Meantone family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 240715343 - Original comment: ** |
Wikispaces>keenanpepper **Imported revision 246435339 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-17 04:31:57 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>246435339</tt>.<br> | ||
: The revision comment was: <tt></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> | 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> | <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]] | <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]] | ||
The [[5-limit]] parent [[Comma|comma]] of the [[meantone]] family is the Didymus or [[http://en.wikipedia.org/wiki/Syntonic_comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval. | The [[5-limit]] parent [[Comma|comma]] of the [[meantone]] family is the Didymus or [[http://en.wikipedia.org/wiki/Syntonic_comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval. | ||
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[[Badness]]: 0.00736 | [[Badness]]: 0.00736 | ||
==Seven limit children== | ==Seven limit children== | ||
The [[7-limit]] children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1>, |-13 10 0 -1>], flattone, with normal list [|-4 4 -1>, |-17 9 0 1>], dominant, with normal list [|-4 4 -1>, |6 -2 0 -1>], sharptone, with normal list [|-4 4 -1>, |2 -3 0 1>], injera, with normal list [|-4 4 -1>, |-7 8 0 -2>], mohajira, with normal list [|-4 4 -1>, |-23 11 0 2>], godzilla, with normal list [|-4 4 -1>, |-4 -1 0 2>], mothra, with normal list [|-4 4 -1>, |-10 1 0 3>], squares, with normal list [|-4 4 -1>, |-3 9 0 -4>], and liese, with normal list [|-4 4 -1>, |-9 11 0 -3>]. | The [[7-limit]] children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1>, |-13 10 0 -1>], flattone, with normal list [|-4 4 -1>, |-17 9 0 1>], dominant, with normal list [|-4 4 -1>, |6 -2 0 -1>], sharptone, with normal list [|-4 4 -1>, |2 -3 0 1>], injera, with normal list [|-4 4 -1>, |-7 8 0 -2>], mohajira, with normal list [|-4 4 -1>, |-23 11 0 2>], godzilla, with normal list [|-4 4 -1>, |-4 -1 0 2>], mothra, with normal list [|-4 4 -1>, |-10 1 0 3>], squares, with normal list [|-4 4 -1>, |-3 9 0 -4>], and liese, with normal list [|-4 4 -1>, |-9 11 0 -3>]. | ||
=Septimal meantone= | =Septimal meantone= | ||
The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the [[7_4|7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and [[7_5|7/5]], C-F#, the tritone. The [[Wedgies and Multivals|wedgie]] for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo]] is a good tuning for it. | The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the [[7_4|7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and [[7_5|7/5]], C-F#, the tritone. The [[Wedgies and Multivals|wedgie]] for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo]] is a good tuning for it. | ||
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[[Badness]]: 0.0137 | [[Badness]]: 0.0137 | ||
==Unidecimal meantone aka Huygens== | ==Unidecimal meantone aka Huygens== | ||
[[Comma]]s: 81/80, 126/125, 99/98 | [[Comma]]s: 81/80, 126/125, 99/98 | ||
[[11-limit]] minimax | [[11-limit]] minimax | ||
[|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>, | [|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>, | ||
|21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>] | |21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>] | ||
[[Eigenmonzo]]s: 2, 11/9 | [[Eigenmonzo]]s: 2, 11/9 | ||
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[[Badness]]: 0.0170 | [[Badness]]: 0.0170 | ||
===Tridecimal meantone=== | ===Tridecimal meantone=== | ||
[[Comma]]s: 66/65, 81/80, 99/98, 105/104 | [[Comma]]s: 66/65, 81/80, 99/98, 105/104 | ||
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[[Badness]]: 0.0180 | [[Badness]]: 0.0180 | ||
===Grosstone=== | ===Grosstone=== | ||
Commas: 81/80, 99/98, 126/125, 144/143 | Commas: 81/80, 99/98, 126/125, 144/143 | ||
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Badness: 0.0259 | Badness: 0.0259 | ||
==Meanpop== | ==Meanpop== | ||
[[Comma]]s: 81/80, 126/125, 385/384 | [[Comma]]s: 81/80, 126/125, 385/384 | ||
[[11-limit]] [[minimax]] 1/4 comma | [[11-limit]] [[minimax]] 1/4 comma | ||
[|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, | [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, | ||
|-3 0 5/2 0 0>, |11 0 -13/4 0 0>] | |-3 0 5/2 0 0>, |11 0 -13/4 0 0>] | ||
[[Eigenmonzo]]s: 2, 5 | [[Eigenmonzo]]s: 2, 5 | ||
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[[Badness]]: 0.0215 | [[Badness]]: 0.0215 | ||
===13-limit Meanpop=== | ===13-limit Meanpop=== | ||
[[Comma]]s: 81/80, 105/104, 144/143, 196/195 | [[Comma]]s: 81/80, 105/104, 144/143, 196/195 | ||
