31st-octave temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
__FORCETOC__
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-01 18:23:43 UTC</tt>.<br>
: The original revision id was <tt>281304484</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">[[toc|flat]]
 
 
=31-3-comma=
=31-3-comma=
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.
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=
=Prajapati=
Line 20: Line 11:


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


Line 29: Line 22:


Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 0|]
Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 0|]
EDOs: 31, 93f, 124bf
EDOs: 31, 93f, 124bf
Badness: 0.0379
Badness: 0.0379


Line 38: Line 33:


Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 -1|]
Map: [[&lt;31 49 72 87 107 115|, &lt;0 0 0 0 1 -1|]
EDOs: 31, 62e, 93, 124b, 341b
EDOs: 31, 62e, 93, 124b, 341b
Badness: 0.0486
Badness: 0.0486


Line 49: Line 46:


Map: [&lt;31 49 72 87 107 0|, &lt;0 0 0 0 0 1|]
Map: [&lt;31 49 72 87 107 0|, &lt;0 0 0 0 0 1|]
EDOs: 31, 62, 93e, 155bef
EDOs: 31, 62, 93e, 155bef
Badness: 0.0255
Badness: 0.0255


=Birds=
=Birds=
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.
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
Commas: 3136/3125, 823543/819200
Line 60: Line 59:


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


Line 70: Line 72:


Map: [&lt;31 49 72 87 107|, &lt;0 1 0 0 2|]
Map: [&lt;31 49 72 87 107|, &lt;0 1 0 0 2|]
EDOs: 31, 186e, 217, 248, 961cd
EDOs: 31, 186e, 217, 248, 961cd
Badness: 0.0399
Badness: 0.0399


Line 79: Line 83:


Map: [&lt;31 49 72 87 107 115|, &lt;0 1 0 0 2 -2|]
Map: [&lt;31 49 72 87 107 115|, &lt;0 1 0 0 2 -2|]
EDOs: 31, 186e, 217, 248, 465
EDOs: 31, 186e, 217, 248, 465
Badness: 0.0357
Badness: 0.0357
</pre></div>
<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;&lt;!-- ws:start:WikiTextTocRule:16:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#x31-3-comma"&gt;31-3-comma&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Prajapati"&gt;Prajapati&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Gallium"&gt;Gallium&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Birds"&gt;Birds&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&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="x31-3-comma"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;31-3-comma&lt;/h1&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:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Prajapati"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&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:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Prajapati-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&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:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Prajapati-Kumhar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&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:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Gallium"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&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:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Birds"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&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:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Birds-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&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:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Birds-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&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>

Revision as of 00:00, 17 July 2018

31-3-comma

By the 31-3-comma is meant 617673396283947/562949953421312 = <-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

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

POTE generator: ~176/175 = 6.519

Map: [[<31 49 72 87 107|, <0 0 0 0 1|]

EDOs: 31, 93, 124b, 155b, 186b

Badness: 0.0430

13-limit

Commas: 81/80, 126/125, 105/104, 512/507

POTE generator: ~66/65 = 9.171

Map: [[<31 49 72 87 107 115|, <0 0 0 0 1 0|]

EDOs: 31, 93f, 124bf

Badness: 0.0379

Kumhar

Commas: 81/80, 126/125, 1029/1024, 144/143

POTE generator: ~196/195 = 10.120

Map: [[<31 49 72 87 107 115|, <0 0 0 0 1 -1|]

EDOs: 31, 62e, 93, 124b, 341b

Badness: 0.0486

Gallium

The name of gallium temperament comes from Gallium, the 31st element.

Commas: 81/80, 126/125, 99/98, 121/120

POTE generator: ~16807/16640 = 15.541

Map: [<31 49 72 87 107 0|, <0 0 0 0 0 1|]

EDOs: 31, 62, 93e, 155bef

Badness: 0.0255

Birds

The birds temperament tempers out the 31-5 comma, |72 0 -31>, 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

POTE generator: ~1029/1024 = 5.155

Map: [<31 49 72 87|, <0 1 0 0|]

Wedgie: <<31 0 0 -72 -87 0||

Edos: 31, 124, 155, 186, 217, 248, 465

Badness: 0.1000

11-limit

Commas: 441/440, 3136/3125, 41503/41472

POTE generator: ~385/384 = 4.938

Map: [<31 49 72 87 107|, <0 1 0 0 2|]

EDOs: 31, 186e, 217, 248, 961cd

Badness: 0.0399

13-limit

Commas: 441/440, 3136/3125, 1001/1000, 13720/13689

POTE generator: ~385/384 = 5.170

Map: [<31 49 72 87 107 115|, <0 1 0 0 2 -2|]

EDOs: 31, 186e, 217, 248, 465

Badness: 0.0357

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