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31st-octave temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{Infobox fractional-octave|31}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-01 16:29:31 UTC</tt>.<br>
This page collects rank-2 temperaments with a period that is 1/31 of an octave.
: The original revision id was <tt>281263302</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">By the //31-3-comma// is meant 617673396283947/562949953421312 = &lt;-49 31|, the amount (160.605 cents) by which 31 just (3/2) fifths exceed 18 octaves. This may not seem like much of a comma, but since 31et is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.


=Prajapati=
By the ''31-3-comma'' is meant 617673396283947/562949953421312 = {{monzo| -49 31 }}, the amount (160.605 [[cent]]s) by which 31 just perfect fifths ([[3/2]]) exceed 18 [[octave]]s. This may not seem like much of a comma, but since [[31edo]] is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense. This approach leads to the prajapati temperament and the gallium temperament.  
The Hindu  god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31. People with an interest in 31 may want to try this method themselves.


Commas: 81/80, 126/125, 1029/1024
31edo is accurate for harmonics 5 and 7, the 31-5-comma ({{monzo| 72 0 -31 }}, the amount by which 31 just major thirds ([[5/4]]) fall short of 10 octaves) and the 31-7-comma ({{monzo| -87 0 0 31 }}, the amount by which 31 septimal whole tones ([[8/7]]) fall short of 6 octaves) is tempered out by the following ETs: {{Optimal ET sequence| 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, and 527 }}. Tempering out these commas leads to the birds temperament.


POTE generator: ~176/175 = 6.519
== 31-commatic ==
Subgroup: 2.3.5


Map: [[&lt;31 49 72 87 107|, &lt;0 0 0 0 1|]
Comma list: {{monzo| -49 31 }}
EDOs: 31, 93, 124b, 155b, 186b
Badness: 0.0430


==13-limit==
{{Mapping|legend=1| -31 -49 0 | 0 0 1 }}
Commas: 81/80, 126/125, 105/104, 512/507
 
: mapping generators: ~531441/524288 = 1\31, ~5
 
[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.314
 
[[Support]]ing [[ET]]s: {{EDOs|31, 62, 93}}
 
== 31-5-commatic ==
Subgroup: 2.3.5
 
Comma list: {{monzo| 72 0 -31 }}
 
{{Mapping|legend=1| 31 31 72 | 0 1 0 }}
 
[[Optimal tuning]] ([[CWE]]): ~128/125 = 1\31, ~3/2 = 702.133
 
[[Support]]ing [[ET]]s: 31, 217, 186, 248, 155, 465, 403, 279, 124, 93c, 62c, 682, 310, 620
 
== 31-17/13-commatic ==
A circle of 31 [[17/13]]'s closes at the octave with an error of only 2.74 cents.
 
Subgroup: 2.13.17
 
Comma list: {{Monzo|12 0 0 0 0 31 -31}}
 
{{Mapping|31 0 12|0 1 1|legend=2}}
 
: sval mapping generators: ~2.13.17 {{monzo|-5 -13 13}} = 1\31, ~13
 
[[Optimal tuning]] ([[CTE]]): ~13/8 = 840.488
 
== Birds ==
The birds temperament tempers out the 31-5 comma, {{monzo| 72 0 -31 }}, and the 31-7 comma, ({{monzo| -87 0 0 31 }}. The name comes from Isaiah 31:5 "As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it."
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 3136/3125, 823543/819200
 
[[Mapping]]: [{{val| 31 49 72 87 }}, {{val| 0 1 0 0 }}]
 
[[POTE generator]]: ~1029/1024 = 5.1551
 
{{Optimal ET sequence|legend=1| 31, 124, 155, 186, 217, 248, 465 }}
 
[[Badness]]: 0.099928
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3136/3125, 41503/41472
 
Mapping: [{{val| 31 49 72 87 107 }}, {{val| 0 1 0 0 2 }}]
 
POTE generator: ~385/384 = 4.9377
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 961cd }}
 
Badness: 0.039921
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 1001/1000, 3136/3125, 13720/13689
 
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 1 0 0 2 -2 }}]
 
POTE generator: ~385/384 = 5.1703
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}
 
Badness: 0.035680
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125
 
Mapping: [{{val| 31 49 72 87 107 115 127 }}, {{val| 0 1 0 0 2 -2 -2 }}]
 
POTE generator: ~385/384 = 5.2248
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}
 
Badness: 0.025890
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 343/342, 441/440, 476/475, 833/832, 1001/1000, 1445/1444
 
Mapping: [{{val| 31 49 72 87 107 115 127 132 }}, {{val| 0 1 0 0 2 -2 -2 -2 }}]
 
POTE generator: ~385/384 = 5.3169
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248h, 465h }}
 
Badness: 0.021271
 
== 217 & 1178 ==
The 217 & 1178 temperament combines two multiples of 31, which are large equal divisions consistent in the 21-odd-limit. 1395edo, also consistent in 21-odd-limit, is also a tuning.
 
