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| <h2>IMPORTED REVISION FROM WIKISPACES</h2> | | <span style="display: block; text-align: right;"> |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | __FORCETOC__ |
| : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2017-07-11 09:16:01 UTC</tt>.<br>
| | [[:de:mitteltönig Deutsch]]</span> |
| : The original revision id was <tt>615523693</tt>.<br>
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| : The revision comment was: <tt></tt><br>
| | The [[5-limit|5-limit]] parent [[Comma|comma]] of the [[Meantone|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 revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
| | [[POTE_tuning|POTE generator]]: ~3/2 = 696.239 |
| <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"><span style="display: block; text-align: right;">[[toc]]
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| [[xenharmonie/mitteltönig|Deutsch]] | |
| </span>
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| The [[5-limit]] parent [[Comma|comma]] of the [[meantone]] family is the Didymus or [[http://en.wikipedia.org/wiki/Syntonic_comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval. | |
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| [[POTE tuning|POTE generator]]: ~3/2 = 696.239
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| [[Tuning Ranges of Regular Temperaments|valid range]]: [685.714, 720.000] (7 to 5) | | [[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [685.714, 720.000] (7 to 5) |
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| nice range: [694.786, 701.955] (1/3 comma to Pythagorean) | | nice range: [694.786, 701.955] (1/3 comma to Pythagorean) |
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| strict range: [694.786, 701.955] | | strict range: [694.786, 701.955] |
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| [[Map]]: [<1 0 -4|, <0 1 4|] | | [[Map|Map]]: [<1 0 -4|, <0 1 4|] |
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| EDOs (patent val edo list is complete): [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[24edo|24]], [[26edo|26]], [[31edo|31]], [[36edo|36]], [[38edo|38]], [[43edo|43]], [[45edo|45]], [[50edo|50]], [[55edo|55]], [[57edo|57]], [[62edo|62]], [[67edo|67]], [[69edo|69]], [[74edo|74]], [[76edo|76]], [[81edo|81]], [[86edo|86]], [[88edo|88]], [[93edo|93]], [[98edo|98]], [[100edo|100]], [[105edo|105]], [[117edo|117]], [[129edo|129]], [[212edo|212b]] | | EDOs (patent val edo list is complete): [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[24edo|24]], [[26edo|26]], [[31edo|31]], [[36edo|36]], [[38edo|38]], [[43edo|43]], [[45edo|45]], [[50edo|50]], [[55edo|55]], [[57edo|57]], [[62edo|62]], [[67edo|67]], [[69edo|69]], [[74edo|74]], [[76edo|76]], [[81edo|81]], [[86edo|86]], [[88edo|88]], [[93edo|93]], [[98edo|98]], [[100edo|100]], [[105edo|105]], [[117edo|117]], [[129edo|129]], [[212edo|212b]] |
| [[Badness]]: 0.00736
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| ==Seven limit children==
| | [[Badness|Badness]]: 0.00736 |
| 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>].
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| =Septimal meantone= | | ==Seven limit children== |
| <span style="display: block; text-align: right;">[[xenharmonie/septimal-mitteltönig|Deutsch]] | | The [[7-limit|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>]. |
| </span> | |
| [[https://en.wikipedia.org/wiki/Septimal_meantone_temperament|Wikipedia article]] | |
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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|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.
| | =Septimal meantone= |
| | <span style="display: block; text-align: right;">[[:de:septimal-mitteltönig Deutsch]]</span> |
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| [[Comma]]s: 81/80, 126/125 | | [https://en.wikipedia.org/wiki/Septimal_meantone_temperament Wikipedia article] |
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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|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|31edo]] is a good tuning for it. |
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| | [[Comma|Comma]]s: 81/80, 126/125 |
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| | 7 and [[9-limit|9-limit]] minimax |
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| 7 and [[9-limit]] minimax
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| [|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
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| [[Tuning Ranges of Regular Temperaments|valid range]]: [694.737, 700.000] (19 to 12) | | [[Eigenmonzo|Eigenmonzo]]s: 2, 5 |
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| | [[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [694.737, 700.000] (19 to 12) |
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| nice range: [694.786, 701.955] | | nice range: [694.786, 701.955] |
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| strict range: [694.786, 700.000] | | strict range: [694.786, 700.000] |
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| [[POTE tuning|POTE generator]]: 696.495 | | [[POTE_tuning|POTE generator]]: 696.495 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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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. | | Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly. |
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| [[Map]]: [<1 0 -4 -13|, <0 1 4 10|] | | [[Map|Map]]: [<1 0 -4 -13|, <0 1 4 10|] |
| [[Generator]]s: 2, 3 | | |
| [[Wedgie]]: <<1 4 10 4 13 12|| | | [[generator|Generator]]s: 2, 3 |
| | |
| | [[wedgie|Wedgie]]: <<1 4 10 4 13 12|| |
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| EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[143edo|143b]] | | EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[143edo|143b]] |
| [[Badness]]: 0.0137
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| ==Bimeantone== | | [[Badness|Badness]]: 0.0137 |
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| | ==Bimeantone== |
| Commas: 81/80, 126/125, 245/242 | | Commas: 81/80, 126/125, 245/242 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 696.016 | | [[POTE_tuning|POTE generator]]: ~3/2 = 696.016 |
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| Map: [<2 0 -8 -26 -31|, <0 1 4 10 12|] | | Map: [<2 0 -8 -26 -31|, <0 1 4 10 12|] |
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| EDOs: 12, 38d, 50 | | EDOs: 12, 38d, 50 |
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| Badness: 0.0381 | | Badness: 0.0381 |
