Meantone family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 208854920 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 208972928 - Original comment: some links added** |
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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:xenwolf|xenwolf]] and made on <tt>2011-03-09 16:40:29 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>208972928</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>some links added</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> | ||
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Map: [<1 0 -4|, <0 1 4|] | Map: [<1 0 -4|, <0 1 4|] | ||
EDOs: 5, 7, 12, 19, 31, 50, 81 | EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]] | ||
Badness: 0.00736 | [[Badness]]: 0.00736 | ||
==Seven limit children== | ==Seven limit children== | ||
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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 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. | ||
[[Comma]]s: 81/80, 126/125 | |||
7 and 9 limit minimax | 7 and [[9-limit]] minimax | ||
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>] | ||
[[Eigenmonzo]]s: 2, 5 | |||
[[POTE tuning|POTE generator]]: 696.495 | [[POTE tuning|POTE generator]]: 696.495 | ||
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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: 12, 19, 31, 81 | EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]] | ||
Badness: 0.0137 | [[Badness]]: 0.0137 | ||
==Unidecimal meantone aka Huygens== | ==Unidecimal meantone aka Huygens== | ||
[[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 | |||
[[POTE tuning|POTE generator]]: 696.967 | [[POTE tuning|POTE generator]]: 696.967 | ||
| Line 49: | Line 49: | ||
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: 7, 12, 31, [[105edo|105]], [[198edo|198]] | EDOs: [[7edo|7]], [[12edo|12]], [[31edo|31]], [[105edo|105]], [[198edo|198]] | ||
Badness: 0.0170 | [[Badness]]: 0.0170 | ||
===Tridecimal meantone=== | ===Tridecimal meantone=== | ||
[[Comma]]s: 66/65, 81/80, 99/98, 105/104 | |||
POTE generator: ~3/2 = 696.642 | POTE generator: ~3/2 = 696.642 | ||
Map: Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] | Map: Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] | ||
EDOs: 12, 19, 31, 267, 298 | EDOs: [[12edo|12]], [[19edo|19], [[31edo|31]], [[267edo|267]], [[298edo|298]] | ||
Badness: 0.0180 | [[Badness]]: 0.0180 | ||
==Meanpop== | ==Meanpop== | ||
[[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 | |||
[[POTE tuning|POTE generator]]: 696.434 | [[POTE tuning|POTE generator]]: 696.434 | ||
| Line 75: | Line 75: | ||
Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] | Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] | ||
[[Generator]]s: 2, 3 | |||
EDOs: 12, 19, 31, 81, 112 | EDOs: 12, 19, 31, 81, [[112edo|112]] | ||
Badness: 0.0215 | [[Badness]]: 0.0215 | ||
===13-limit Meanpop=== | ===13-limit Meanpop=== | ||
[[Comma]]s: 81/80, 105/104, 144/143, 196/195 | |||
POTE generator: ~3/2 = 696.211 | POTE generator: ~3/2 = 696.211 | ||
Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] | Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] | ||
EDOS: 7, 12, 19, 31, 50, 81, 131 | EDOS: 7, 12, 19, 31, 50, 81, [[131edo|131]] | ||
Badness: 0.0209 | [[Badness]]: 0.0209 | ||
==Meanenneadecal== | ==Meanenneadecal== | ||
[[Comma]]s: 45/44, 56/55, 81/80 | |||
[[POTE tuning|POTE generator]]: ~3/2 = 696.250 | [[POTE tuning|POTE generator]]: ~3/2 = 696.250 | ||
| Line 95: | Line 95: | ||
Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|] | Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|] | ||
EDOs: 7, 12, 19, 31, 50, 81 | EDOs: 7, 12, 19, 31, 50, 81 | ||
Badness: 0.0214 | [[Badness]]: 0.0214 | ||
===13-limit=== | ===13-limit=== | ||
[[Comma]]s: 45/44, 56/55, 78/77, 81/80 | |||
[[POTE tuning|POTE generator]]: ~3/2 = 696.146 | [[POTE tuning|POTE generator]]: ~3/2 = 696.146 | ||
Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|] | Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|] | ||
EDOs: 7, 12, 19, 31, 50, 131, 181 | EDOs: 7, 12, 19, 31, 50, [[131edo|131]], [[181edo|181]] | ||
Badness: 0.0212 | [[Badness]]: 0.0212 | ||
==Flattone== | ==Flattone== | ||
[[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 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]]. | ||
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 | |||
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 | |||
