31st-octave temperaments
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-12-01 16:29:31 UTC.
- The original revision id was 281263302.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
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
Original HTML content:
<html><head><title>31 comma temperaments</title></head><body>By the <em>31-3-comma</em> 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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Prajapati"></a><!-- ws:end:WikiTextHeadingRule:0 -->Prajapati</h1> 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.<br /> <br /> Commas: 81/80, 126/125, 1029/1024<br /> <br /> POTE generator: ~176/175 = 6.519<br /> <br /> Map: [[<31 49 72 87 107|, <0 0 0 0 1|]<br /> EDOs: 31, 93, 124b, 155b, 186b<br /> Badness: 0.0430<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Prajapati-13-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->13-limit</h2> Commas: 81/80, 126/125, 105/104, 512/507<br /> <br /> POTE generator: ~66/65 = 9.171<br /> <br /> Map: [[<31 49 72 87 107 115|, <0 0 0 0 1 0|]<br /> EDOs: 31, 93f, 124bf<br /> Badness: 0.0379<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Prajapati-Kumhar"></a><!-- ws:end:WikiTextHeadingRule:4 -->Kumhar</h2> Commas: 81/80, 126/125, 1029/1024, 144/143<br /> <br /> POTE generator: ~196/195 = 10.120<br /> <br /> Map: [[<31 49 72 87 107 115|, <0 0 0 0 1 -1|]<br /> EDOs: 31, 62e, 93, 124b, 341b<br /> Badness: 0.0486<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Gallium"></a><!-- ws:end:WikiTextHeadingRule:6 -->Gallium</h1> The name of gallium temperament comes from Gallium, the 31st element.<br /> <br /> Commas: 81/80, 126/125, 99/98, 121/120<br /> <br /> POTE generator: ~16807/16640 = 15.541<br /> <br /> Map: [<31 49 72 87 107 0|, <0 0 0 0 0 1|]<br /> EDOs: 31, 62, 93e, 155bef<br /> Badness: 0.0255<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Birds"></a><!-- ws:end:WikiTextHeadingRule:8 -->Birds</h1> The birds temperament tempers out the <em>31-5 comma</em>, |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.<br /> <br /> Commas: 3136/3125, 823543/819200<br /> <br /> POTE generator: ~1029/1024 = 5.155<br /> <br /> Map: [<31 49 72 87|, <0 1 0 0|]<br /> Wedgie: <<31 0 0 -72 -87 0||<br /> Edos: 31, 124, 155, 186, 217, 248, 465<br /> Badness: 0.1000<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Birds-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h2> Commas: 441/440, 3136/3125, 41503/41472<br /> <br /> POTE generator: ~385/384 = 4.938<br /> <br /> Map: [<31 49 72 87 107|, <0 1 0 0 2|]<br /> EDOs: 31, 186e, 217, 248, 961cd<br /> Badness: 0.0399<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Birds-13-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit</h2> Commas: 441/440, 3136/3125, 1001/1000, 13720/13689<br /> <br /> POTE generator: ~385/384 = 5.170<br /> <br /> Map: [<31 49 72 87 107 115|, <0 1 0 0 2 -2|]<br /> EDOs: 31, 186e, 217, 248, 465<br /> Badness: 0.0357</body></html>