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| ===13-limit=== | | ===13-limit=== |
| Commas: 81/80, 105/104, 126/125, 245/242 | | Commas: 81/80, 105/104, 126/125, 245/242 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 695.836 | | [[POTE_tuning|POTE generator]]: ~3/2 = 695.836 |
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| Map: [<2 0 -8 -26 -31 -40|, <0 1 4 10 12 15|] | | Map: [<2 0 -8 -26 -31 -40|, <0 1 4 10 12 15|] |
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| EDOs: 12f, 50 | | EDOs: 12f, 50 |
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| Badness: 0.0288 | | Badness: 0.0288 |
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| ==Unidecimal meantone aka Huygens== | | ==Unidecimal meantone aka Huygens== |
| See also [[Meantone vs meanpop]] | | See also [[Meantone_vs_meanpop|Meantone vs meanpop]] |
| [[Comma]]s: 81/80, 126/125, 99/98 | | |
| | [[Comma|Comma]]s: 81/80, 126/125, 99/98 |
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| | [[11-limit|11-limit]] minimax |
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| [[11-limit]] minimax
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| [|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>, | | [|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>, |
| |21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>] | | |21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>] |
| [[Eigenmonzo]]s: 2, 11/9 | | |
| | [[Eigenmonzo|Eigenmonzo]]s: 2, 11/9 |
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| valid range: [696.774, 700.000] (31 to 12) | | valid range: [696.774, 700.000] (31 to 12) |
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| nice range: [691.202, 701.955] | | nice range: [691.202, 701.955] |
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| strict range: [696.774, 700.000] | | strict range: [696.774, 700.000] |
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| [[POTE tuning|POTE generator]]: 696.967 | | [[POTE_tuning|POTE generator]]: 696.967 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| [[Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents. | | [[Algebraic_generator|Algebraic generator]]: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents. |
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| | [[Map|Map]]: [<1 0 -4 -13 -25|, <0 1 4 10 18|] |
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| | [[generator|Generator]]s: 2, 3 |
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| [[Map]]: [<1 0 -4 -13 -25|, <0 1 4 10 18|]
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| [[Generator]]s: 2, 3
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| EDOs: [[7edo|7]], [[12edo|12]], [[31edo|31]], [[105edo|105]], [[198edo|198be]] | | EDOs: [[7edo|7]], [[12edo|12]], [[31edo|31]], [[105edo|105]], [[198edo|198be]] |
| [[Badness]]: 0.0170
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| [[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3|Twinkle canon – 74 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]] | | [[Badness|Badness]]: 0.0170 |
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| ===Tridecimal meantone=== | | [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 Twinkle canon – 74 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin] |
| [[Comma]]s: 66/65, 81/80, 99/98, 105/104 | | |
| | ===Tridecimal meantone=== |
| | [[Comma|Comma]]s: 66/65, 81/80, 99/98, 105/104 |
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| valid range: 697.674 (43) | | valid range: 697.674 (43) |
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| nice range: [691.202, 701.955] | | nice range: [691.202, 701.955] |
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| strict range: 697.674 | | strict range: 697.674 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 696.642 | | [[POTE_tuning|POTE generator]]: ~3/2 = 696.642 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| 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|] |
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| 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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| ===Grosstone=== | | [[Badness|Badness]]: 0.0180 |
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| | ===Grosstone=== |
| Commas: 81/80, 99/98, 126/125, 144/143 | | Commas: 81/80, 99/98, 126/125, 144/143 |
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| POTE generator: ~3/2 = 697.264 | | POTE generator: ~3/2 = 697.264 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -13 -25 29|, <0 1 4 10 18 -16|] | | Map: [<1 0 -4 -13 -25 29|, <0 1 4 10 18 -16|] |
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| EDOs: 12, 31, 43, 74 | | EDOs: 12, 31, 43, 74 |
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| Badness: 0.0259 | | Badness: 0.0259 |
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| ===Meridetone=== | | ===Meridetone=== |
| Commas: 78/77, 81/80, 99/98, 126/125 | | Commas: 78/77, 81/80, 99/98, 126/125 |
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| POTE generator: ~3/2 = 697.529 | | POTE generator: ~3/2 = 697.529 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -13 -25 -39|, <0 1 4 10 18 27|] | | Map: [<1 0 -4 -13 -25 -39|, <0 1 4 10 18 27|] |
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| EDOs: 43, 117df, 160bdf, 203bcdef | | EDOs: 43, 117df, 160bdf, 203bcdef |
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| Badness: 0.0264 | | Badness: 0.0264 |
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| ===Hemimeantone=== | | ===Hemimeantone=== |
| Commas: 81/80, 99/98, 126/125, 169/168 | | Commas: 81/80, 99/98, 126/125, 169/168 |
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| POTE generator: ~52/45 = 250.304 | | POTE generator: ~52/45 = 250.304 |
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| Mapping generator: ~26/15 | | Mapping generator: ~26/15 |
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| Map: [<1 0 -4 -13 -25 -5|, <0 2 8 20 36 11|] | | Map: [<1 0 -4 -13 -25 -5|, <0 2 8 20 36 11|] |
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| EDOs: 43, 62, 167bef, 229bef | | EDOs: 43, 62, 167bef, 229bef |
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| Badness: 0.0314 | | Badness: 0.0314 |
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| ==Meanpop== | | ==Meanpop== |
| See also [[Meantone vs meanpop]] | | See also [[Meantone_vs_meanpop|Meantone vs meanpop]] |
| [[Comma]]s: 81/80, 126/125, 385/384 | | |
| | [[Comma|Comma]]s: 81/80, 126/125, 385/384 |
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| | [[11-limit|11-limit]] [[minimax|minimax]] 1/4 comma |