[[POTE tuning|POTE generator]]: 693.779 | [[POTE tuning|POTE generator]]: 693.779 | ||
| Line 126: | Line 126: | ||
Map: [<1 0 -4 17|, <0 1 4 -9|] | Map: [<1 0 -4 17|, <0 1 4 -9|] | ||
Wedgie: <<1 4 -9 4 -17 -32|| | [[Wedgie]]: <<1 4 -9 4 -17 -32|| | ||
[[Generator]]s: 2, 3 | |||
EDOs: 7, 19, [[45edo|45]], [[64edo|64]] | EDOs: 7, 19, [[45edo|45]], [[64edo|64]] | ||
Badness: 0.0386 | [[Badness]]: 0.0386 | ||
==Dominant== | ==Dominant== | ||
[[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 fifths, and with [[29edo]], [[41edo]], or [[53edo]]. | ||
| Line 139: | Line 139: | ||
Map: [<1 0 -4 6|, <0 1 4 -2|] | Map: [<1 0 -4 6|, <0 1 4 -2|] | ||
Wedgie: <<1 4 -2 4 -6 -16|| | [[Wedgie]]: <<1 4 -2 4 -6 -16|| | ||
EDOs: 5, 7, 12, [[53edo|53]], [[65edo|65]] | EDOs: 5, 7, 12, [[53edo|53]], [[65edo|65]] | ||
Badness: 0.0207 | [[Badness]]: 0.0207 | ||
==Sharptone== | ==Sharptone== | ||
[[Comma]]s: 21/20, 28/27 | |||
Sharptone, with a wedgie <<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. [[12edo]] tuning does sharptone about as well as such a thing can be done. | Sharptone, with a wedgie <<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. [[12edo]] tuning does sharptone about as well as such a thing can be done. | ||
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Map: [<1 0 -4 -2|, <0 1 4 3|] | Map: [<1 0 -4 -2|, <0 1 4 3|] | ||
Wedgie: <<1 4 3 4 2 -4|| | [[Wedgie]]: <<1 4 3 4 2 -4|| | ||
EDOs: 5, 12 | EDOs: 5, 12 | ||
Badness: 0.0248 | [[Badness]]: 0.0248 | ||
==Injera== | ==Injera== | ||
[[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 19edos, is an excellent tuning for injera. | ||
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Map: [<2 0 -8 -7|, <0 1 4 4|] | Map: [<2 0 -8 -7|, <0 1 4 4|] | ||
Wedgie: <<2 8 8 8 7 -4|| | [[Wedgie]]: <<2 8 8 8 7 -4|| | ||
EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[140edo|140]], [[178edo|178]] | EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[140edo|140]], [[178edo|178]] | ||
Badness: 0.0311 | [[Badness]]: 0.0311 | ||
==Godzilla== | ==Godzilla== | ||
[[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-half intervals these represent give a fourth, and so step-and-a-half 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. | 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-half intervals these represent give a fourth, and so step-and-a-half 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. | ||
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Map: [<1 0 -4 2|, <0 2 8 1|] | Map: [<1 0 -4 2|, <0 2 8 1|] | ||
Wedgie: <<2 8 1 8 -4 -20|| | [[Wedgie]]: <<2 8 1 8 -4 -20|| | ||
EDOs: 5, 9, 14, 19 | EDOs: [[5edo|5]], [[9edo|9]], [[14edo|14]], 19 | ||
Badness: 0.0267 | [[Badness]]: 0.0267 | ||
Music: Igliashon Jones, [[http://tinyurl.com/4uyumk9|"Change is on the Wind"]], in Godzilla[9] | Music: Igliashon Jones, [[http://tinyurl.com/4uyumk9|"Change is on the Wind"]], in Godzilla[9] | ||
==Mohajira== | ==Mohajira== | ||
[[Comma]]s: 81/80, 6144/6125 | |||
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. | 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 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>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>] | ||
[[Eigenmonzo]]s: 2, 5 | |||
[[POTE tuning|POTE generator]]: 348.415 | [[POTE tuning|POTE generator]]: 348.415 | ||
| Line 194: | Line 194: | ||
Map: [<1 1 0 6|, <0 2 8 -11|] | Map: [<1 1 0 6|, <0 2 8 -11|] | ||
[[Generator]]s: 2, 128/105 | |||
Wedgie: <<2 8 -11 8 -23 -48|| | [[Wedgie]]: <<2 8 -11 8 -23 -48|| | ||
EDOs: 7, 24, 31 | EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]] | ||
Badness: 0.0557 | [[Badness]]: 0.0557 | ||
===11-limit=== | ===11-limit=== | ||
[[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 | |||
[[POTE tuning|POTE generator]]: 348.477 | [[POTE tuning|POTE generator]]: 348.477 | ||
Map: [<1 1 0 6 2|, <0 2 8 -11 5|] | Map: [<1 1 0 6 2|, <0 2 8 -11 5|] | ||
[[Generator]]s: 2, 11/9 | |||
EDOs: 7, 24, 31 | EDOs: 7, 24, 31 | ||
Badness: 0.0261 | [[Badness]]: 0.0261 | ||
==Mothra== | ==Mothra== | ||
| Line 228: | Line 228: | ||
Map: [<1 1 0 3|, <0 3 12 -1|] | Map: [<1 1 0 3|, <0 3 12 -1|] | ||
[[Generator]]s: 2, 8/7 | |||
Wedgie: <<3 12 -1 12 -10 -36|| | [[Wedgie]]: <<3 12 -1 12 -10 -36|| | ||
EDOs: 5, 26, 31 | EDOs: 5, [[26edo|26]], 31 | ||
Badness: 0.0371 | [[Badness]]: 0.0371 | ||
===11-limit=== | ===11-limit=== | ||
[[Comma]]s: 81/80, 99/98, 385/384 | |||
POTE generator: ~63/55 = 232.031 | POTE generator: ~63/55 = 232.031 | ||