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[[Badness]]: 0.0209 | [[Badness]]: 0.0209 | ||
==Meanenneadecal== | ==Meanenneadecal== | ||
[[Comma]]s: 45/44, 56/55, 81/80 | [[Comma]]s: 45/44, 56/55, 81/80 | ||
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[[Badness]]: 0.0214 | [[Badness]]: 0.0214 | ||
===13-limit=== | ===13-limit=== | ||
[[Comma]]s: 45/44, 56/55, 78/77, 81/80 | [[Comma]]s: 45/44, 56/55, 78/77, 81/80 | ||
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[[Badness]]: 0.0212 | [[Badness]]: 0.0212 | ||
=Flattone= | =Flattone= | ||
[[Comma]]s: 81/80, 525/512 | [[Comma]]s: 81/80, 525/512 | ||
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[[7-limit]] minimax | [[7-limit]] minimax | ||
[|1 0 0 0>, |21/13 0 1/13 -1/13>, | [|1 0 0 0>, |21/13 0 1/13 -1/13>, | ||
|32/13 0 4/13 -4/13>, |32/13 0 -9/13 9/13>] | |32/13 0 4/13 -4/13>, |32/13 0 -9/13 9/13>] | ||
[[Eigenmonzo]]s: 2, 7/5 | [[Eigenmonzo]]s: 2, 7/5 | ||
[[9-limit]] minimax | [[9-limit]] minimax | ||
[|1 0 0 0>, |17/11 2/11 0 -1/11>, | [|1 0 0 0>, |17/11 2/11 0 -1/11>, | ||
|24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>] | |24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>] | ||
[[Eigenmonzo]]s: 2, 9/7 | [[Eigenmonzo]]s: 2, 9/7 | ||
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[[Badness]]: 0.0386 | [[Badness]]: 0.0386 | ||
=Dominant= | =Dominant= | ||
[[Comma]]s: 36/35, 64/63 | [[Comma]]s: 36/35, 64/63 | ||
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[[Badness]]: 0.0207 | [[Badness]]: 0.0207 | ||
=Sharptone= | =Sharptone= | ||
[[Comma]]s: 21/20, 28/27 | [[Comma]]s: 21/20, 28/27 | ||
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[[Badness]]: 0.0248 | [[Badness]]: 0.0248 | ||
=Injera= | =Injera= | ||
[[Comma]]s: 50/49, 81/80 | [[Comma]]s: 50/49, 81/80 | ||
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[[Badness]]: 0.0311 | [[Badness]]: 0.0311 | ||
=Godzilla= | =Godzilla= | ||
[[Comma]]s: 49/48, 81/80 | [[Comma]]s: 49/48, 81/80 | ||
Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a- | Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo]] is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes. | ||
[[POTE tuning|POTE generator]]: 252.635 | [[POTE tuning|POTE generator]]: 252.635 | ||
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[[http://tinyurl.com/4uyumk9|"Change is on the Wind"]] in Godzilla[9] by [[Igliashon Jones]] | [[http://tinyurl.com/4uyumk9|"Change is on the Wind"]] in Godzilla[9] by [[Igliashon Jones]] | ||
=Mohajira= | =Mohajira= | ||
[[Comma]]s: 81/80, 6144/6125 | [[Comma]]s: 81/80, 6144/6125 | ||
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[[Badness]]: 0.0557 | [[Badness]]: 0.0557 | ||
==11-limit== | ==11-limit== | ||
[[Comma]]s: 81/80, 121/120, 176/175 | [[Comma]]s: 81/80, 121/120, 176/175 | ||
[[11-limit]] minimax 1/4 comma | [[11-limit]] minimax 1/4 comma | ||
[|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, | [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, | ||
|6 0 -11/8 0 0>, |2 0 5/8 0 0>] | |6 0 -11/8 0 0>, |2 0 5/8 0 0>] | ||
[[Eigenmonzo]]s: 2, 5 | [[Eigenmonzo]]s: 2, 5 | ||
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[[Badness]]: 0.0261 | [[Badness]]: 0.0261 | ||
=Mothra= | =Mothra= | ||
[[Comma]]s: 81/80, 1029/1024 | [[Comma]]s: 81/80, 1029/1024 | ||
Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. | Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. | ||
[[7-limit|7]] and [[9-limit]] minimax 1/4 comma | [[7-limit|7]] and [[9-limit]] minimax 1/4 comma | ||
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>] | ||
[[Eigenmonzo]]s: 2, 5 | [[Eigenmonzo]]s: 2, 5 | ||
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[[Badness]]: 0.0371 | [[Badness]]: 0.0371 | ||
==11-limit== | ==11-limit== | ||
[[Comma]]s: 81/80, 99/98, 385/384 | [[Comma]]s: 81/80, 99/98, 385/384 | ||
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[[Badness]]: 0.0256 | [[Badness]]: 0.0256 | ||
=Squares= | =Squares= | ||
[[Comma]]s: 81/80, 2401/2400 | [[Comma]]s: 81/80, 2401/2400 | ||
Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third ([[9_7|9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a | Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third ([[9_7|9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401. | ||
7 and 9 limit minimax 1/4 comma | 7 and 9 limit minimax 1/4 comma | ||
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[[http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3|Square 8]] | [[http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3|Square 8]] | ||
==11-limit== | ==11-limit== | ||
[[Comma]]s: 81/80, 385/384, 1375/1372 | [[Comma]]s: 81/80, 385/384, 1375/1372 | ||
