Meantone family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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<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.

[[POTE tuning|POTE generator]]: 696.239

Map: [<1 0 -4|, <0 1 4|]

[[Map]]: [<1 0 -4|, <0 1 4|]

EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]

[[Badness]]: 0.00736

==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=

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 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and 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.

[[Comma]]s: 81/80, 126/125

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Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.

Map: [<1 0 -4 -13|, <0 1 4 10|]

[[Map]]: [<1 0 -4 -13|, <0 1 4 10|]

[[Generator]]s: 2, 3

Wedgie: <<1 4 10 4 13 12||

[[Wedgie]]: <<1 4 10 4 13 12||

EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]]

[[Badness]]: 0.0137

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[[POTE tuning|POTE generator]]: 696.967

Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.

[[Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.

Map: [<1 0 -4 -13 -25|, <0 1 4 10 18|]

[[Map]]: [<1 0 -4 -13 -25|, <0 1 4 10 18|]

[[Generator]]s: 2, 3

EDOs: [[7edo|7]], [[12edo|12]], [[31edo|31]], [[105edo|105]], [[198edo|198]]

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[[Comma]]s: 66/65, 81/80, 99/98, 105/104

POTE generator: ~3/2 = 696.642

[[POTE tuning|POTE generator]]: ~3/2 = 696.642

~~Map:~~ Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]

Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]

EDOs: [[12edo|12]], [[19edo|19], [[31edo|31]], [[267edo|267]], [[298edo|298]]

EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[267edo|267]], [[298edo|298]]

[[Badness]]: 0.0180

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[[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>,

|-3 0 5/2 0 0>, |11 0 -13/4 0 0>]

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[[POTE tuning|POTE generator]]: 696.434

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

[[Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]

[[Generator]]s: 2, 3

EDOs: 12, 19, 31, 81, [[112edo|112]]

EDOs: [[12edo|12]], [[19edo|19]], [[31|31]], [[81edo|81]], [[112edo|112]]

[[Badness]]: 0.0215

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Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]

EDOS: 7, 12, 19, 31, 50, 81, [[131edo|131]]

EDOS: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131]]

[[Badness]]: 0.0209

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Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]

EDOs: 7, 12, 19, 31, 50, 81

EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]

[[Badness]]: 0.0214

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Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]

EDOs: 7, 12, 19, 31, 50, [[131edo|131]], [[181edo|181]]

EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[131edo|131]], [[181edo|181]]

[[Badness]]: 0.0212

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[[Comma]]s: 81/80, 525/512

The [[wedgie]] for flattone is <<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 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].

The [[wedgie]] for flattone is <<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 [[7_4|7/4]] is a diminished minor seventh interval. Other intervals are [[7_6|7/6]], a diminished minor third, and [[7_5|7/5]], a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].

[[7-limit]] minimax

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[[Wedgie]]: <<1 4 -9 4 -17 -32||

[[Generator]]s: 2, 3

EDOs: 7, 19, [[45edo|45]], [[64edo|64]]

EDOs: [[7edo|7]], [[19edo|19]], [[45edo|45]], [[64edo|64]]

[[Badness]]: 0.0386

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[[Comma]]s: 36/35, 64/63

The wedgie for dominant is <<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 [[12edo]], but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with [[29edo]], [[41edo]], or [[53edo]].

The wedgie for dominant is <<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 [[12edo]], but it also works well with the Pythagorean tuning of pure [[3_2|3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].

[[POTE tuning|POTE generator]]: 701.573

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Map: [<1 0 -4 6|, <0 1 4 -2|]

[[Wedgie]]: <<1 4 -2 4 -6 -16||

EDOs: 5, 7, 12, [[53edo|53]], [[65edo|65]]

EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[53edo|53]], [[65edo|65]]

[[Badness]]: 0.0207

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Map: [<1 0 -4 -2|, <0 1 4 3|]

[[Wedgie]]: <<1 4 3 4 2 -4||

EDOs: 5, 12

EDOs: [[5edo|5]], [[12edo|12]]

[[Badness]]: 0.0248

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[[Comma]]s: 50/49, 81/80

The wedgie for injera is <<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. [[38edo]], which is two parallel ~~19edos~~, is an excellent tuning for injera.

The wedgie for injera is <<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. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera.

[[POTE tuning|POTE generator]]: 694.375

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Mohajira, with wedgie <<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. [[31edo]] 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.

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>, |6 0 -11/8 0>]

[[Eigenmonzo]]s: 2, 5

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Map: [<1 1 0 6 2|, <0 2 8 -11 5|]

[[Generator]]s: 2, 11/9

EDOs: 7, 24, 31

EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]

[[Badness]]: 0.0261

=Mothra=

~~Commas~~: 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.

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>]

~~Eigenmonzos~~: 2, 5

[[Eigenmonzo]]s: 2, 5

[[POTE tuning|POTE generator]]: 232.193

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[[Generator]]s: 2, 8/7

[[Wedgie]]: <<3 12 -1 12 -10 -36||

EDOs: 5, [[26edo|26]], 31

EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]]

[[Badness]]: 0.0371

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Map: [<1 1 0 3 5|, <0 3 12 -1 -8|]

EDOs: 5, [[26edo|26]], 31, [[88edo|88]], [[150edo|150]], [[181edo|181]]

EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]]

[[Badness]]: 0.0256

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[[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) 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.