Map: [<1 1 0 3 5|, <0 3 12 -1 -8|] | Map: [<1 1 0 3 5|, <0 3 12 -1 -8|] | ||
EDOs: 5, 26, 31, 88, 150, 181 | EDOs: 5, [[26edo|26]], 31, [[88edo|88]], [[150edo|150]], [[181edo|181]] | ||
Badness: 0.0256 | [[Badness[[: 0.0256 | ||
==Squares== | ==Squares== | ||
[[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) 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. | ||
| Line 249: | Line 249: | ||
7 and 9 limit minimax 1/4 comma | 7 and 9 limit minimax 1/4 comma | ||
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>] | ||
[[Eigenmonzo]]s: 2, 5 | |||
[[POTE tuning|POTE generator]]: 425.942 | [[POTE tuning|POTE generator]]: 425.942 | ||
| Line 256: | Line 256: | ||
Map: [<1 3 8 6|, <0 -4 -16 -9|] | Map: [<1 3 8 6|, <0 -4 -16 -9|] | ||
[[Generator]]s: 2, 9/7 | |||
EDOs: 14, 31, 262, 293 | EDOs: [[14edo|14]], 31, [[262edo|262]], [[293edo|293]] | ||
Badness: 0.0460 | [[Badness]]: 0.0460 | ||
Music: | Music: | ||
By Chris Vaisvil | By [[Chris Vaisvil]] | ||
http://tinyurl.com/25kv7cq | http://tinyurl.com/25kv7cq | ||
http://tinyurl.com/24cbxse | http://tinyurl.com/24cbxse | ||
===11-limit=== | ===11-limit=== | ||
[[Comma]]s: 81/80, 385/384, 1375/1372 | |||
[[POTE tuning|POTE generator]]: 425.993 | [[POTE tuning|POTE generator]]: 425.993 | ||
Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|] | Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|] | ||
EDOs: 14, 31, 200 | EDOs: [[14edo|14]], 31, [[200edo|200]] | ||
Badness: 0.0568 | [[Badness]]: 0.0568 | ||
==Liese== | ==Liese== | ||
[[Comma]]s: 81/80, 686/675 | |||
Liese, with wedgie <<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. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55. | Liese, with wedgie <<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. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55. | ||
| Line 281: | Line 281: | ||
7 and 9 limit minimax 1/4 comma | 7 and 9 limit minimax 1/4 comma | ||
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |2/3 0 11/12 0>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |2/3 0 11/12 0>] | ||
[[Eigenmonzo]]s: 2, 5 | |||
[[POTE tuning|POTE generator]]: 632.406 | [[POTE tuning|POTE generator]]: 632.406 | ||
| Line 288: | Line 288: | ||
Map: [<1 0 -4 -3|, <0 3 12 11|] | Map: [<1 0 -4 -3|, <0 3 12 11|] | ||
[[Generator]]s: 2, 10/7 | |||
EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]] | EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]] | ||
Badness: 0.0467</pre></div> | [[Badness]]: 0.0467</pre></div> | ||
<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>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.<br /> | <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>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.<br /> | ||
| Line 297: | Line 297: | ||
<br /> | <br /> | ||
Map: [&lt;1 0 -4|, &lt;0 1 4|]<br /> | Map: [&lt;1 0 -4|, &lt;0 1 4|]<br /> | ||
EDOs: 5, 7, 12, 19, 31, 50, 81<br /> | 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><br /> | ||
Badness: 0.00736<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.00736<br /> | ||
<br /> | <br /> | ||
<!-- 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> | ||
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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.<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 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.<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125<br /> | |||
<br /> | <br /> | ||
7 and 9 limit minimax<br /> | 7 and <a class="wiki_link" href="/9-limit">9-limit</a> minimax<br /> | ||
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |-3 0 5/2 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |-3 0 5/2 0&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.495<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.495<br /> | ||
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<br /> | <br /> | ||
Map: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]<br /> | Map: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]<br /> | ||
<a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> | |||
Wedgie: &lt;&lt;1 4 10 4 13 12||<br /> | Wedgie: &lt;&lt;1 4 10 4 13 12||<br /> | ||
EDOs: 12, 19, 31, 81<br /> | 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><br /> | ||
Badness: 0.0137<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0137<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-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="x-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 /> | |||
<br /> | <br /> | ||