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| [[11-limit]] [[minimax]] 1/4 comma
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| [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, | | [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |
| |-3 0 5/2 0 0>, |11 0 -13/4 0 0>] | | |-3 0 5/2 0 0>, |11 0 -13/4 0 0>] |
| [[Eigenmonzo]]s: 2, 5 | | |
| | [[Eigenmonzo|Eigenmonzo]]s: 2, 5 |
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| valid range: [694.737, 696.774] (19 to 31) | | valid range: [694.737, 696.774] (19 to 31) |
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| nice range: [691.202, 701.955] | | nice range: [691.202, 701.955] |
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| strict range: [694.737, 696.774] | | strict range: [694.737, 696.774] |
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| [[POTE tuning|POTE generator]]: 696.434 | | [[POTE_tuning|POTE generator]]: 696.434 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| [[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|Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge. |
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| [[@http://soonlabel.com/xenharmonic/archives/607|Scott Joplin's "The Entertainer" tuned into meanpop]] | | [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop] |
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| 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 | | |
| | [[generator|Generator]]s: 2, 3 |
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| EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[112edo|112]] | | EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[112edo|112]] |
| [[Badness]]: 0.0215
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| [[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3|Twinkle canon – 50 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]] | | [[Badness|Badness]]: 0.0215 |
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| ===13-limit Meanpop=== | | [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin] |
| [[Comma]]s: 81/80, 105/104, 144/143, 196/195 | | |
| | ===13-limit Meanpop=== |
| | [[Comma|Comma]]s: 81/80, 105/104, 144/143, 196/195 |
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| valid range: [694.737, 696.774] (19 to 31) | | valid range: [694.737, 696.774] (19 to 31) |
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| nice range: [691.202, 701.955] | | nice range: [691.202, 701.955] |
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| strict range: [694.737, 696.774] | | strict range: [694.737, 696.774] |
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| POTE generator: ~3/2 = 696.211 | | POTE generator: ~3/2 = 696.211 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|] | | Map: [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|] |
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| EDOS: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131bd]], [[212edo|212bdf]] | | EDOS: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131bd]], [[212edo|212bdf]] |
| [[Badness]]: 0.0209
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| ===Meanplop=== | | [[Badness|Badness]]: 0.0209 |
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| | ===Meanplop=== |
| Commas: 65/64, 78/77, 81/80, 91/90 | | Commas: 65/64, 78/77, 81/80, 91/90 |
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| POTE generator: ~3/2 = 696.202 | | POTE generator: ~3/2 = 696.202 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -13 24 10|, <0 1 4 10 -13 -4|] | | Map: [<1 0 -4 -13 24 10|, <0 1 4 10 -13 -4|] |
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| EDOs: 12e, 19, 31f, 50f | | EDOs: 12e, 19, 31f, 50f |
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| Badness: 0.0277 | | Badness: 0.0277 |
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| ==Meanenneadecal== | | ==Meanenneadecal== |
| [[Comma]]s: 45/44, 56/55, 81/80 | | [[Comma|Comma]]s: 45/44, 56/55, 81/80 |
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| | [[POTE_tuning|POTE generator]]: ~3/2 = 696.250 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 696.250
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|] | | Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|] |
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| EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31e]], [[50edo|50e]] | | EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31e]], [[50edo|50e]] |
| [[Badness]]: 0.0214
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| ===13-limit=== | | [[Badness|Badness]]: 0.0214 |
| [[Comma]]s: 45/44, 56/55, 78/77, 81/80 | | |
| | ===13-limit=== |
| | [[Comma|Comma]]s: 45/44, 56/55, 78/77, 81/80 |
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| | [[POTE_tuning|POTE generator]]: ~3/2 = 696.146 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 696.146
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| 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: [[19edo|19]], [[31edo|31e]], [[50edo|50e]]] | | EDOs: [[19edo|19]], [[31edo|31e]], [[50edo|50e]]] |
| [[Badness]]: 0.0212
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| ===Vincenzo=== | | [[Badness|Badness]]: 0.0212 |
| | |
| | ===Vincenzo=== |
| Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22 | | Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22 |
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| POTE generator: ~3/2 | | POTE generator: ~3/2 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -13 ... |, <0 1 4 10 6 -4 -5 -3 -6|] | | Map: [<1 0 -4 -13 ... |, <0 1 4 10 6 -4 -5 -3 -6|] |
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| EDOs: 12 | | EDOs: 12 |
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| Badness: | | Badness: |
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| ==Meanundeci== | | ==Meanundeci== |
| Commas: 33/32, 55/54, 77/75 | | Commas: 33/32, 55/54, 77/75 |
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| POTE generator: ~3/2 = 694.689 | | POTE generator: ~3/2 = 694.689 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -13 5|, <0 1 4 10 -1|] | | Map: [<1 0 -4 -13 5|, <0 1 4 10 -1|] |
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| EDOs: 12e, 19e | | EDOs: 12e, 19e |
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| Badness: 0.0315 | | Badness: 0.0315 |
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| ===13-limit=== | | ===13-limit=== |
| Commas: 33/32, 55/54, 77/75, 729/728 | | Commas: 33/32, 55/54, 77/75, 729/728 |