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

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Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|]

EDOs: [[14edo|14]], 31, [[200edo|200]]

EDOs: [[14edo|14]], [[31edo|31]], [[200edo|200]]

[[Badness]]: 0.0568

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POTE generator: ~9/7 = 425.942

Map: [<1 3 8 6|, <0 -4 -16 -9|]

[[Map]]: [<1 3 8 6|, <0 -4 -16 -9|]

Wedgie: <<4 16 9 16 3 -24||

[[Wedgie]]: <<4 16 9 16 3 -24||

EDOs: 5, 8, 11, 14, 17, 31

EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]]

Badness: 0.0460

[[Badness]]: 0.0460

==11-limit==

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Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|]

EDOs: 5, 8, 11, 14, 17, 31

EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]]

Badness: 0.0216

[[Badness]]: 0.0216

==13-limit==

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Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|]

EDOs: 5, 8, 11, 14, 17, 31

EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]]

Badness: 0.0255</pre></div>

[[Badness]]: 0.0255</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"><html><head><title>Meantone family</title></head><body> | <a href="#Septimal meantone">Septimal meantone</a> | <a href="#Flattone">Flattone</a> | <a href="#Dominant">Dominant</a> | <a href="#Sharptone">Sharptone</a> | <a href="#Injera">Injera</a> | <a href="#Godzilla">Godzilla</a> | <a href="#Mohajira">Mohajira</a> | <a href="#Mothra">Mothra</a> | <a href="#Squares">Squares</a> | <a href="#Liese">Liese</a> | <a href="#Squares">Squares</a>

The 5-limit 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.

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.

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.239

Map: [&lt;1 0 -4|, &lt;0 1 4|]

<a class="wiki_link" href="/Map">Map</a>: [&lt;1 0 -4|, &lt;0 1 4|]

EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/81edo">81</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.00736

<h2 id="toc0"><a name="x-Seven limit children"></a>Seven limit children</h2>

The 7-limit 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;].

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;].

<h1 id="toc1"><a name="Septimal meantone"></a>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 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, 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.

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.

<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125

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Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.

Map: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]

<a class="wiki_link" href="/Map">Map</a>: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]

<a class="wiki_link" href="/Generator">Generator</a>s: 2, 3

Wedgie: &lt;&lt;1 4 10 4 13 12||

<a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 10 4 13 12||

EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/81edo">81</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0137

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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.967

Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.

<a class="wiki_link" href="/Algebraic%20generator">Algebraic generator</a>: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.

Map: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]

<a class="wiki_link" href="/Map">Map</a>: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]

<a class="wiki_link" href="/Generator">Generator</a>s: 2, 3

EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/105edo">105</a>, <a class="wiki_link" href="/198edo">198</a>

Line 376:

<a class="wiki_link" href="/Comma">Comma</a>s: 66/65, 81/80, 99/98, 105/104

POTE generator: ~3/2 = 696.642

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.642

~~Map:~~ Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]

Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]

EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19], [[31edo|31</a>, <a class="wiki_link" href="/267edo">267</a>, <a class="wiki_link" href="/298edo">298</a>

EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/267edo">267</a>, <a class="wiki_link" href="/298edo">298</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0180

Line 385:

<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 385/384

11-limit minimax 1/4 comma

<a class="wiki_link" href="/11-limit">11-limit</a> <a class="wiki_link" href="/minimax">minimax</a> 1/4 comma

[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,

|-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]

Line 392:

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.434

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

<a class="wiki_link" href="/Algebraic%20generator">Algebraic generator</a>: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]

<a class="wiki_link" href="/Generator">Generator</a>s: 2, 3

EDOs: 12, 19, 31, 81, <a class="wiki_link" href="/112edo">112</a>

EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31">31</a>, <a class="wiki_link" href="/81edo">81</a>, <a class="wiki_link" href="/112edo">112</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0215

Line 405:

Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]

EDOS: 7, 12, 19, 31, 50, 81, <a class="wiki_link" href="/131edo">131</a>

EDOS: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/81edo">81</a>, <a class="wiki_link" href="/131edo">131</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0209

Line 414:

Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]

EDOs: 7, 12, 19, 31, 50, 81

EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/81edo">81</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0214

Line 423:

Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]

EDOs: 7, 12, 19, 31, 50, <a class="wiki_link" href="/131edo">131</a>, <a class="wiki_link" href="/181edo">181</a>

EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/131edo">131</a>, <a class="wiki_link" href="/181edo">181</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0212

Line 429:

<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 525/512

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 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, 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>.

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>.

<a class="wiki_link" href="/7-limit">7-limit</a> minimax

Line 448:

<a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 -9 4 -17 -32||

<a class="wiki_link" href="/Generator">Generator</a>s: 2, 3

EDOs: 7, 19, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/64edo">64</a>

EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/64edo">64</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0386

Line 454:

<a class="wiki_link" href="/Comma">Comma</a>s: 36/35, 64/63

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 3/2 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>.

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>.

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.573

Line 460:

Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]

<a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 -2 4 -6 -16||

EDOs: 5, 7, 12, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/65edo">65</a>

EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/65edo">65</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0207

Line 472:

Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]

<a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 3 4 2 -4||

EDOs: 5, 12

EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/12edo">12</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0248

Line 478:

<a class="wiki_link" href="/Comma">Comma</a>s: 50/49, 81/80

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 ~~19edos~~, is an excellent tuning for injera.

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.

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 694.375

Line 508:

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.

7 and 9-limit minimax 1/4 comma

<a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> minimax 1/4 comma

[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]

<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5

Line 534:

Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]

<a class="wiki_link" href="/Generator">Generator</a>s: 2, 11/9

EDOs: 7, 24, 31

EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/24edo">24</a>, <a class="wiki_link" href="/31edo">31</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0261

<h1 id="toc16"><a name="Mothra"></a>Mothra</h1>

~~Commas~~: 81/80, 1029/1024

<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 1029/1024

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.