Subgroup: 2.3.5.7
 
Comma list: 4375/4374, {{monzo|-153 42 7 25}}
 
{{Mapping|legend=1| 31 2 -38 197 | 0 3 7 -7 }}
 
: mapping generators: ~562711519881/549755813888 = 1\31, ~67108864/47258883 = 608.167
 
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
 
[[Support]]ing [[ET]]s: {{EDOs|217, 744c, 961, 1178, 1395, 1612, 2573}}
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 820125/819896, {{monzo|-37 12 -1  6  1}}
 
{{Mapping|legend=1| 31 2 -38 197 -97 | 0 3 7 -7 13 }}
 
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167
 
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
 
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 4375/4374, 225000/224939, 18753525/18743296
 
{{Mapping|legend=1| 31 2 -38 197 -97 99 | 0 3 7 -7 13 1 }}
 
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167
 
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
 
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
 
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 4225/4224, 4375/4374, 14400/14399, 14875/14872, 56595/56576
 
{{Mapping|legend=1| 31 2 -38 197 -97 99 111 | 0 3 7 -7 13 1 1 }}
 
: mapping generators: ~45/44 = 1\31, ~1989/1400 = 608.167
 
[[Optimal tuning]] ([[CTE]]): ~1989/1400 = 608.167
 
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
 
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 4200/4199, 4225/4224, 4375/4374, 5929/5928, 5985/5984, 14875/14872
 
{{Mapping|legend=1| 31 2 -38 197 -97 99 111 6 | 0 3 7 -7 13 1 1 8 }}
 
: mapping generators: ~112651/110160 = 1\31, ~665/468 = 608.166
 
[[Optimal tuning]] ([[CTE]]): ~665/468 = 608.166
 
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
 
; Music
 
* ''[https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening]'' by [[Eliora]] (2023) - 217 & 1178 and enneadecal in 1178edo tuning
 
== Prajapati ==
The Hindu god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31.
 
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 81/80, 126/125, 1029/1024
 
[[Mapping]]: [{{val| 31 49 72 87 107 }}, {{val| 0 0 0 0 1 }}]
 
[[POTE generator]]: ~176/175 = 6.519
 
{{Optimal ET sequence|legend=1| 31, 93, 124b, 155b, 186b }}
 
[[Badness]]: 0.042959
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 126/125, 105/104, 512/507
 
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 1 0 }}]


POTE generator: ~66/65 = 9.171
POTE generator: ~66/65 = 9.171


Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 0|]
{{Optimal ET sequence|legend=1| 31, 93f, 124bf }}
EDOs: 31, 93f, 124bf
Badness: 0.0379


==Kumhar==
Badness: 0.037885
Commas: 81/80, 126/125, 1029/1024, 144/143
 
=== Kumhar ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 126/125, 144/143, 1029/1024
 
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 1 -1 }}]


POTE generator: ~196/195 = 10.120
POTE generator: ~196/195 = 10.120


Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 -1|]
{{Optimal ET sequence|legend=1| 31, 62e, 93, 124b, 341b }}
EDOs: 31, 62e, 93, 124b, 341b
 
Badness: 0.0486
Badness: 0.048582


=Gallium=
== Gallium ==
The name of gallium temperament comes from Gallium, the 31st element.
The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31et, while adding 13, 17, and 19 on an independent generator chain, and this considerably improves the qualities of 13-limit and beyond.  


Commas: 81/80, 126/125, 99/98, 121/120
[[Subgroup]]: 2.3.5.7.11.13


POTE generator: ~16807/16640 = 15.541
[[Comma list]]: 81/80, 99/98, 121/120, 126/125


Map: [&lt;31 49 72 87 107 0|, &lt;0 0 0 0 0 1|]
{{Mapping|legend=1| 31 49 72 87 107 115 | 0 0 0 0 0 -1 }}
EDOs: 31, 62, 93e, 155bef
Badness: 0.0255


=Birds=
[[Optimal tuning]] ([[CTE]]): ~45/44 = 1\31, ~13/8 = 840.5276 (~144/143 = 11.0853)
The birds temperament tempers out the //31-5 comma//, |72 0 -31&gt;, which is the amount (24.275 cents) by which 31 just (5/4) major thirds fall short of ten octaves. The name comes from Isaiah 31:5 "As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it." It also tempers out the 31-7 comma, but sadly, combining the two commas leads to torsion.