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| POTE generator: ~3/2 = 694.764 | | POTE generator: ~3/2 = 694.764 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -13 5 10|, <0 1 4 10 -1 -4|] | | Map: [<1 0 -4 -13 5 10|, <0 1 4 10 -1 -4|] |
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| EDOs: 12e, 19e | | EDOs: 12e, 19e |
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| Badness: 0.0263 | | Badness: 0.0263 |
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| ==Meanundec== | | ==Meanundec== |
| Commas: 27/26, 40/39, 45/44, 56/55 | | Commas: 27/26, 40/39, 45/44, 56/55 |
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| POTE generator: ~3/2 = 697.254 | | POTE generator: ~3/2 = 697.254 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -13 -6 -1|, <0 1 4 10 6 3|] | | Map: [<1 0 -4 -13 -6 -1|, <0 1 4 10 6 3|] |
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| EDOS: 12f, 19f, 31ef | | EDOS: 12f, 19f, 31ef |
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| Badness: 0.0242 | | Badness: 0.0242 |
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| =Flattone= | | =Flattone= |
| [[Comma]]s: 81/80, 525/512 | | [[Comma|Comma]]s: 81/80, 525/512 |
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| 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 seventh interval. Other intervals are [[7_6|7/6]], a diminished third, and [[7_5|7/5]], a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]]. | | The [[wedgie|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 seventh interval. Other intervals are [[7/6|7/6]], a diminished third, and [[7/5|7/5]], a doubly diminshed fifth. Good tunings for flattone are [[26edo|26edo]], [[45edo|45edo]] and [[64edo|64edo]]. |
| | |
| | [[7-limit|7-limit]] minimax |
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| [[7-limit]] minimax
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| [|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
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| [[9-limit]] minimax | | [[Eigenmonzo|Eigenmonzo]]s: 2, 7/5 |
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| | [[9-limit|9-limit]] minimax |
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| [|1 0 0 0>, |17/11 2/11 0 -1/11>, | | [|1 0 0 0>, |17/11 2/11 0 -1/11>, |
| |24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>] | | |24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>] |
| [[Eigenmonzo]]s: 2, 9/7 | | |
| | [[Eigenmonzo|Eigenmonzo]]s: 2, 9/7 |
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| valid range: [692.308, 694.737] (26 to 19) | | valid range: [692.308, 694.737] (26 to 19) |
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| nice range: [692.353, 701.955] | | nice range: [692.353, 701.955] |
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| strict range: [692.353, 694.737] | | strict range: [692.353, 694.737] |
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| [[POTE tuning|POTE generator]]: 693.779 | | [[POTE_tuning|POTE generator]]: 693.779 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Line 276: |
Line 353: |
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| 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|| | | |
| [[Generator]]s: 2, 3 | | [[wedgie|Wedgie]]: <<1 4 -9 4 -17 -32|| |
| | |
| | [[generator|Generator]]s: 2, 3 |
| | |
| EDOs: [[7edo|7]], [[19edo|19]], [[45edo|45]], [[64edo|64]] | | EDOs: [[7edo|7]], [[19edo|19]], [[45edo|45]], [[64edo|64]] |
| [[Badness]]: 0.0386
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| ==11-limit== | | [[Badness|Badness]]: 0.0386 |
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| | ==11-limit== |
| Commas: 45/44, 81/80, 385/384 | | Commas: 45/44, 81/80, 385/384 |
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| valid range: [692.308, 694.737] (26 to 19) | | valid range: [692.308, 694.737] (26 to 19) |
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| nice range: [682.502, 701.955] | | nice range: [682.502, 701.955] |
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| strict range: [692.308, 694.737] | | strict range: [692.308, 694.737] |
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| POTE generator: ~3/2 = 693.126 | | POTE generator: ~3/2 = 693.126 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 17 -6|, <0 1 4 -9 6|] | | Map: [<1 0 -4 17 -6|, <0 1 4 -9 6|] |
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| EDOs: 7, 19, 26, 45, 71bc, 116bcde | | EDOs: 7, 19, 26, 45, 71bc, 116bcde |
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| Badness: 0.0338 | | Badness: 0.0338 |
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| ==13-limit== | | ==13-limit== |
| 45/44, 65/64, 78/77, 81/80 | | 45/44, 65/64, 78/77, 81/80 |
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| valid range: [692.308, 694.737] (26 to 19) | | valid range: [692.308, 694.737] (26 to 19) |
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| nice range: [682.502, 701.955] | | nice range: [682.502, 701.955] |
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| strict range: [692.308, 694.737] | | strict range: [692.308, 694.737] |
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| POTE generator: ~3/2 = 693.058 | | POTE generator: ~3/2 = 693.058 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 17 -6 10|, <0 1 4 -9 6 -4|] | | Map: [<1 0 -4 17 -6 10|, <0 1 4 -9 6 -4|] |
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| EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef | | EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef |
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| Badness: 0.0223 | | Badness: 0.0223 |
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| =Dominant= | | =Dominant= |
| [[Comma]]s: 36/35, 64/63 | | [[Comma|Comma]]s: 36/35, 64/63 |
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| 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]]. | | 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|12edo]], but it also works well with the Pythagorean tuning of pure [[3/2|3/2]] fifths, and with [[29edo|29edo]], [[41edo|41edo]], or [[53edo|53edo]]. |
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| valid range: [700.000, 720.000] (12 to 5) | | valid range: [700.000, 720.000] (12 to 5) |
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| nice range: [694.786, 715.587] | | nice range: [694.786, 715.587] |
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| strict range: [700.000, 715.587] | | strict range: [700.000, 715.587] |
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| [[POTE tuning|POTE generator]]: 701.573 | | [[POTE_tuning|POTE generator]]: 701.573 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| 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|Wedgie]]: <<1 4 -2 4 -6 -16|| |