7 and 9 limit minimax 1/4 comma

<a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> minimax 1/4 comma

[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]

~~Eigenmonzos~~: 2, 5

<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 232.193

Line 553:

<a class="wiki_link" href="/Generator">Generator</a>s: 2, 8/7

<a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;3 12 -1 12 -10 -36||

EDOs: 5, <a class="wiki_link" href="/26edo">26</a>, 31

EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/31edo">31</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0371

Line 562:

Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]

EDOs: 5, <a class="wiki_link" href="/26edo">26</a>, 31, <a class="wiki_link" href="/88edo">88</a>, <a class="wiki_link" href="/150edo">150</a>, <a class="wiki_link" href="/181edo">181</a>

EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/88edo">88</a>, <a class="wiki_link" href="/150edo">150</a>, <a class="wiki_link" href="/181edo">181</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0256

Line 568:

<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 2401/2400

Squares, with wedgie &lt;&lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) 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.

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.

7 and 9 limit minimax 1/4 comma

Line 593:

Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]

EDOs: <a class="wiki_link" href="/14edo">14</a>, 31, <a class="wiki_link" href="/200edo">200</a>

EDOs: <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/200edo">200</a>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0568

Line 619:

POTE generator: ~9/7 = 425.942

Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]

<a class="wiki_link" href="/Map">Map</a>: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]

Wedgie: &lt;&lt;4 16 9 16 3 -24||

<a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;4 16 9 16 3 -24||

EDOs: 5, 8, 11, 14, 17, 31

EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/8edo">8</a>, <a class="wiki_link" href="/11edo">11</a>, <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/31edo">31</a>

Badness: 0.0460

<a class="wiki_link" href="/Badness">Badness</a>: 0.0460

<h2 id="toc22"><a name="Squares-11-limit"></a>11-limit</h2>

Line 630:

Map: [&lt;1 3 8 6 7|, &lt;0 -4 -16 -9 -10|]

EDOs: 5, 8, 11, 14, 17, 31

EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/8edo">8</a>, <a class="wiki_link" href="/11edo">11</a>, <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/31edo">31</a>

Badness: 0.0216

<a class="wiki_link" href="/Badness">Badness</a>: 0.0216

<h2 id="toc23"><a name="Squares-13-limit"></a>13-limit</h2>

Line 639:

Map: [&lt;1 3 8 6 7 3|, &lt;0 -4 -16 -9 -10 2|]

EDOs: 5, 8, 11, 14, 17, 31

EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/8edo">8</a>, <a class="wiki_link" href="/11edo">11</a>, <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/31edo">31</a>