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[[Badness]]: 0.0568 | [[Badness]]: 0.0568 | ||
=Liese= | =Liese= | ||
[[Comma]]s: 81/80, 686/675 | [[Comma]]s: 81/80, 686/675 | ||
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[[Badness]]: 0.0467 | [[Badness]]: 0.0467 | ||
=Squares= | =Squares= | ||
Commas: 81/80, 2401/2400 | Commas: 81/80, 2401/2400 | ||
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[[Badness]]: 0.0460 | [[Badness]]: 0.0460 | ||
==11-limit== | ==11-limit== | ||
Commas: 81/80, 99/98, 121/120 | Commas: 81/80, 99/98, 121/120 | ||
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[[Badness]]: 0.0216 | [[Badness]]: 0.0216 | ||
==13-limit== | ==13-limit== | ||
Commas: 81/80, 99/98, 121/120, 66/65 | Commas: 81/80, 99/98, 121/120, 66/65 | ||
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<h4>Original HTML content:</h4> | <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"><html><head><title>Meantone family</title></head><body><!-- ws:start:WikiTextTocRule:50:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --> | <a href="#Septimal meantone">Septimal meantone</a><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --> | <a href="#Flattone">Flattone</a><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --> | <a href="#Dominant">Dominant</a><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Sharptone">Sharptone</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --> | <a href="#Injera">Injera</a><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --> | <a href="#Godzilla">Godzilla</a><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --> | <a href="#Mohajira">Mohajira</a><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --> | <a href="#Mothra">Mothra</a><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --> | <a href="#Squares">Squares</a><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --> | <a href="#Liese">Liese</a><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --> | <a href="#Squares">Squares</a><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --> | <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"><html><head><title>Meantone family</title></head><body><!-- ws:start:WikiTextTocRule:50:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --> | <a href="#Septimal meantone">Septimal meantone</a><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --> | <a href="#Flattone">Flattone</a><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --> | <a href="#Dominant">Dominant</a><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --> | <a href="#Sharptone">Sharptone</a><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --> | <a href="#Injera">Injera</a><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --> | <a href="#Godzilla">Godzilla</a><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --> | <a href="#Mohajira">Mohajira</a><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --> | <a href="#Mothra">Mothra</a><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --> | <a href="#Squares">Squares</a><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --> | <a href="#Liese">Liese</a><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --> | <a href="#Squares">Squares</a><!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --> | ||
<!-- ws:end:WikiTextTocRule:76 --> | <!-- ws:end:WikiTextTocRule:76 -->The <a class="wiki_link" href="/5-limit">5-limit</a> parent <a class="wiki_link" href="/Comma">comma</a> of the <a class="wiki_link" href="/meantone">meantone</a> family is the Didymus or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br /> | ||
The <a class="wiki_link" href="/5-limit">5-limit</a> parent <a class="wiki_link" href="/Comma">comma</a> of the <a class="wiki_link" href="/meantone">meantone</a> family is the Didymus or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br /> | |||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.239<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.239<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> | ||