| | |
| EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[53edo|53]], [[65edo|65]] | | EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[53edo|53]], [[65edo|65]] |
| [[Badness]]: 0.0207
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| ==11-limit== | | [[Badness|Badness]]: 0.0207 |
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| | ==11-limit== |
| Commas: 36/35, 64/63, 56/55 | | Commas: 36/35, 64/63, 56/55 |
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| valid range: [700.000, 705.882] (12 to 17) | | valid range: [700.000, 705.882] (12 to 17) |
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| nice range: [691.202, 715.587] | | nice range: [691.202, 715.587] |
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| strict range: [700.000, 705.882] | | strict range: [700.000, 705.882] |
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| POTE generator: ~3/2 = 703.254 | | POTE generator: ~3/2 = 703.254 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 6 13|, <0 1 4 -2 -6|] | | Map: [<1 0 -4 6 13|, <0 1 4 -2 -6|] |
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| EDOs: 5, 12, 17c, 29cde | | EDOs: 5, 12, 17c, 29cde |
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| Badness: 0.0242 | | Badness: 0.0242 |
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| ==13-limit== | | ==13-limit== |
| Commas: 36/35, 56/55, 64/63, 66/65 | | Commas: 36/35, 56/55, 64/63, 66/65 |
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| valid range: 705.882 (17) | | valid range: 705.882 (17) |
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| nice range: [691.202, 715.587] | | nice range: [691.202, 715.587] |
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| strict range:705.882 | | strict range:705.882 |
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| Line 350: |
Line 454: |
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| Map: [<1 0 -4 6 13 18|, <0 1 4 -2 -6 -9|] | | Map: [<1 0 -4 6 13 18|, <0 1 4 -2 -6 -9|] |
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| EDOs: 12f, 17c, 29cdef | | EDOs: 12f, 17c, 29cdef |
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| Badness: 0.0241 | | Badness: 0.0241 |
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| ==Dominion== | | ==Dominion== |
| Commas: 26/25, 36/35, 56/55, 64/63 | | Commas: 26/25, 36/35, 56/55, 64/63 |
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| Line 359: |
Line 465: |
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| Map: [<1 0 -4 6 13 -9|, <0 1 4 -2 -6 8|] | | Map: [<1 0 -4 6 13 -9|, <0 1 4 -2 -6 8|] |
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| EDOs: 5, 12, 17c, 46cde | | EDOs: 5, 12, 17c, 46cde |
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| Badness: 0.0273 | | Badness: 0.0273 |
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| ==Domineering== | | ==Domineering== |
| Commas: 36/35, 45/44, 64/63 | | Commas: 36/35, 45/44, 64/63 |
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| POTE generator: ~3/2 = 698.776 | | POTE generator: ~3/2 = 698.776 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 6 -6|, <0 1 4 -2 6|] | | Map: [<1 0 -4 6 -6|, <0 1 4 -2 6|] |
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| EDOs: 7, 12, 43de | | EDOs: 7, 12, 43de |
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| Badness: 0.0220 | | Badness: 0.0220 |
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| ==Domination== | | ==Domination== |
| Commas: 36/35, 64/63, 77/75 | | Commas: 36/35, 64/63, 77/75 |
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| POTE generator: ~3/2 = 705.004 | | POTE generator: ~3/2 = 705.004 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 6 -14|, <0 1 4 -2 11|] | | Map: [<1 0 -4 6 -14|, <0 1 4 -2 11|] |
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| EDOs: 17c, 46cd | | EDOs: 17c, 46cd |
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| Badness: 0.0366 | | Badness: 0.0366 |
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| ===13-limit=== | | ===13-limit=== |
| Commas: 26/25, 36/35, 64/63, 66/65 | | Commas: 26/25, 36/35, 64/63, 66/65 |
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| POTE generator: ~3/2 = 705.496 | | POTE generator: ~3/2 = 705.496 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 6 -14 -9|, <0 1 4 -2 11 8|] | | Map: [<1 0 -4 6 -14 -9|, <0 1 4 -2 11 8|] |
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| EDOs: 17c | | EDOs: 17c |
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| Badness: 0.0274 | | Badness: 0.0274 |
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| ==Twelve== | | ==Twelve== |
| Commas: 81/80 64/63 45/44 65/64 256/255 153/152 | | Commas: 81/80 64/63 45/44 65/64 256/255 153/152 |
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| POTE generator: ~3/2 = 696.217 | | POTE generator: ~3/2 = 696.217 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 6 -6 10 12 9|, <0 1 4 -2 6 -4 -5 -3|] | | Map: [<1 0 -4 6 -6 10 12 9|, <0 1 4 -2 6 -4 -5 -3|] |
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| EDOs: 7, 12, 19d, 31def | | EDOs: 7, 12, 19d, 31def |
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| Badness: 0.0204 | | Badness: 0.0204 |
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| ==Arnold== | | ==Arnold== |
| Commas: 22/21, 33/32, 36/35 | | Commas: 22/21, 33/32, 36/35 |
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| POTE generator: ~3/2 = 698.491 | | POTE generator: ~3/2 = 698.491 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 6 5|, <0 1 4 -2 -1|] | | Map: [<1 0 -4 6 5|, <0 1 4 -2 -1|] |
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| EDOs: 5, 7, 12e | | EDOs: 5, 7, 12e |
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| Badness: 0.0261 | | Badness: 0.0261 |
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| ==13-limit== | | ==13-limit== |
| Commas: 22/21, 27/26, 33/32, 40/39 | | Commas: 22/21, 27/26, 33/32, 40/39 |
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| POTE generator: ~3/2 = 696.743 | | POTE generator: ~3/2 = 696.743 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 6 5 -1|, <0 1 4 -2 -1 3|] | | Map: [<1 0 -4 6 5 -1|, <0 1 4 -2 -1 3|] |
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| EDOs: 5, 7, 12ef, 19def, 31def | | EDOs: 5, 7, 12ef, 19def, 31def |
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| Badness: 0.0233 | | Badness: 0.0233 |
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| ==Dominatrix== | | ==Dominatrix== |
| Commas: 27/26 36/35 45/44 64/63 | | Commas: 27/26 36/35 45/44 64/63 |