Badness: 0.0255</body></html></pre></div>

<a class="wiki_link" href="/Badness">Badness</a>: 0.0255</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-06-07 18:12:21 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-11 12:33:01 UTC</tt>.<br>
-: The original revision id was <tt>234981240</tt>.<br>
+: The original revision id was <tt>235941270</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>
@@ Line 8: / Line 8: @@
 <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&gt;, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], &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.
+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&gt;, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], &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.
 [[POTE tuning|POTE generator]]: 696.239
-Map: [&lt;1 0 -4|, &lt;0 1 4|]
+[[Map]]: [&lt;1 0 -4|, &lt;0 1 4|]
 EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]
 [[Badness]]: 0.00736
 ==Seven limit children==
-The 7-limit 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;].
+The [[7-limit]] 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;].
 =Septimal meantone=
-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 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, C-F#, the tritone. The [[Wedgies and Multivals|wedgie]] 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 [[31edo]] is a good tuning for it.
+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 [[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 &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 [[31edo]] is a good tuning for it.
 [[Comma]]s: 81/80, 126/125
@@ Line 32: / Line 32: @@
 Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.
-Map: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
+[[Map]]: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]
 [[Generator]]s: 2, 3
-Wedgie: &lt;&lt;1 4 10 4 13 12||
+[[Wedgie]]: &lt;&lt;1 4 10 4 13 12||
 EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]]
 [[Badness]]: 0.0137
@@ Line 48: / Line 48: @@
 [[POTE tuning|POTE generator]]: 696.967
-Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.
+[[Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.
-Map: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
+[[Map]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
 [[Generator]]s: 2, 3
 EDOs: [[7edo|7]], [[12edo|12]], [[31edo|31]], [[105edo|105]], [[198edo|198]]
@@ Line 58: / Line 58: @@
 [[Comma]]s: 66/65, 81/80, 99/98, 105/104
-POTE generator: ~3/2 = 696.642
+[[POTE tuning|POTE generator]]: ~3/2 = 696.642
-Map: Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
+Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]
-EDOs: [[12edo|12]], [[19edo|19], [[31edo|31]], [[267edo|267]], [[298edo|298]]
+EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[267edo|267]], [[298edo|298]]
 [[Badness]]: 0.0180
@@ Line 67: / Line 67: @@
 [[Comma]]s: 81/80, 126/125, 385/384
--limit minimax 1/4 comma
+[[11-limit]] [[minimax]] 1/4 comma
 [|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
 |-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]
@@ Line 74: / Line 74: @@
 [[POTE tuning|POTE generator]]: 696.434
-Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
+[[Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
 Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
 [[Generator]]s: 2, 3
-EDOs: 12, 19, 31, 81, [[112edo|112]]
+EDOs: [[12edo|12]], [[19edo|19]], [[31|31]], [[81edo|81]], [[112edo|112]]
 [[Badness]]: 0.0215
@@ Line 87: / Line 87: @@
 Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]
-EDOS: 7, 12, 19, 31, 50, 81, [[131edo|131]]
+EDOS: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131]]
 [[Badness]]: 0.0209
@@ Line 96: / Line 96: @@
 Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]
-EDOs: 7, 12, 19, 31, 50, 81
+EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]]
 [[Badness]]: 0.0214
@@ Line 105: / Line 105: @@
 Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]
-EDOs: 7, 12, 19, 31, 50, [[131edo|131]], [[181edo|181]]
+EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[131edo|131]], [[181edo|181]]
 [[Badness]]: 0.0212
@@ Line 111: / Line 111: @@
 [[Comma]]s: 81/80, 525/512
-The [[wedgie]] 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 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].
+The [[wedgie]] 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 [[7_4|7/4]] is a diminished minor seventh interval. Other intervals are [[7_6|7/6]], a diminished minor third, and [[7_5|7/5]], a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].
 [[7-limit]] minimax
@@ Line 130: / Line 130: @@
 [[Wedgie]]: &lt;&lt;1 4 -9 4 -17 -32||
 [[Generator]]s: 2, 3
-EDOs: 7, 19, [[45edo|45]], [[64edo|64]]
+EDOs: [[7edo|7]], [[19edo|19]], [[45edo|45]], [[64edo|64]]
 [[Badness]]: 0.0386
@@ Line 136: / Line 136: @@
 [[Comma]]s: 36/35, 64/63
-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 [[12edo]], but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with [[29edo]], [[41edo]], or [[53edo]].
+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 [[12edo]], but it also works well with the Pythagorean tuning of pure [[3_2|3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].
 [[POTE tuning|POTE generator]]: 701.573
@@ Line 142: / Line 142: @@
 Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]
 [[Wedgie]]: &lt;&lt;1 4 -2 4 -6 -16||
-EDOs: 5, 7, 12, [[53edo|53]], [[65edo|65]]
+EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[53edo|53]], [[65edo|65]]
 [[Badness]]: 0.0207
@@ Line 154: / Line 154: @@
 Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]
 [[Wedgie]]: &lt;&lt;1 4 3 4 2 -4||
-EDOs: 5, 12
+EDOs: [[5edo|5]], [[12edo|12]]
 [[Badness]]: 0.0248
@@ Line 160: / Line 160: @@
 [[Comma]]s: 50/49, 81/80
-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. [[38edo]], which is two parallel 19edos, is an excellent tuning for injera.