The <a class="wiki_link" href="/7-limit">7-limit</a> children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1&gt;, |-13 10 0 -1&gt;], flattone, with normal list [|-4 4 -1&gt;, |-17 9 0 1&gt;], dominant, with normal list [|-4 4 -1&gt;, |6 -2 0 -1&gt;], sharptone, with normal list [|-4 4 -1&gt;, |2 -3 0 1&gt;], injera, with normal list [|-4 4 -1&gt;, |-7 8 0 -2&gt;], mohajira, with normal list [|-4 4 -1&gt;, |-23 11 0 2&gt;], godzilla, with normal list [|-4 4 -1&gt;, |-4 -1 0 2&gt;], mothra, with normal list [|-4 4 -1&gt;, |-10 1 0 3&gt;], squares, with normal list [|-4 4 -1&gt;, |-3 9 0 -4&gt;], and liese, with normal list [|-4 4 -1&gt;, |-9 11 0 -3&gt;].<br /> | The <a class="wiki_link" href="/7-limit">7-limit</a> children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1&gt;, |-13 10 0 -1&gt;], flattone, with normal list [|-4 4 -1&gt;, |-17 9 0 1&gt;], dominant, with normal list [|-4 4 -1&gt;, |6 -2 0 -1&gt;], sharptone, with normal list [|-4 4 -1&gt;, |2 -3 0 1&gt;], injera, with normal list [|-4 4 -1&gt;, |-7 8 0 -2&gt;], mohajira, with normal list [|-4 4 -1&gt;, |-23 11 0 2&gt;], godzilla, with normal list [|-4 4 -1&gt;, |-4 -1 0 2&gt;], mothra, with normal list [|-4 4 -1&gt;, |-10 1 0 3&gt;], squares, with normal list [|-4 4 -1&gt;, |-3 9 0 -4&gt;], and liese, with normal list [|-4 4 -1&gt;, |-9 11 0 -3&gt;].<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Septimal meantone"></a><!-- ws:end:WikiTextHeadingRule:2 -->Septimal meantone</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Septimal meantone"></a><!-- ws:end:WikiTextHeadingRule:2 -->Septimal meantone</h1> | ||
The comma |-13 10 0 -1&gt; for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the <a class="wiki_link" href="/7_4">7/4</a> of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and <a class="wiki_link" href="/7_5">7/5</a>, C-F#, the tritone. The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for septimal meantone is &lt;&lt;1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and <a class="wiki_link" href="/31edo">31edo</a> is a good tuning for it.<br /> | The comma |-13 10 0 -1&gt; for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the <a class="wiki_link" href="/7_4">7/4</a> of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and <a class="wiki_link" href="/7_5">7/5</a>, C-F#, the tritone. The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for septimal meantone is &lt;&lt;1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and <a class="wiki_link" href="/31edo">31edo</a> is a good tuning for it.<br /> | ||
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<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Septimal meantone-Unidecimal meantone aka Huygens"></a><!-- ws:end:WikiTextHeadingRule:4 -->Unidecimal meantone aka Huygens</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Septimal meantone-Unidecimal meantone aka Huygens"></a><!-- ws:end:WikiTextHeadingRule:4 -->Unidecimal meantone aka Huygens</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 99/98<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 99/98<br /> | ||
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<a class="wiki_link" href="/11-limit">11-limit</a> minimax<br /> | <a class="wiki_link" href="/11-limit">11-limit</a> minimax<br /> | ||
[|1 0 0 0 0&gt;, |25/16 -1/8 0 0 1/16&gt;, |9/4 -1/2 0 0 1/4&gt;, <br /> | [|1 0 0 0 0&gt;, |25/16 -1/8 0 0 1/16&gt;, |9/4 -1/2 0 0 1/4&gt;,<br /> | ||
|21/8 -5/4 0 0 5/8&gt;, |25/8 -9/4 0 0 9/8&gt;]<br /> | |21/8 -5/4 0 0 5/8&gt;, |25/8 -9/4 0 0 9/8&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 11/9<br /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 11/9<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tridecimal meantone</h3> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tridecimal meantone</h3> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 66/65, 81/80, 99/98, 105/104<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 66/65, 81/80, 99/98, 105/104<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.642<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.642<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Grosstone"></a><!-- ws:end:WikiTextHeadingRule:8 -->Grosstone</h3> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Grosstone"></a><!-- ws:end:WikiTextHeadingRule:8 -->Grosstone</h3> | ||
Commas: 81/80, 99/98, 126/125, 144/143<br /> | Commas: 81/80, 99/98, 126/125, 144/143<br /> | ||
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POTE generator: ~3/2 = 697.264<br /> | POTE generator: ~3/2 = 697.264<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Septimal meantone-Meanpop"></a><!-- ws:end:WikiTextHeadingRule:10 -->Meanpop</h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Septimal meantone-Meanpop"></a><!-- ws:end:WikiTextHeadingRule:10 -->Meanpop</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 385/384<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 385/384<br /> | ||
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<a class="wiki_link" href="/11-limit">11-limit</a> <a class="wiki_link" href="/minimax">minimax</a> 1/4 comma<br /> | <a class="wiki_link" href="/11-limit">11-limit</a> <a class="wiki_link" href="/minimax">minimax</a> 1/4 comma<br /> | ||
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;, <br /> | [|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,<br /> | ||