11-limit 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 /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.967<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.967<br /> | ||
| Line 335: | Line 335: | ||
<br /> | <br /> | ||
Map: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]<br /> | Map: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]<br /> | ||
<a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> | |||
EDOs: 7, 12, 31, <a class="wiki_link" href="/105edo">105</a>, <a class="wiki_link" href="/198edo">198</a><br /> | 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><br /> | ||
Badness: 0.0170<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0170<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-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="x-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 /> | |||
<br /> | <br /> | ||
POTE generator: ~3/2 = 696.642<br /> | POTE generator: ~3/2 = 696.642<br /> | ||
<br /> | <br /> | ||
Map: Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]<br /> | Map: Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]<br /> | ||
EDOs: 12, 19, 31, 267, 298<br /> | 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><br /> | ||
Badness: 0.0180<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0180<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-Meanpop"></a><!-- ws:end:WikiTextHeadingRule:8 -->Meanpop</h2> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-Meanpop"></a><!-- ws:end:WikiTextHeadingRule:8 -->Meanpop</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 385/384<br /> | |||
<br /> | <br /> | ||
11-limit minimax 1/4 comma<br /> | 11-limit 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 /> | ||
|-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 /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.434<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.434<br /> | ||
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<br /> | <br /> | ||
Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]<br /> | Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]<br /> | ||
<a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> | |||
EDOs: 12, 19, 31, 81, 112<br /> | EDOs: 12, 19, 31, 81, <a class="wiki_link" href="/112edo">112</a><br /> | ||
Badness: 0.0215<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0215<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Meanpop-13-limit Meanpop"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit Meanpop</h3> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Meanpop-13-limit Meanpop"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit Meanpop</h3> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 105/104, 144/143, 196/195<br /> | |||
<br /> | <br /> | ||
POTE generator: ~3/2 = 696.211<br /> | POTE generator: ~3/2 = 696.211<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]<br /> | Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]<br /> | ||
EDOS: 7, 12, 19, 31, 50, 81, 131<br /> | EDOS: 7, 12, 19, 31, 50, 81, <a class="wiki_link" href="/131edo">131</a><br /> | ||
Badness: 0.0209<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0209<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x-Meanenneadecal"></a><!-- ws:end:WikiTextHeadingRule:12 -->Meanenneadecal</h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x-Meanenneadecal"></a><!-- ws:end:WikiTextHeadingRule:12 -->Meanenneadecal</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 81/80<br /> | |||
<br /> | <br /> | ||
<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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Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]<br /> | Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]<br /> | ||
EDOs: 7, 12, 19, 31, 50, 81<br /> | EDOs: 7, 12, 19, 31, 50, 81<br /> | ||
Badness: 0.0214<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0214<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Meanenneadecal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h3> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Meanenneadecal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h3> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 78/77, 81/80<br /> | |||
<br /> | <br /> | ||
<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 /> | ||
<br /> | <br /> | ||
Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]<br /> | Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]<br /> | ||