+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. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera.
 [[POTE tuning|POTE generator]]: 694.375
@@ Line 190: / Line 190: @@
 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. [[31edo]] 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.
-and 9-limit minimax 1/4 comma
+[[7-limit|7]] and [[9-limit]] minimax 1/4 comma
 [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]
 [[Eigenmonzo]]s: 2, 5
@@ Line 216: / Line 216: @@
 Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]
 [[Generator]]s: 2, 11/9
-EDOs: 7, 24, 31
+EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]]
 [[Badness]]: 0.0261
 =Mothra=
-Commas: 81/80, 1029/1024
+[[Comma]]s: 81/80, 1029/1024
 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 [[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.
-and 9 limit minimax 1/4 comma
+[[7-limit|7]] and [[9-limit]] minimax 1/4 comma
 [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]
-Eigenmonzos: 2, 5
+[[Eigenmonzo]]s: 2, 5
 [[POTE tuning|POTE generator]]: 232.193
@@ Line 235: / Line 235: @@
 [[Generator]]s: 2, 8/7
 [[Wedgie]]: &lt;&lt;3 12 -1 12 -10 -36||
-EDOs: 5, [[26edo|26]], 31
+EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]]
 [[Badness]]: 0.0371
@@ Line 244: / Line 244: @@
 Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]
-EDOs: 5, [[26edo|26]], 31, [[88edo|88]], [[150edo|150]], [[181edo|181]]
+EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]]
 [[Badness]]: 0.0256
@@ Line 250: / Line 250: @@
 [[Comma]]s: 81/80, 2401/2400
-Squares, with wedgie &lt;&lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (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.
+Squares, with wedgie &lt;&lt;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.
 and 9 limit minimax 1/4 comma
@@ Line 275: / Line 275: @@
 Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]
-EDOs: [[14edo|14]], 31, [[200edo|200]]
+EDOs: [[14edo|14]], [[31edo|31]], [[200edo|200]]
 [[Badness]]: 0.0568
@@ Line 301: / Line 301: @@
 POTE generator: ~9/7 = 425.942
-Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]
+[[Map]]: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]
-Wedgie: &lt;&lt;4 16 9 16 3 -24||
+[[Wedgie]]: &lt;&lt;4 16 9 16 3 -24||
-EDOs: 5, 8, 11, 14, 17, 31
+EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]]
-Badness: 0.0460
+[[Badness]]: 0.0460
 ==11-limit==
@@ Line 312: / Line 312: @@
 Map: [&lt;1 3 8 6 7|, &lt;0 -4 -16 -9 -10|]
-EDOs: 5, 8, 11, 14, 17, 31
+EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]]
-Badness: 0.0216
+[[Badness]]: 0.0216
 ==13-limit==
@@ Line 321: / Line 321: @@
 Map: [&lt;1 3 8 6 7 3|, &lt;0 -4 -16 -9 -10 2|]
-EDOs: 5, 8, 11, 14, 17, 31
+EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]]
-Badness: 0.0255</pre></div>
+[[Badness]]: 0.0255</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;Meantone family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:48:&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:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt; | &lt;a href="#Septimal meantone"&gt;Septimal meantone&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt; | &lt;a href="#Flattone"&gt;Flattone&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt; | &lt;a href="#Dominant"&gt;Dominant&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt; | &lt;a href="#Sharptone"&gt;Sharptone&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt; | &lt;a href="#Injera"&gt;Injera&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt; | &lt;a href="#Godzilla"&gt;Godzilla&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt; | &lt;a href="#Mohajira"&gt;Mohajira&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt; | &lt;a href="#Mothra"&gt;Mothra&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt;&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt; | &lt;a href="#Squares"&gt;Squares&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt;&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt; | &lt;a href="#Liese"&gt;Liese&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt; | &lt;a href="#Squares"&gt;Squares&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;!-- ws:start:WikiTextTocRule:71: --&gt;&lt;!-- ws:end:WikiTextTocRule:71 --&gt;&lt;!-- ws:start:WikiTextTocRule:72: --&gt;&lt;!-- ws:end:WikiTextTocRule:72 --&gt;&lt;!-- ws:start:WikiTextTocRule:73: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:73 --&gt;&lt;br /&gt;
-The 5-limit parent &lt;a class="wiki_link" href="/Comma"&gt;comma&lt;/a&gt; of the &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; family is the Didymus or &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow"&gt;syntonic comma&lt;/a&gt;, 81/80. This is the one they all temper out. The &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzo&lt;/a&gt; for 81/80 goes |-4 4 -1&amp;gt;, and that can be flipped around to the corresponding &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;, &amp;lt;&amp;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.&lt;br /&gt;
+The &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; parent &lt;a class="wiki_link" href="/Comma"&gt;comma&lt;/a&gt; of the &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; family is the Didymus or &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow"&gt;syntonic comma&lt;/a&gt;, 81/80. This is the one they all temper out. The &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzo&lt;/a&gt; for 81/80 goes |-4 4 -1&amp;gt;, and that can be flipped around to the corresponding &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;, &amp;lt;&amp;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.&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 696.239&lt;br /&gt;
 &lt;br /&gt;
-Map: [&amp;lt;1 0 -4|, &amp;lt;0 1 4|]&lt;br /&gt;
+&lt;a class="wiki_link" href="/Map"&gt;Map&lt;/a&gt;: [&amp;lt;1 0 -4|, &amp;lt;0 1 4|]&lt;br /&gt;
