supersets ===
&lt;br /&gt;
Since 121 factors into 11<sup>2</sup>, 121edo contains [[11edo]] as its only nontrivial subset.
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x13-limit detempering of 121et"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;13-limit detempering of 121et&lt;/h1&gt;
 
[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]&lt;/body&gt;&lt;/html&gt;</pre></div>
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| −0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| −0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| −0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| −0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| −0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| −0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| −0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]].
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperament
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121<br />(5\121)
| 495.87<br />(49.59)
| 4/3<br />(36/35)
| [[Hendecatonic]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== 13-limit detempering of 121et ==
{{See also| Detempering }}
 
[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]
 
== Miscellany ==
Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.
 
[[Category:Grendel]]
[[Category:Quintupole]]
Retrieved from "https://en.xen.wiki/w/121edo"