EDOs: 7, 12, 19, 31, 50, 131, 181<br /> | EDOs: 7, 12, 19, 31, 50, <a class="wiki_link" href="/131edo">131</a>, <a class="wiki_link" href="/181edo">181</a><br /> | ||
Badness: 0.0212<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0212<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="x-Flattone"></a><!-- ws:end:WikiTextHeadingRule:16 -->Flattone</h2> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="x-Flattone"></a><!-- ws:end:WikiTextHeadingRule:16 -->Flattone</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 525/512<br /> | |||
<br /> | <br /> | ||
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 <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 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>.<br /> | ||
<br /> | <br /> | ||
7-limit 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 /> | |||
<br /> | <br /> | ||
9-limit 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 /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 693.779<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 693.779<br /> | ||
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<br /> | <br /> | ||
Map: [&lt;1 0 -4 17|, &lt;0 1 4 -9|]<br /> | Map: [&lt;1 0 -4 17|, &lt;0 1 4 -9|]<br /> | ||
Wedgie: &lt;&lt;1 4 -9 4 -17 -32||<br /> | <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 -9 4 -17 -32||<br /> | ||
<a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> | |||
EDOs: 7, 19, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/64edo">64</a><br /> | EDOs: 7, 19, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/64edo">64</a><br /> | ||
Badness: 0.0386<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0386<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="x-Dominant"></a><!-- ws:end:WikiTextHeadingRule:18 -->Dominant</h2> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="x-Dominant"></a><!-- ws:end:WikiTextHeadingRule:18 -->Dominant</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 36/35, 64/63<br /> | |||
<br /> | <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 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>.<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 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>.<br /> | ||
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<br /> | <br /> | ||
Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]<br /> | Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]<br /> | ||
Wedgie: &lt;&lt;1 4 -2 4 -6 -16||<br /> | <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 -2 4 -6 -16||<br /> | ||
EDOs: 5, 7, 12, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/65edo">65</a><br /> | EDOs: 5, 7, 12, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/65edo">65</a><br /> | ||
Badness: 0.0207<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0207<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x-Sharptone"></a><!-- ws:end:WikiTextHeadingRule:20 -->Sharptone</h2> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x-Sharptone"></a><!-- ws:end:WikiTextHeadingRule:20 -->Sharptone</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 21/20, 28/27<br /> | |||
<br /> | <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 /> | 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 /> | ||
| Line 437: | Line 437: | ||
<br /> | <br /> | ||
Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]<br /> | Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]<br /> | ||
Wedgie: &lt;&lt;1 4 3 4 2 -4||<br /> | <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 3 4 2 -4||<br /> | ||
EDOs: 5, 12<br /> | EDOs: 5, 12<br /> | ||
Badness: 0.0248<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0248<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="x-Injera"></a><!-- ws:end:WikiTextHeadingRule:22 -->Injera</h2> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="x-Injera"></a><!-- ws:end:WikiTextHeadingRule:22 -->Injera</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 50/49, 81/80<br /> | |||
<br /> | <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 19edos, 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 19edos, is an excellent tuning for injera.<br /> | ||
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Map: [&lt;2 0 -8 -7|, &lt;0 1 4 4|]<br /> | Map: [&lt;2 0 -8 -7|, &lt;0 1 4 4|]<br /> | ||
Wedgie: &lt;&lt;2 8 8 8 7 -4||<br /> | <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;2 8 8 8 7 -4||<br /> | ||
EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/38edo">38</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/178edo">178</a><br /> | EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/38edo">38</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/178edo">178</a><br /> | ||
Badness: 0.0311<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0311<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="x-Godzilla"></a><!-- ws:end:WikiTextHeadingRule:24 -->Godzilla</h2> | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="x-Godzilla"></a><!-- ws:end:WikiTextHeadingRule:24 -->Godzilla</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 49/48, 81/80<br /> | |||
<br /> | <br /> | ||
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-half intervals these represent give a fourth, and so step-and-a-half 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 /> | 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-half intervals these represent give a fourth, and so step-and-a-half 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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<br /> | <br /> | ||
Map: [&lt;1 0 -4 2|, &lt;0 2 8 1|]<br /> | Map: [&lt;1 0 -4 2|, &lt;0 2 8 1|]<br /> | ||
Wedgie: &lt;&lt;2 8 1 8 -4 -20||<br /> | <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;2 8 1 8 -4 -20||<br /> | ||
EDOs: 5, 9, 14, 19<br /> | EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/9edo">9</a>, <a class="wiki_link" href="/14edo">14</a>, 19<br /> | ||
Badness: 0.0267<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0267<br /> | ||
<br /> | <br /> | ||
Music: Igliashon Jones, <a class="wiki_link_ext" href="http://tinyurl.com/4uyumk9" rel="nofollow">&quot;Change is on the Wind&quot;</a>, in Godzilla[9]<br /> | Music: Igliashon Jones, <a class="wiki_link_ext" href="http://tinyurl.com/4uyumk9" rel="nofollow">&quot;Change is on the Wind&quot;</a>, in Godzilla[9]<br /> | ||
<!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="x-Mohajira"></a><!-- ws:end:WikiTextHeadingRule:26 -->Mohajira</h2> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="x-Mohajira"></a><!-- ws:end:WikiTextHeadingRule:26 -->Mohajira</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 6144/6125<br /> | |||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
7 and 9 limit minimax 1/4 comma<br /> | 7 and 9-limit minimax 1/4 comma<br /> | ||
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.415<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.415<br /> | ||
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Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]<br /> | Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]<br /> | ||
<a class="wiki_link" href="/Generator">Generator</a>s: 2, 128/105<br /> | |||
Wedgie: &lt;&lt;2 8 -11 8 -23 -48||<br /> | <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;2 8 -11 8 -23 -48||<br /> | ||
EDOs: 7, 24, 31<br /> | 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><br /> | ||
Badness: 0.0557<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0557<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="x-Mohajira-11-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->11-limit</h3> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="x-Mohajira-11-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->11-limit</h3> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 121/120, 176/175<br /> | |||
<br /> | <br /> | ||
11-limit 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 /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.477<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.477<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]<br /> | Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]<br /> | ||
<a class="wiki_link" href="/Generator">Generator</a>s: 2, 11/9<br /> | |||
EDOs: 7, 24, 31<br /> | EDOs: 7, 24, 31<br /> | ||
Badness: 0.0261<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0261<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="x-Mothra"></a><!-- ws:end:WikiTextHeadingRule:30 -->Mothra</h2> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="x-Mothra"></a><!-- ws:end:WikiTextHeadingRule:30 -->Mothra</h2> | ||
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Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]<br /> | Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]<br /> | ||
<a class="wiki_link" href="/Generator">Generator</a>s: 2, 8/7<br /> | |||
Wedgie: &lt;&lt;3 12 -1 12 -10 -36||<br /> | <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;3 12 -1 12 -10 -36||<br /> | ||