 EDOs: &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50&lt;/a&gt;, &lt;a class="wiki_link" href="/81edo"&gt;81&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.00736&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
-The 7-limit children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1&amp;gt;, |-13 10 0 -1&amp;gt;], flattone, with normal list [|-4 4 -1&amp;gt;, |-17 9 0 1&amp;gt;], dominant, with normal list [|-4 4 -1&amp;gt;, |6 -2 0 -1&amp;gt;], sharptone, with normal list [|-4 4 -1&amp;gt;, |2 -3 0 1&amp;gt;], injera, with normal list [|-4 4 -1&amp;gt;, |-7 8 0 -2&amp;gt;], mohajira, with normal list [|-4 4 -1&amp;gt;, |-23 11 0 2&amp;gt;], godzilla, with normal list [|-4 4 -1&amp;gt;, |-4 -1 0 2&amp;gt;], mothra, with normal list [|-4 4 -1&amp;gt;, |-10 1 0 3&amp;gt;], squares, with normal list [|-4 4 -1&amp;gt;, |-3 9 0 -4&amp;gt;], and liese, with normal list [|-4 4 -1&amp;gt;, |-9 11 0 -3&amp;gt;].&lt;br /&gt;
+The &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1&amp;gt;, |-13 10 0 -1&amp;gt;], flattone, with normal list [|-4 4 -1&amp;gt;, |-17 9 0 1&amp;gt;], dominant, with normal list [|-4 4 -1&amp;gt;, |6 -2 0 -1&amp;gt;], sharptone, with normal list [|-4 4 -1&amp;gt;, |2 -3 0 1&amp;gt;], injera, with normal list [|-4 4 -1&amp;gt;, |-7 8 0 -2&amp;gt;], mohajira, with normal list [|-4 4 -1&amp;gt;, |-23 11 0 2&amp;gt;], godzilla, with normal list [|-4 4 -1&amp;gt;, |-4 -1 0 2&amp;gt;], mothra, with normal list [|-4 4 -1&amp;gt;, |-10 1 0 3&amp;gt;], squares, with normal list [|-4 4 -1&amp;gt;, |-3 9 0 -4&amp;gt;], and liese, with normal list [|-4 4 -1&amp;gt;, |-9 11 0 -3&amp;gt;].&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="Septimal meantone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Septimal meantone&lt;/h1&gt;
-The comma |-13 10 0 -1&amp;gt; for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the 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, C-F#, the tritone. The &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt; for septimal meantone is &amp;lt;&amp;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 &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; is a good tuning for it.&lt;br /&gt;
+The comma |-13 10 0 -1&amp;gt; for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt;, C-F#, the tritone. The &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt; for septimal meantone is &amp;lt;&amp;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 &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; is a good tuning for it.&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 81/80, 126/125&lt;br /&gt;
@@ Line 350: / Line 350: @@
 Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.&lt;br /&gt;
 &lt;br /&gt;
-Map: [&amp;lt;1 0 -4 -13|, &amp;lt;0 1 4 10|]&lt;br /&gt;
+&lt;a class="wiki_link" href="/Map"&gt;Map&lt;/a&gt;: [&amp;lt;1 0 -4 -13|, &amp;lt;0 1 4 10|]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Generator"&gt;Generator&lt;/a&gt;s: 2, 3&lt;br /&gt;
-Wedgie: &amp;lt;&amp;lt;1 4 10 4 13 12||&lt;br /&gt;
+&lt;a class="wiki_link" href="/Wedgie"&gt;Wedgie&lt;/a&gt;: &amp;lt;&amp;lt;1 4 10 4 13 12||&lt;br /&gt;
 EDOs: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/81edo"&gt;81&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0137&lt;br /&gt;
@@ Line 366: / Line 366: @@
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 696.967&lt;br /&gt;
 &lt;br /&gt;
-Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.&lt;br /&gt;
+&lt;a class="wiki_link" href="/Algebraic%20generator"&gt;Algebraic generator&lt;/a&gt;: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.&lt;br /&gt;
 &lt;br /&gt;
-Map: [&amp;lt;1 0 -4 -13 -25|, &amp;lt;0 1 4 10 18|]&lt;br /&gt;
+&lt;a class="wiki_link" href="/Map"&gt;Map&lt;/a&gt;: [&amp;lt;1 0 -4 -13 -25|, &amp;lt;0 1 4 10 18|]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Generator"&gt;Generator&lt;/a&gt;s: 2, 3&lt;br /&gt;
 EDOs: &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/105edo"&gt;105&lt;/a&gt;, &lt;a class="wiki_link" href="/198edo"&gt;198&lt;/a&gt;&lt;br /&gt;
@@ Line 376: / Line 376: @@
 &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 66/65, 81/80, 99/98, 105/104&lt;br /&gt;
 &lt;br /&gt;
-POTE generator: ~3/2 = 696.642&lt;br /&gt;
+&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 696.642&lt;br /&gt;
 &lt;br /&gt;
-Map: Map: [&amp;lt;1 0 -4 -13 -25 -20|, &amp;lt;0 1 4 10 18 15|]&lt;br /&gt;
+Map: [&amp;lt;1 0 -4 -13 -25 -20|, &amp;lt;0 1 4 10 18 15|]&lt;br /&gt;
-EDOs: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19], [[31edo|31&lt;/a&gt;, &lt;a class="wiki_link" href="/267edo"&gt;267&lt;/a&gt;, &lt;a class="wiki_link" href="/298edo"&gt;298&lt;/a&gt;&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/267edo"&gt;267&lt;/a&gt;, &lt;a class="wiki_link" href="/298edo"&gt;298&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0180&lt;br /&gt;
 &lt;br /&gt;
@@ Line 385: / Line 385: @@
 &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 81/80, 126/125, 385/384&lt;br /&gt;
 &lt;br /&gt;
--limit minimax 1/4 comma&lt;br /&gt;
+&lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; &lt;a class="wiki_link" href="/minimax"&gt;minimax&lt;/a&gt; 1/4 comma&lt;br /&gt;
 [|1 0 0 0 0&amp;gt;, |1 0 1/4 0 0&amp;gt;, |0 0 1 0 0&amp;gt;, &lt;br /&gt;
 |-3 0 5/2 0 0&amp;gt;, |11 0 -13/4 0 0&amp;gt;]&lt;br /&gt;
@@ Line 392: / Line 392: @@
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 696.434&lt;br /&gt;
 &lt;br /&gt;
-Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.&lt;br /&gt;
+&lt;a class="wiki_link" href="/Algebraic%20generator"&gt;Algebraic generator&lt;/a&gt;: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;1 0 -4 -13 24|, &amp;lt;0 1 4 10 -13|]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Generator"&gt;Generator&lt;/a&gt;s: 2, 3&lt;br /&gt;