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| POTE generator: ~3/2 = 698.544 | | POTE generator: ~3/2 = 698.544 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 6 -6 -1|, <0 1 4 -2 6 3|] | | Map: [<1 0 -4 6 -6 -1|, <0 1 4 -2 6 3|] |
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| EDOs: 7, 12f | | EDOs: 7, 12f |
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| Badness: 0.0183 | | Badness: 0.0183 |
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| =Sharptone= | | =Sharptone= |
| [[Comma]]s: 21/20, 28/27 | | [[Comma|Comma]]s: 21/20, 28/27 |
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|
| 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|12edo]] tuning does sharptone about as well as such a thing can be done. |
| | |
| | [[POTE_tuning|POTE generator]]: 700.140 |
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| [[POTE tuning|POTE generator]]: 700.140
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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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|Wedgie]]: <<1 4 3 4 2 -4|| |
| | |
| EDOs: [[5edo|5]], [[12edo|12]] | | EDOs: [[5edo|5]], [[12edo|12]] |
| [[Badness]]: 0.0248
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| =Meansept= | | [[Badness|Badness]]: 0.0248 |
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| | =Meansept= |
| Commas: 15/14, 81/80 | | Commas: 15/14, 81/80 |
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| POTE generator: ~3/2 = 682.895 | | POTE generator: ~3/2 = 682.895 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -5|, <0 1 4 5|] | | Map: [<1 0 -4 -5|, <0 1 4 5|] |
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| Wedgie: <<1 4 5 4 5 0|| | | Wedgie: <<1 4 5 4 5 0|| |
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| EDOs: 7 | | EDOs: 7 |
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| Badness: 0.0453 | | Badness: 0.0453 |
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| ==11-limit== | | ==11-limit== |
| Commas: 15/14, 22/21, 125/121 | | Commas: 15/14, 22/21, 125/121 |
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| POTE generator: ~3/2 = 685.234 | | POTE generator: ~3/2 = 685.234 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 -4 -5 -6|, <0 1 4 5 6|] | | Map: [<1 0 -4 -5 -6|, <0 1 4 5 6|] |
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| EDOs: 7 | | EDOs: 7 |
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| Badness: 0.0325 | | Badness: 0.0325 |
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| =Supermean= | | =Supermean= |
| Commas: 81/80, 672/625 | | Commas: 81/80, 672/625 |
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| Line 472: |
Line 612: |
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| Map: [<1 0 -4 -21|, <0 1 4 15|] | | Map: [<1 0 -4 -21|, <0 1 4 15|] |
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| EDOs: 17c, 46c | | EDOs: 17c, 46c |
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| Badness: 0.1342 | | Badness: 0.1342 |
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| ==11-limit== | | ==11-limit== |
| Commas: 56/55, 81/80, 132/125 | | Commas: 56/55, 81/80, 132/125 |
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| Line 481: |
Line 623: |
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| Map: [<1 0 -4 -21 -14|, <0 1 4 15 11|] | | Map: [<1 0 -4 -21 -14|, <0 1 4 15 11|] |
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| EDOs: 17c, 46c | | EDOs: 17c, 46c |
| | |
| Badness: 0.0633 | | Badness: 0.0633 |
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| ==13-limit== | | ==13-limit== |
| Commas: 26/25, 56/55, 66/65, 81/80 | | Commas: 26/25, 56/55, 66/65, 81/80 |
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| Line 490: |
Line 634: |
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| Map: [<1 0 -4 -21 -14 -9|, <0 1 4 15 11 8|] | | Map: [<1 0 -4 -21 -14 -9|, <0 1 4 15 11 8|] |
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| EDOs: 17c, 46c | | EDOs: 17c, 46c |
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| =Injera= | | =Injera= |
| [[Comma]]s: 50/49, 81/80 | | [[Comma|Comma]]s: 50/49, 81/80 |
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| |
|
| 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. | | 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|38edo]], which is two parallel [[19edo|19edo]]s, is an excellent tuning for injera. |
|
| |
|
| [[http://tech.groups.yahoo.com/group/tuning-math/message/3091|Origin of the name]]
| | [http://tech.groups.yahoo.com/group/tuning-math/message/3091 Origin of the name] |
|
| |
|
| valid range: [685.714, 700.000] (14c to 12) | | valid range: [685.714, 700.000] (14c to 12) |
| | |
| nice range: [688.957, 701.955] | | nice range: [688.957, 701.955] |
| | |
| strict range: [688.957, 700.000] | | strict range: [688.957, 700.000] |
|
| |
|
| [[POTE tuning|POTE generator]]: 694.375 | | [[POTE_tuning|POTE generator]]: 694.375 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| 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|Wedgie]]: <<2 8 8 8 7 -4|| |
| | |
| EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[102edo|102bcd]], [[140edo|140bcd]], [[178edo|178bcd]] | | EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[102edo|102bcd]], [[140edo|140bcd]], [[178edo|178bcd]] |
| [[Badness]]: 0.0311
| |
|
| |
|
| [[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3|Two Pairs of Socks]] (in [[26edo]]) by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]] | | [[Badness|Badness]]: 0.0311 |
| [[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Injera%20Jam.mp3|Injera Jam]] (in [[26edo]]) by [[Zach Curley]]
| |
|
| |
|
| ==11-limit== | | [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26edo]]) by [[Igliashon_Jones|Igliashon Calvin Jones-Coolidge]] |
| | |
| | [http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Injera%20Jam.mp3 Injera Jam] (in [[26edo|26edo]]) by [[Zach_Curley|Zach Curley]] |
| | |
| | ==11-limit== |
| Commas: 45/44, 50/49, 81/80 | | Commas: 45/44, 50/49, 81/80 |
|
| |
|
| valid range: [685.714, 700.000] (14c to 12) | | valid range: [685.714, 700.000] (14c to 12) |
| | |
| nice range: [682.458, 701.955] | | nice range: [682.458, 701.955] |
| | |
| strict range: [685.714, 700.000] | | strict range: [685.714, 700.000] |
|
| |
|
| POTE generator: ~3/2 = 692.840 | | POTE generator: ~3/2 = 692.840 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<2 0 -8 -7 -12|, <0 1 4 4 6|] | | Map: [<2 0 -8 -7 -12|, <0 1 4 4 6|] |
| | |
| EDOs: 12, 14c, 26. 90bce, 116bce | | EDOs: 12, 14c, 26. 90bce, 116bce |
| | |
| Badness: 0.0231 | | Badness: 0.0231 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 45/44, 50/49, 81/80, 78/77 | | Commas: 45/44, 50/49, 81/80, 78/77 |