Commas: 3136/3125, 823543/819200
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}


POTE generator: ~1029/1024 = 5.155
[[Badness]]: 0.025484


Map: [&lt;31 49 72 87|, &lt;0 1 0 0|]
=== 17-limit ===
Wedgie: &lt;&lt;31 0 0 -72 -87 0||
Subgroup: 2.3.5.7.11.13.17
Edos: 31, 124, 155, 186, 217, 248, 465
Badness: 0.1000


==11-limit==
Comma list: 81/80, 99/98, 121/120, 126/125, 273/272
Commas: 441/440, 3136/3125, 41503/41472


POTE generator: ~385/384 = 4.938
Mapping: {{mapping| 31 49 72 87 107 115 127 | 0 0 0 0 0 -1 -1 }}


Map: [&lt;31 49 72 87 107|, &lt;0 1 0 0 2|]
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)
EDOs: 31, 186e, 217, 248, 961cd
Badness: 0.0399


==13-limit==
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}
Commas: 441/440, 3136/3125, 1001/1000, 13720/13689


POTE generator: ~385/384 = 5.170
Badness: 0.023421


Map: [&lt;31 49 72 87 107 115|, &lt;0 1 0 0 2 -2|]
=== 19-limit ===
EDOs: 31, 186e, 217, 248, 465
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0357


Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272


Mapping: {{mapping| 31 49 72 87 107 115 127 132 | 0 0 0 0 0 -1 -1 -1 }}


Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)


{{Optimal ET sequence|legend=1| 31, 62, 155befg }}


Badness: 0.019963


</pre></div>
{{Navbox fractional-octave}}
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;31 comma temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;By the &lt;em&gt;31-3-comma&lt;/em&gt; is meant 617673396283947/562949953421312 = &amp;lt;-49 31|, the amount (160.605 cents) by which 31 just (3/2) fifths exceed 18 octaves. This may not seem like much of a comma, but since 31et is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Prajapati"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Prajapati&lt;/h1&gt;
The Hindu  god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31. People with an interest in 31 may want to try this method themselves.&lt;br /&gt;
&lt;br /&gt;
Commas: 81/80, 126/125, 1029/1024&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~176/175 = 6.519&lt;br /&gt;
&lt;br /&gt;
Map: [[&amp;lt;31 49 72 87 107|, &amp;lt;0 0 0 0 1|]&lt;br /&gt;
EDOs: 31, 93, 124b, 155b, 186b&lt;br /&gt;
Badness: 0.0430&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Prajapati-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;13-limit&lt;/h2&gt;
Commas: 81/80, 126/125, 105/104, 512/507&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~66/65 = 9.171&lt;br /&gt;
&lt;br /&gt;
Map: [[&amp;lt;31 49 72 87 107 115|, &amp;lt;0 0 0 0 1 0|]&lt;br /&gt;
EDOs: 31, 93f, 124bf&lt;br /&gt;
Badness: 0.0379&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Prajapati-Kumhar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Kumhar&lt;/h2&gt;
Commas: 81/80, 126/125, 1029/1024, 144/143&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~196/195 = 10.120&lt;br /&gt;
&lt;br /&gt;
Map: [[&amp;lt;31 49 72 87 107 115|, &amp;lt;0 0 0 0 1 -1|]&lt;br /&gt;
EDOs: 31, 62e, 93, 124b, 341b&lt;br /&gt;
Badness: 0.0486&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Gallium"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Gallium&lt;/h1&gt;
The name of gallium temperament comes from Gallium, the 31st element.&lt;br /&gt;
&lt;br /&gt;
Commas: 81/80, 126/125, 99/98, 121/120&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~16807/16640 = 15.541&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87 107 0|, &amp;lt;0 0 0 0 0 1|]&lt;br /&gt;
EDOs: 31, 62, 93e, 155bef&lt;br /&gt;
Badness: 0.0255&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Birds"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Birds&lt;/h1&gt;
The birds temperament tempers out the &lt;em&gt;31-5 comma&lt;/em&gt;, |72 0 -31&amp;gt;, which is the amount (24.275 cents) by which 31 just (5/4) major thirds fall short of ten octaves. The name comes from Isaiah 31:5 &amp;quot;As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it.&amp;quot; It also tempers out the 31-7 comma, but sadly, combining the two commas leads to torsion.&lt;br /&gt;
&lt;br /&gt;
Commas: 3136/3125, 823543/819200&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~1029/1024 = 5.155&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87|, &amp;lt;0 1 0 0|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;31 0 0 -72 -87 0||&lt;br /&gt;
Edos: 31, 124, 155, 186, 217, 248, 465&lt;br /&gt;
Badness: 0.1000&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Birds-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
Commas: 441/440, 3136/3125, 41503/41472&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~385/384 = 4.938&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87 107|, &amp;lt;0 1 0 0 2|]&lt;br /&gt;
EDOs: 31, 186e, 217, 248, 961cd&lt;br /&gt;
Badness: 0.0399&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Birds-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h2&gt;
Commas: 441/440, 3136/3125, 1001/1000, 13720/13689&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~385/384 = 5.170&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;31 49 72 87 107 115|, &amp;lt;0 1 0 0 2 -2|]&lt;br /&gt;
EDOs: 31, 186e, 217, 248, 465&lt;br /&gt;
Badness: 0.0357&lt;/body&gt;&lt;/html&gt;</pre></div>
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