|-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]<br /> | |-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Septimal meantone-Meanpop-13-limit Meanpop"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit Meanpop</h3> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Septimal meantone-Meanpop-13-limit Meanpop"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit Meanpop</h3> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 105/104, 144/143, 196/195<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 105/104, 144/143, 196/195<br /> | ||
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POTE generator: ~3/2 = 696.211<br /> | POTE generator: ~3/2 = 696.211<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Septimal meantone-Meanenneadecal"></a><!-- ws:end:WikiTextHeadingRule:14 -->Meanenneadecal</h2> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Septimal meantone-Meanenneadecal"></a><!-- ws:end:WikiTextHeadingRule:14 -->Meanenneadecal</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 81/80<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 81/80<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.250<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.250<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Septimal meantone-Meanenneadecal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit</h3> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Septimal meantone-Meanenneadecal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit</h3> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 78/77, 81/80<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 78/77, 81/80<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.146<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.146<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Flattone"></a><!-- ws:end:WikiTextHeadingRule:18 -->Flattone</h1> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Flattone"></a><!-- ws:end:WikiTextHeadingRule:18 -->Flattone</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 525/512<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 525/512<br /> | ||
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The <a class="wiki_link" href="/wedgie">wedgie</a> for flattone is &lt;&lt;1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that <a class="wiki_link" href="/7_4">7/4</a> is a diminished minor seventh interval. Other intervals are <a class="wiki_link" href="/7_6">7/6</a>, a diminished minor third, and <a class="wiki_link" href="/7_5">7/5</a>, a doubly diminshed fifth. Good tunings for flattone are <a class="wiki_link" href="/26edo">26edo</a>, <a class="wiki_link" href="/45edo">45edo</a> and <a class="wiki_link" href="/64edo">64edo</a>.<br /> | The <a class="wiki_link" href="/wedgie">wedgie</a> for flattone is &lt;&lt;1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that <a class="wiki_link" href="/7_4">7/4</a> is a diminished minor seventh interval. Other intervals are <a class="wiki_link" href="/7_6">7/6</a>, a diminished minor third, and <a class="wiki_link" href="/7_5">7/5</a>, a doubly diminshed fifth. Good tunings for flattone are <a class="wiki_link" href="/26edo">26edo</a>, <a class="wiki_link" href="/45edo">45edo</a> and <a class="wiki_link" href="/64edo">64edo</a>.<br /> | ||
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<a class="wiki_link" href="/7-limit">7-limit</a> minimax<br /> | <a class="wiki_link" href="/7-limit">7-limit</a> minimax<br /> | ||
[|1 0 0 0&gt;, |21/13 0 1/13 -1/13&gt;, <br /> | [|1 0 0 0&gt;, |21/13 0 1/13 -1/13&gt;,<br /> | ||
|32/13 0 4/13 -4/13&gt;, |32/13 0 -9/13 9/13&gt;]<br /> | |32/13 0 4/13 -4/13&gt;, |32/13 0 -9/13 9/13&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 7/5<br /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 7/5<br /> | ||
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<a class="wiki_link" href="/9-limit">9-limit</a> minimax<br /> | <a class="wiki_link" href="/9-limit">9-limit</a> minimax<br /> | ||
[|1 0 0 0&gt;, |17/11 2/11 0 -1/11&gt;, <br /> | [|1 0 0 0&gt;, |17/11 2/11 0 -1/11&gt;,<br /> | ||
|24/11 8/11 0 -4/11&gt;, |34/11 -18/11 0 9/11&gt;]<br /> | |24/11 8/11 0 -4/11&gt;, |34/11 -18/11 0 9/11&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 9/7<br /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 9/7<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Dominant"></a><!-- ws:end:WikiTextHeadingRule:20 -->Dominant</h1> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Dominant"></a><!-- ws:end:WikiTextHeadingRule:20 -->Dominant</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 36/35, 64/63<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 36/35, 64/63<br /> | ||