|
| |
|
| valid range: 692.308 (26) | | valid range: 692.308 (26) |
| | |
| nice range: [682.458, 701.955] | | nice range: [682.458, 701.955] |
| | |
| strict range: 692.308 (26) | | strict range: 692.308 (26) |
|
| |
|
| POTE generator: ~3/2 = 692.673 | | POTE generator: ~3/2 = 692.673 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<2 0 -8 -7 -12 -21|, <0 1 4 4 6 9|] | | Map: [<2 0 -8 -7 -12 -21|, <0 1 4 4 6 9|] |
| | |
| EDOs: 26, 104bcf | | EDOs: 26, 104bcf |
| | |
| Badness: 0.0216 | | Badness: 0.0216 |
|
| |
|
| ==Enjera== | | ==Enjera== |
| Commas: 27/26, 40/39, 45/44, 99/98 | | Commas: 27/26, 40/39, 45/44, 99/98 |
|
| |
|
| POTE generator: ~3/2 = 694.121 | | POTE generator: ~3/2 = 694.121 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<2 0 -8 -7 -12 -2|, <0 1 4 4 6 3|] | | Map: [<2 0 -8 -7 -12 -2|, <0 1 4 4 6 3|] |
| | |
| EDOs: 12f, 26f, 38ef | | EDOs: 12f, 26f, 38ef |
| | |
| Badness: 0.0265 | | Badness: 0.0265 |
|
| |
|
| ==Injerous== | | ==Injerous== |
| Commas: 33/32, 50/49, 55/54 | | Commas: 33/32, 50/49, 55/54 |
|
| |
|
| POTE generator: ~3/2 = 690.548 | | POTE generator: ~3/2 = 690.548 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<2 0 -8 -7 10|, <0 1 4 4 -1|] | | Map: [<2 0 -8 -7 10|, <0 1 4 4 -1|] |
| | |
| EDOs: 12e, 14c, 26e, 40ce | | EDOs: 12e, 14c, 26e, 40ce |
| | |
| Badness: 0.0386 | | Badness: 0.0386 |
|
| |
|
| ==Lahoh== | | ==Lahoh== |
| Commas: 50/49, 56/55, 81/77 | | Commas: 50/49, 56/55, 81/77 |
|
| |
|
| POTE generator: ~3/2 = 699.001 | | POTE generator: ~3/2 = 699.001 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<2 0 -8 -7 7|, <0 1 4 4 0|] | | Map: [<2 0 -8 -7 7|, <0 1 4 4 0|] |
| | |
| EDOs: 12 | | EDOs: 12 |
| | |
| Badness: 0.0431 | | Badness: 0.0431 |
|
| |
|
| =Godzilla= | | =Godzilla= |
| <span style="display: block; text-align: right;">[[xenharmonie/Semiphor, Semaphor, Godzilla|Deutsch]]
| | <span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla Deutsch]]</span> |
| </span>
| | |
| Main article: [[Semaphore and Godzilla]] | | Main article: [[Semaphore_and_Godzilla|Semaphore and 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-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo]] is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes. | | [[Comma|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-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo|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. |
|
| |
|
| valid range: [240.000, 257.143] (5 to 14c) | | valid range: [240.000, 257.143] (5 to 14c) |
| | |
| nice range: [231.174, 266.871] | | nice range: [231.174, 266.871] |
| | |
| strict range: [240.000, 257.143] | | strict range: [240.000, 257.143] |
|
| |
|
| [[POTE tuning|POTE generator]]: ~8/7 = 252.635 | | [[POTE_tuning|POTE generator]]: ~8/7 = 252.635 |
| | |
| Mapping generator: ~7/4 | | Mapping generator: ~7/4 |
|
| |
|
| 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|Wedgie]]: <<2 8 1 8 -4 -20|| |
| | |
| EDOs: [[5edo|5]], [[9edo|9c]], [[14edo|14c]], [[19edo|19]], [[62edo|62d]], [[81edo|81d]], 143bd | | EDOs: [[5edo|5]], [[9edo|9c]], [[14edo|14c]], [[19edo|19]], [[62edo|62d]], [[81edo|81d]], 143bd |
| [[Badness]]: 0.0267
| |
|
| |
|
| ==11-limit== | | [[Badness|Badness]]: 0.0267 |
| | |
| | ==11-limit== |
| Commas: 45/44, 49/48, 81/80 | | Commas: 45/44, 49/48, 81/80 |
|
| |
|
| valid range: [252.632, 257.143] (19 to 14c) | | valid range: [252.632, 257.143] (19 to 14c) |
| | |
| nice range: [231.174, 266.871] | | nice range: [231.174, 266.871] |
| | |
| strict range: [252.632, 257.143] | | strict range: [252.632, 257.143] |
|
| |
|
| POTE generator: ~8/7 = 254.027 | | POTE generator: ~8/7 = 254.027 |
| | |
| Mapping generator: ~7/4 | | Mapping generator: ~7/4 |
|
| |
|
| Map: [<1 0 -4 2 -6|, <0 2 8 1 12|] | | Map: [<1 0 -4 2 -6|, <0 2 8 1 12|] |
| | |
| EDOs: 14c, 19, 33cd, 52cd | | EDOs: 14c, 19, 33cd, 52cd |
| | |
| Badness: 0.0290 | | Badness: 0.0290 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 45/44, 49/48, 78/77, 81/80 | | Commas: 45/44, 49/48, 78/77, 81/80 |
|
| |
|
| valid range: 694.737 (19) | | valid range: 694.737 (19) |
| | |
| nice range: [621.581, 737.652] | | nice range: [621.581, 737.652] |
| | |
| strict range: 694.737 | | strict range: 694.737 |
|
| |
|
| POTE generator: ~8/7 = 253.603 | | POTE generator: ~8/7 = 253.603 |
| | |
| Mapping generator: ~7/4 | | Mapping generator: ~7/4 |
|
| |
|
| Map: [<1 0 -4 2 -6 -5|, <0 2 8 1 12 11|] | | Map: [<1 0 -4 2 -6 -5|, <0 2 8 1 12 11|] |
| | |
| EDOs: 14cf, 19, 33cdf, 52cdf | | EDOs: 14cf, 19, 33cdf, 52cdf |
| | |
| Badness: 0.0225 | | Badness: 0.0225 |
|
| |
|
| ==Semafour== | | ==Semafour== |
| Commas: 33/32, 49/48, 55/54 | | Commas: 33/32, 49/48, 55/54 |
|
| |
|
| POTE generator: ~8/7 = 254.042 | | POTE generator: ~8/7 = 254.042 |
| | |
| Mapping generator: ~7/4 | | Mapping generator: ~7/4 |
|
| |
|
| Map: [<1 0 -4 2 5|, <0 2 8 1 -2|] | | Map: [<1 0 -4 2 5|, <0 2 8 1 -2|] |
| | |
| EDOs: 5, 14c, 19e, 33cde | | EDOs: 5, 14c, 19e, 33cde |
| | |
| Badness: 0.0285 | | Badness: 0.0285 |
|
| |
|
| ==Varan== | | ==Varan== |
| Commas: 49/48, 77/75, 81/80 | | Commas: 49/48, 77/75, 81/80 |
|
| |
|
| POTE generator: ~8/7 = 251.079 | | POTE generator: ~8/7 = 251.079 |
| | |
| Mapping generator: ~7/4 | | Mapping generator: ~7/4 |
|
| |
|
| Map: [<1 0 -4 2 -10|, <0 2 8 1 17|] | | Map: [<1 0 -4 2 -10|, <0 2 8 1 17|] |
| | |
| EDOs: 19e, 24, 43de | | EDOs: 19e, 24, 43de |
| | |
| Badness: 0.0396 | | Badness: 0.0396 |
|
| |
|
| ===13-limit=== | | ===13-limit=== |
| Commas: 49/48, 66/65, 77/75, 81/80 | | Commas: 49/48, 66/65, 77/75, 81/80 |
|
| |
|
| POTE generator: ~8/7 = 251.165 | | POTE generator: ~8/7 = 251.165 |
| | |
| Mapping generator: ~7/4 | | Mapping generator: ~7/4 |
|
| |
|
| Map: [<1 0 -4 2 -10 -5|, <0 2 8 1 17 11|] | | Map: [<1 0 -4 2 -10 -5|, <0 2 8 1 17 11|] |
| | |
| EDOs: 19e, 24, 43de | | EDOs: 19e, 24, 43de |
| | |
| Badness: 0.0257 | | Badness: 0.0257 |
|
| |
|
| ==Baragon== | | ==Baragon== |
| Commas: 49/48, 56/55, 81/80 | | Commas: 49/48, 56/55, 81/80 |
|
| |
|
| POTE generator: ~8/7 = 251.173 | | POTE generator: ~8/7 = 251.173 |
| | |
| Mapping generator: ~7/4 | | Mapping generator: ~7/4 |
|
| |
|
| Map: [<1 0 -4 2 9|, <0 2 8 1 -7|] | | Map: [<1 0 -4 2 9|, <0 2 8 1 -7|] |
| | |
| EDOs: 19, 24, 43d | | EDOs: 19, 24, 43d |
| | |
| Badness: 0.0357 | | Badness: 0.0357 |
|
| |
|
| ==Music== | | ==Music== |
| [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3|Godzilla Example]] by [[Cameron Bobro]]
| | [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3 Godzilla Example] by [[Cameron_Bobro|Cameron Bobro]] |