-EDOs: 12, 19, 31, 81, &lt;a class="wiki_link" href="/112edo"&gt;112&lt;/a&gt;&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/81edo"&gt;81&lt;/a&gt;, &lt;a class="wiki_link" href="/112edo"&gt;112&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0215&lt;br /&gt;
 &lt;br /&gt;
@@ Line 405: / Line 405: @@
 &lt;br /&gt;
 Map: [&amp;lt;1 0 -4 -13 24|, &amp;lt;0 1 4 10 -13|]&lt;br /&gt;
-EDOS: 7, 12, 19, 31, 50, 81, &lt;a class="wiki_link" href="/131edo"&gt;131&lt;/a&gt;&lt;br /&gt;
+EDOS: &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50&lt;/a&gt;, &lt;a class="wiki_link" href="/81edo"&gt;81&lt;/a&gt;, &lt;a class="wiki_link" href="/131edo"&gt;131&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0209&lt;br /&gt;
 &lt;br /&gt;
@@ Line 414: / Line 414: @@
 &lt;br /&gt;
 Map: [&amp;lt;1 0 -4 -13 -6|, &amp;lt;0 1 4 10 6|]&lt;br /&gt;
-EDOs: 7, 12, 19, 31, 50, 81&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50&lt;/a&gt;, &lt;a class="wiki_link" href="/81edo"&gt;81&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0214&lt;br /&gt;
 &lt;br /&gt;
@@ Line 423: / Line 423: @@
 &lt;br /&gt;
 Map: [&amp;lt;1 0 -4 -13 -6 -20|, &amp;lt;0 1 4 10 6 15|]&lt;br /&gt;
-EDOs: 7, 12, 19, 31, 50, &lt;a class="wiki_link" href="/131edo"&gt;131&lt;/a&gt;, &lt;a class="wiki_link" href="/181edo"&gt;181&lt;/a&gt;&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50&lt;/a&gt;, &lt;a class="wiki_link" href="/131edo"&gt;131&lt;/a&gt;, &lt;a class="wiki_link" href="/181edo"&gt;181&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0212&lt;br /&gt;
 &lt;br /&gt;
@@ Line 429: / Line 429: @@
 &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 81/80, 525/512&lt;br /&gt;
 &lt;br /&gt;
-The &lt;a class="wiki_link" href="/wedgie"&gt;wedgie&lt;/a&gt; for flattone is &amp;lt;&amp;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 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are &lt;a class="wiki_link" href="/26edo"&gt;26edo&lt;/a&gt;, &lt;a class="wiki_link" href="/45edo"&gt;45edo&lt;/a&gt; and &lt;a class="wiki_link" href="/64edo"&gt;64edo&lt;/a&gt;.&lt;br /&gt;
+The &lt;a class="wiki_link" href="/wedgie"&gt;wedgie&lt;/a&gt; for flattone is &amp;lt;&amp;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 &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; is a diminished minor seventh interval. Other intervals are &lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;, a diminished minor third, and &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt;, a doubly diminshed fifth. Good tunings for flattone are &lt;a class="wiki_link" href="/26edo"&gt;26edo&lt;/a&gt;, &lt;a class="wiki_link" href="/45edo"&gt;45edo&lt;/a&gt; and &lt;a class="wiki_link" href="/64edo"&gt;64edo&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; minimax&lt;br /&gt;
@@ Line 448: / Line 448: @@
 &lt;a class="wiki_link" href="/Wedgie"&gt;Wedgie&lt;/a&gt;: &amp;lt;&amp;lt;1 4 -9 4 -17 -32||&lt;br /&gt;
 &lt;a class="wiki_link" href="/Generator"&gt;Generator&lt;/a&gt;s: 2, 3&lt;br /&gt;
-EDOs: 7, 19, &lt;a class="wiki_link" href="/45edo"&gt;45&lt;/a&gt;, &lt;a class="wiki_link" href="/64edo"&gt;64&lt;/a&gt;&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/45edo"&gt;45&lt;/a&gt;, &lt;a class="wiki_link" href="/64edo"&gt;64&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0386&lt;br /&gt;
 &lt;br /&gt;
@@ Line 454: / Line 454: @@
 &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 36/35, 64/63&lt;br /&gt;
 &lt;br /&gt;
-The wedgie for dominant is &amp;lt;&amp;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 &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt;, or &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;.&lt;br /&gt;
+The wedgie for dominant is &amp;lt;&amp;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 &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, but it also works well with the Pythagorean tuning of pure &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt; fifths, and with &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt;, or &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 701.573&lt;br /&gt;
@@ Line 460: / Line 460: @@
 Map: [&amp;lt;1 0 -4 6|, &amp;lt;0 1 4 -2|]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Wedgie"&gt;Wedgie&lt;/a&gt;: &amp;lt;&amp;lt;1 4 -2 4 -6 -16||&lt;br /&gt;
-EDOs: 5, 7, 12, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/65edo"&gt;65&lt;/a&gt;&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/65edo"&gt;65&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0207&lt;br /&gt;
 &lt;br /&gt;
@@ Line 472: / Line 472: @@
 Map: [&amp;lt;1 0 -4 -2|, &amp;lt;0 1 4 3|]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Wedgie"&gt;Wedgie&lt;/a&gt;: &amp;lt;&amp;lt;1 4 3 4 2 -4||&lt;br /&gt;
-EDOs: 5, 12&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0248&lt;br /&gt;
 &lt;br /&gt;
@@ Line 478: / Line 478: @@
 &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 50/49, 81/80&lt;br /&gt;
 &lt;br /&gt;
-The wedgie for injera is &amp;lt;&amp;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. &lt;a class="wiki_link" href="/38edo"&gt;38edo&lt;/a&gt;, which is two parallel 19edos, is an excellent tuning for injera.&lt;br /&gt;
+The wedgie for injera is &amp;lt;&amp;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. &lt;a class="wiki_link" href="/38edo"&gt;38edo&lt;/a&gt;, which is two parallel &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;s, is an excellent tuning for injera.&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 694.375&lt;br /&gt;
@@ Line 508: / Line 508: @@
 Mohajira, with wedgie &amp;lt;&amp;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. &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; 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.&lt;br /&gt;
 &lt;br /&gt;
-and 9-limit minimax 1/4 comma&lt;br /&gt;
+&lt;a class="wiki_link" href="/7-limit"&gt;7&lt;/a&gt; and &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt; minimax 1/4 comma&lt;br /&gt;