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The wedgie for dominant is &lt;&lt;1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is <a class="wiki_link" href="/12edo">12edo</a>, but it also works well with the Pythagorean tuning of pure <a class="wiki_link" href="/3_2">3/2</a> fifths, and with <a class="wiki_link" href="/29edo">29edo</a>, <a class="wiki_link" href="/41edo">41edo</a>, or <a class="wiki_link" href="/53edo">53edo</a>.<br /> | The wedgie for dominant is &lt;&lt;1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is <a class="wiki_link" href="/12edo">12edo</a>, but it also works well with the Pythagorean tuning of pure <a class="wiki_link" href="/3_2">3/2</a> fifths, and with <a class="wiki_link" href="/29edo">29edo</a>, <a class="wiki_link" href="/41edo">41edo</a>, or <a class="wiki_link" href="/53edo">53edo</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Sharptone"></a><!-- ws:end:WikiTextHeadingRule:22 -->Sharptone</h1> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Sharptone"></a><!-- ws:end:WikiTextHeadingRule:22 -->Sharptone</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 21/20, 28/27<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 21/20, 28/27<br /> | ||
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Sharptone, with a wedgie &lt;&lt;1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. <a class="wiki_link" href="/12edo">12edo</a> tuning does sharptone about as well as such a thing can be done.<br /> | Sharptone, with a wedgie &lt;&lt;1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. <a class="wiki_link" href="/12edo">12edo</a> tuning does sharptone about as well as such a thing can be done.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Injera"></a><!-- ws:end:WikiTextHeadingRule:24 -->Injera</h1> | <!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Injera"></a><!-- ws:end:WikiTextHeadingRule:24 -->Injera</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 50/49, 81/80<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 50/49, 81/80<br /> | ||
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The wedgie for injera is &lt;&lt;2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. <a class="wiki_link" href="/38edo">38edo</a>, which is two parallel <a class="wiki_link" href="/19edo">19edo</a>s, is an excellent tuning for injera.<br /> | The wedgie for injera is &lt;&lt;2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. <a class="wiki_link" href="/38edo">38edo</a>, which is two parallel <a class="wiki_link" href="/19edo">19edo</a>s, is an excellent tuning for injera.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Godzilla"></a><!-- ws:end:WikiTextHeadingRule:26 -->Godzilla</h1> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc13"><a name="Godzilla"></a><!-- ws:end:WikiTextHeadingRule:26 -->Godzilla</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 49/48, 81/80<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 49/48, 81/80<br /> | ||
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Godzilla has wedgie &lt;&lt;2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a- | Godzilla has wedgie &lt;&lt;2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. <a class="wiki_link" href="/19edo">19edo</a> is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 252.635<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 252.635<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Mohajira"></a><!-- ws:end:WikiTextHeadingRule:30 -->Mohajira</h1> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Mohajira"></a><!-- ws:end:WikiTextHeadingRule:30 -->Mohajira</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 6144/6125<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 6144/6125<br /> | ||
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Mohajira, with wedgie &lt;&lt;2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. <a class="wiki_link" href="/31edo">31edo</a> makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.<br /> | Mohajira, with wedgie &lt;&lt;2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. <a class="wiki_link" href="/31edo">31edo</a> makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Mohajira-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Mohajira-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 121/120, 176/175<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 121/120, 176/175<br /> | ||
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<a class="wiki_link" href="/11-limit">11-limit</a> minimax 1/4 comma<br /> | <a class="wiki_link" href="/11-limit">11-limit</a> minimax 1/4 comma<br /> | ||
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;, <br /> | [|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,<br /> | ||