EDOs: 5, 26, 31<br /> | EDOs: 5, <a class="wiki_link" href="/26edo">26</a>, 31<br /> | ||
Badness: 0.0371<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0371<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="x-Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h3> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="x-Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h3> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 99/98, 385/384<br /> | |||
<br /> | <br /> | ||
POTE generator: ~63/55 = 232.031<br /> | POTE generator: ~63/55 = 232.031<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]<br /> | Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]<br /> | ||
EDOs: 5, | 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><br /> | ||
[[Badness[[: 0.0256==Squares==<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 2401/2400<br /> | |||
< | |||
< | |||
<br /> | <br /> | ||
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.<br /> | 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.<br /> | ||
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7 and 9 limit minimax 1/4 comma<br /> | 7 and 9 limit minimax 1/4 comma<br /> | ||
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3/2 0 9/16 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3/2 0 9/16 0&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.942<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.942<br /> | ||
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<br /> | <br /> | ||
Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]<br /> | Map: [&lt;1 3 8 6|, &lt;0 -4 -16 -9|]<br /> | ||
<a class="wiki_link" href="/Generator">Generator</a>s: 2, 9/7<br /> | |||
EDOs: 14, 31, 262, 293<br /> | EDOs: <a class="wiki_link" href="/14edo">14</a>, 31, <a class="wiki_link" href="/262edo">262</a>, <a class="wiki_link" href="/293edo">293</a><br /> | ||
Badness: 0.0460<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0460<br /> | ||
<br /> | <br /> | ||
Music:<br /> | Music:<br /> | ||
By Chris Vaisvil<br /> | By <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:462:http://tinyurl.com/25kv7cq --><a class="wiki_link_ext" href="http://tinyurl.com/25kv7cq" rel="nofollow">http://tinyurl.com/25kv7cq</a><!-- ws:end:WikiTextUrlRule:462 --><br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:463:http://tinyurl.com/24cbxse --><a class="wiki_link_ext" href="http://tinyurl.com/24cbxse" rel="nofollow">http://tinyurl.com/24cbxse</a><!-- ws:end:WikiTextUrlRule:463 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="x-Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->11-limit</h3> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 385/384, 1375/1372<br /> | |||
<br /> | <br /> | ||
<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 /> | ||
<br /> | <br /> | ||
Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]<br /> | Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]<br /> | ||
EDOs: 14, 31, 200<br /> | EDOs: <a class="wiki_link" href="/14edo">14</a>, 31, <a class="wiki_link" href="/200edo">200</a><br /> | ||
Badness: 0.0568<br /> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0568<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="x-Liese"></a><!-- ws:end:WikiTextHeadingRule:36 -->Liese</h2> | ||
<a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 686/675<br /> | |||
<br /> | <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 /> | 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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7 and 9 limit minimax 1/4 comma<br /> | 7 and 9 limit minimax 1/4 comma<br /> | ||
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |2/3 0 11/12 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |2/3 0 11/12 0&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 632.406<br /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 632.406<br /> | ||
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<br /> | <br /> | ||
Map: [&lt;1 0 -4 -3|, &lt;0 3 12 11|]<br /> | Map: [&lt;1 0 -4 -3|, &lt;0 3 12 11|]<br /> | ||
<a class="wiki_link" href="/Generator">Generator</a>s: 2, 10/7<br /> | |||
EDOs: <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/55edo">55</a>, <a class="wiki_link" href="/74edo">74</a><br /> | EDOs: <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/55edo">55</a>, <a class="wiki_link" href="/74edo">74</a><br /> | ||
Badness: 0.0467</body></html></pre></div> | <a class="wiki_link" href="/Badness">Badness</a>: 0.0467</body></html></pre></div> | ||