 [|1 0 0 0&amp;gt;, |1 0 1/4 0&amp;gt;, |0 0 1 0&amp;gt;, |6 0 -11/8 0&amp;gt;]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzo&lt;/a&gt;s: 2, 5&lt;br /&gt;
@@ Line 534: / Line 534: @@
 Map: [&amp;lt;1 1 0 6 2|, &amp;lt;0 2 8 -11 5|]&lt;br /&gt;
 &lt;a class="wiki_link" href="/Generator"&gt;Generator&lt;/a&gt;s: 2, 11/9&lt;br /&gt;
-EDOs: 7, 24, 31&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/24edo"&gt;24&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0261&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc16"&gt;&lt;a name="Mothra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Mothra&lt;/h1&gt;
-Commas: 81/80, 1029/1024&lt;br /&gt;
+&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 81/80, 1029/1024&lt;br /&gt;
 &lt;br /&gt;
 Mothra, with wedgie &amp;lt;&amp;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 &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; 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.&lt;br /&gt;
 &lt;br /&gt;
-and 9 limit minimax 1/4 comma &lt;br /&gt;
+&lt;a class="wiki_link" href="/7-limit"&gt;7&lt;/a&gt; and &lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt; minimax 1/4 comma &lt;br /&gt;
 [|1 0 0 0&amp;gt;, |1 0 1/4 0&amp;gt;, |0 0 1 0&amp;gt;, |3 0 -1/12 0&amp;gt;]&lt;br /&gt;
-Eigenmonzos: 2, 5&lt;br /&gt;
+&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzo&lt;/a&gt;s: 2, 5&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 232.193&lt;br /&gt;
@@ Line 553: / Line 553: @@
 &lt;a class="wiki_link" href="/Generator"&gt;Generator&lt;/a&gt;s: 2, 8/7&lt;br /&gt;
 &lt;a class="wiki_link" href="/Wedgie"&gt;Wedgie&lt;/a&gt;: &amp;lt;&amp;lt;3 12 -1 12 -10 -36||&lt;br /&gt;
-EDOs: 5, &lt;a class="wiki_link" href="/26edo"&gt;26&lt;/a&gt;, 31&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/26edo"&gt;26&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0371&lt;br /&gt;
 &lt;br /&gt;
@@ Line 562: / Line 562: @@
 &lt;br /&gt;
 Map: [&amp;lt;1 1 0 3 5|, &amp;lt;0 3 12 -1 -8|]&lt;br /&gt;
-EDOs: 5, &lt;a class="wiki_link" href="/26edo"&gt;26&lt;/a&gt;, 31, &lt;a class="wiki_link" href="/88edo"&gt;88&lt;/a&gt;, &lt;a class="wiki_link" href="/150edo"&gt;150&lt;/a&gt;, &lt;a class="wiki_link" href="/181edo"&gt;181&lt;/a&gt;&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/26edo"&gt;26&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/88edo"&gt;88&lt;/a&gt;, &lt;a class="wiki_link" href="/150edo"&gt;150&lt;/a&gt;, &lt;a class="wiki_link" href="/181edo"&gt;181&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0256&lt;br /&gt;
 &lt;br /&gt;
@@ Line 568: / Line 568: @@
 &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 81/80, 2401/2400&lt;br /&gt;
 &lt;br /&gt;
-Squares, with wedgie &amp;lt;&amp;lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, 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.&lt;br /&gt;
+Squares, with wedgie &amp;lt;&amp;lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (&lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt;) intervals, and uses it for a generator. &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, 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.&lt;br /&gt;
 &lt;br /&gt;
 and 9 limit minimax 1/4 comma&lt;br /&gt;
@@ Line 593: / Line 593: @@
 &lt;br /&gt;
 Map: [&amp;lt;1 3 8 6 -4|, &amp;lt;0 -4 -16 -9 21|]&lt;br /&gt;
-EDOs: &lt;a class="wiki_link" href="/14edo"&gt;14&lt;/a&gt;, 31, &lt;a class="wiki_link" href="/200edo"&gt;200&lt;/a&gt;&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/14edo"&gt;14&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/200edo"&gt;200&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0568&lt;br /&gt;
 &lt;br /&gt;
@@ Line 619: / Line 619: @@
 POTE generator: ~9/7 = 425.942&lt;br /&gt;
 &lt;br /&gt;
-Map: [&amp;lt;1 3 8 6|, &amp;lt;0 -4 -16 -9|]&lt;br /&gt;
+&lt;a class="wiki_link" href="/Map"&gt;Map&lt;/a&gt;: [&amp;lt;1 3 8 6|, &amp;lt;0 -4 -16 -9|]&lt;br /&gt;
-Wedgie: &amp;lt;&amp;lt;4 16 9 16 3 -24||&lt;br /&gt;
+&lt;a class="wiki_link" href="/Wedgie"&gt;Wedgie&lt;/a&gt;: &amp;lt;&amp;lt;4 16 9 16 3 -24||&lt;br /&gt;
-EDOs: 5, 8, 11, 14, 17, 31&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/8edo"&gt;8&lt;/a&gt;, &lt;a class="wiki_link" href="/11edo"&gt;11&lt;/a&gt;, &lt;a class="wiki_link" href="/14edo"&gt;14&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo"&gt;17&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;&lt;br /&gt;
-Badness: 0.0460&lt;br /&gt;
+&lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0460&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Squares-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;11-limit&lt;/h2&gt;
@@ Line 630: / Line 630: @@
 &lt;br /&gt;
 Map: [&amp;lt;1 3 8 6 7|, &amp;lt;0 -4 -16 -9 -10|]&lt;br /&gt;
-EDOs: 5, 8, 11, 14, 17, 31&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/8edo"&gt;8&lt;/a&gt;, &lt;a class="wiki_link" href="/11edo"&gt;11&lt;/a&gt;, &lt;a class="wiki_link" href="/14edo"&gt;14&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo"&gt;17&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;&lt;br /&gt;
-Badness: 0.0216&lt;br /&gt;
+&lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0216&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Squares-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;13-limit&lt;/h2&gt;
@@ Line 639: / Line 639: @@
 &lt;br /&gt;
 Map: [&amp;lt;1 3 8 6 7 3|, &amp;lt;0 -4 -16 -9 -10 2|]&lt;br /&gt;
-EDOs: 5, 8, 11, 14, 17, 31&lt;br /&gt;
+EDOs: &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/8edo"&gt;8&lt;/a&gt;, &lt;a class="wiki_link" href="/11edo"&gt;11&lt;/a&gt;, &lt;a class="wiki_link" href="/14edo"&gt;14&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo"&gt;17&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;&lt;br /&gt;
-Badness: 0.0255&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0255&lt;/body&gt;&lt;/html&gt;</pre></div>