|6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]<br /> | |6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Mothra"></a><!-- ws:end:WikiTextHeadingRule:34 -->Mothra</h1> | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Mothra"></a><!-- ws:end:WikiTextHeadingRule:34 -->Mothra</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 1029/1024<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 1029/1024<br /> | ||
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Mothra, with wedgie &lt;&lt;3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using <a class="wiki_link" href="/31edo">31edo</a> with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra.<br /> | Mothra, with wedgie &lt;&lt;3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using <a class="wiki_link" href="/31edo">31edo</a> with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra.<br /> | ||
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<a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> minimax 1/4 comma <br /> | <a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> minimax 1/4 comma<br /> | ||
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 99/98, 385/384<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 99/98, 385/384<br /> | ||
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POTE generator: ~63/55 = 232.031<br /> | POTE generator: ~63/55 = 232.031<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:38 -->Squares</h1> | <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:38 -->Squares</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 2401/2400<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 2401/2400<br /> | ||
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Squares, with wedgie &lt;&lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (<a class="wiki_link" href="/9_7">9/7</a>) intervals, and uses it for a generator. <a class="wiki_link" href="/31edo">31edo</a>, with a generator of 11/31, makes for a | Squares, with wedgie &lt;&lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (<a class="wiki_link" href="/9_7">9/7</a>) intervals, and uses it for a generator. <a class="wiki_link" href="/31edo">31edo</a>, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.<br /> | ||
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7 and 9 limit minimax 1/4 comma<br /> | 7 and 9 limit minimax 1/4 comma<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 385/384, 1375/1372<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 385/384, 1375/1372<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.993<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.993<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Liese"></a><!-- ws:end:WikiTextHeadingRule:42 -->Liese</h1> | <!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Liese"></a><!-- ws:end:WikiTextHeadingRule:42 -->Liese</h1> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 686/675<br /> | <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 686/675<br /> | ||
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Liese, with wedgie &lt;&lt;3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. <a class="wiki_link" href="/74edo">74edo</a> makes for a good liese tuning, though <a class="wiki_link" href="/19edo">19edo</a> can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.<br /> | Liese, with wedgie &lt;&lt;3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. <a class="wiki_link" href="/74edo">74edo</a> makes for a good liese tuning, though <a class="wiki_link" href="/19edo">19edo</a> can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:44 -->Squares</h1> | <!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:44 -->Squares</h1> | ||
Commas: 81/80, 2401/2400<br /> | Commas: 81/80, 2401/2400<br /> | ||
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POTE generator: ~9/7 = 425.942<br /> | POTE generator: ~9/7 = 425.942<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->11-limit</h2> | ||
Commas: 81/80, 99/98, 121/120<br /> | Commas: 81/80, 99/98, 121/120<br /> | ||
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POTE generator: ~9/7 = 425.957<br /> | POTE generator: ~9/7 = 425.957<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc24"><a name="Squares-13-limit"></a><!-- ws:end:WikiTextHeadingRule:48 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc24"><a name="Squares-13-limit"></a><!-- ws:end:WikiTextHeadingRule:48 -->13-limit</h2> | ||
Commas: 81/80, 99/98, 121/120, 66/65<br /> | Commas: 81/80, 99/98, 121/120, 66/65<br /> | ||
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POTE generator: ~9/7 = 425.550<br /> | POTE generator: ~9/7 = 425.550<br /> | ||