| [[http://tinyurl.com/4uyumk9|"Change is on the Wind"]] in Godzilla[9] by [[Igliashon Jones]]
| | |
| | [http://tinyurl.com/4uyumk9 "Change is on the Wind"] in Godzilla[9] by [[Igliashon_Jones|Igliashon Jones]] |
|
| |
|
| =Mohajira= | | =Mohajira= |
| <span style="display: block; text-align: right;">[[xenharmonie/Mohajira|Deutsch]]
| | <span style="display: block; text-align: right;">[[:de:Mohajira Deutsch]]</span> |
| </span>
| |
| [[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. | | [[Comma|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|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 can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4. | | Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4. |
|
| |
|
| [[7-limit|7]] and [[9-limit]] minimax 1/4 comma | | [[7-limit|7]] and [[9-limit|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]]: ~128/105 = 348.415 | | [[Eigenmonzo|Eigenmonzo]]s: 2, 5 |
| | |
| | [[POTE_tuning|POTE generator]]: ~128/105 = 348.415 |
| | |
| Mapping generator: ~128/105 | | Mapping generator: ~128/105 |
|
| |
|
| Line 683: |
Line 887: |
|
| |
|
| 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|| | | [[generator|Generator]]s: 2, 128/105 |
| | |
| | [[wedgie|Wedgie]]: <<2 8 -11 8 -23 -48|| |
| | |
| EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]] | | EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]] |
| [[Badness]]: 0.0557
| |
|
| |
|
| ==11-limit== | | [[Badness|Badness]]: 0.0557 |
| [[Comma]]s: 81/80, 121/120, 176/175 | | |
| | ==11-limit== |
| | [[Comma|Comma]]s: 81/80, 121/120, 176/175 |
| | |
| | [[11-limit|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]]: ~11/9 = 348.477 | | [[Eigenmonzo|Eigenmonzo]]s: 2, 5 |
| | |
| | [[POTE_tuning|POTE generator]]: ~11/9 = 348.477 |
| | |
| Mapping generator: ~11/9 | | Mapping generator: ~11/9 |
|
| |
|
| 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 | | |
| | [[generator|Generator]]s: 2, 11/9 |
| | |
| EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]] | | EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]] |
| [[Badness]]: 0.0261
| |
|
| |
|
| ==13-limit== | | [[Badness|Badness]]: 0.0261 |
| | |
| | ==13-limit== |
| Commas: 81/80, 121/120, 105/104, 66/65 | | Commas: 81/80, 121/120, 105/104, 66/65 |
|
| |
|
| POTE generator: ~11/9 = 348.558 | | POTE generator: ~11/9 = 348.558 |
| | |
| Mapping generator: ~11/9 | | Mapping generator: ~11/9 |
|
| |
|
| Map: [<1 1 0 6 2 4|, <0 2 8 -11 5 -1|] | | Map: [<1 1 0 6 2 4|, <0 2 8 -11 5 -1|] |
| | |
| EDOs: 7, 24, 31, 117ef, 148bef | | EDOs: 7, 24, 31, 117ef, 148bef |
| | |
| Badness: 0.0234 | | Badness: 0.0234 |
|
| |
|
| =Ptolemy= | | =Ptolemy= |
| Commas: 81/80, 121/120, 525/512 | | Commas: 81/80, 121/120, 525/512 |
|
| |
|
| Line 720: |
Line 937: |
|
| |
|
| Map: [<1 1 0 8 2|, <0 2 8 -18 5|] | | Map: [<1 1 0 8 2|, <0 2 8 -18 5|] |
| | |
| EDOs: 7, 38d, 45e, 83bcde | | EDOs: 7, 38d, 45e, 83bcde |
| | |
| Badness: 0.0588 | | Badness: 0.0588 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 65/64, 81/80, 105/104, 121/120 | | Commas: 65/64, 81/80, 105/104, 121/120 |
|
| |
|
| Line 729: |
Line 948: |
|
| |
|
| Map: [<1 1 0 8 2 6|, <0 2 8 -18 5 -8|] | | Map: [<1 1 0 8 2 6|, <0 2 8 -18 5 -8|] |
| | |
| EDOs: 7, 38df, 45ef, 83bcdef | | EDOs: 7, 38df, 45ef, 83bcdef |
| | |
| Badness: 0.0343 | | Badness: 0.0343 |
|
| |
|
| =Maqamic= | | =Maqamic= |
| <span style="display: block; text-align: right;">[[xenharmonie/maqamisch|Deutsch]]
| | <span style="display: block; text-align: right;">[[:de:maqamisch Deutsch]]</span> |
| </span>
| | |
| Main article: [[Maqamic]] | | Main article: [[maqamic|Maqamic]] |
| [[Comma]]s: 81/80, 36/35, 121/120 | | |
| | [[Comma|Comma]]s: 81/80, 36/35, 121/120 |
|
| |
|
| Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation. | | Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation. |
|
| |
|
| [[POTE tuning|POTE generator]]: ~11/9 = 350.934 | | [[POTE_tuning|POTE generator]]: ~11/9 = 350.934 |
| | |
| Mapping generator: ~11/9 | | Mapping generator: ~11/9 |
|
| |
|
| Map: [<1 1 0 4 2|, <0 2 8 -4 5|] | | Map: [<1 1 0 4 2|, <0 2 8 -4 5|] |
| [[Generator]]s: 2, 11/9 | | |
| | [[generator|Generator]]s: 2, 11/9 |
| | |
| EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]], [[31edo|31d]] | | EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]], [[31edo|31d]] |
|
| |
|
| ==13-limit== | | ==13-limit== |
| [[Comma]]s: 81/80, 36/35, 121/120, 144/143 | | [[Comma|Comma]]s: 81/80, 36/35, 121/120, 144/143 |
| | |
| | [[POTE_tuning|POTE generator]]: ~11/9 = 350.816 |
|
| |
|
| [[POTE tuning|POTE generator]]: ~11/9 = 350.816
| |
| Mapping generator: ~11/9 | | Mapping generator: ~11/9 |
|
| |
|
| Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|] | | Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|] |
| | |
| Generators: 2, 11/9 | | Generators: 2, 11/9 |
| | |
| EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]],[[31edo| 31d]] | | EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]],[[31edo| 31d]] |
|
| |
|
| =Migration= | | =Migration= |
| Commas: 81/80, 121/120, 126/125 | | Commas: 81/80, 121/120, 126/125 |
|
| |
|
| POTE generator: ~11/9 = 348.182 | | POTE generator: ~11/9 = 348.182 |
| | |
| Mapping generator: ~11/9 | | Mapping generator: ~11/9 |
|
| |
|
| Map: [<1 1 0 -3 2|, <0 2 8 20 5|] | | Map: [<1 1 0 -3 2|, <0 2 8 20 5|] |
| | |
| EDOs: 31, 100de, 131bde, 162bde | | EDOs: 31, 100de, 131bde, 162bde |
| | |
| Badness: 0.0255 | | Badness: 0.0255 |
|
| |
|
| =Mohamaq= | | =Mohamaq= |
| Commas: 81/80, 392/375 | | Commas: 81/80, 392/375 |
|
| |
|
| POTE generator: ~25/21 = 350.586 | | POTE generator: ~25/21 = 350.586 |
| | |
| Mapping generator: ~25/21 | | Mapping generator: ~25/21 |
|
| |
|
| Map: [<1 1 0 -1|, <0 2 8 13|] | | Map: [<1 1 0 -1|, <0 2 8 13|] |
| | |
| EDOs: 17c, 24, 65c, 89cd | | EDOs: 17c, 24, 65c, 89cd |
| | |
| Badness: 0.0777 | | Badness: 0.0777 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 56/55, 77/75, 243/242 | | Commas: 56/55, 77/75, 243/242 |
|
| |
|
| POTE generator: ~11/9 = 350.565 | | POTE generator: ~11/9 = 350.565 |
| | |
| Mapping generator: ~11/9 | | Mapping generator: ~11/9 |
|
| |
|
| Map: [<1 1 0 -1 2|, <0 2 8 13 5|] | | Map: [<1 1 0 -1 2|, <0 2 8 13 5|] |
| | |
| EDOs: 17c, 24, 65c, 89cd | | EDOs: 17c, 24, 65c, 89cd |
| | |
| Badness: 0.0362 | | Badness: 0.0362 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 56/55, 66/65, 77/75, 243/242 | | Commas: 56/55, 66/65, 77/75, 243/242 |
|
| |
|
| POTE generator: ~11/9 = 350.745 | | POTE generator: ~11/9 = 350.745 |
| | |
| Mapping generator: ~11/9 | | Mapping generator: ~11/9 |
|
| |
|
| Map: [<1 1 0 -1 2 4|, <0 2 8 13 5 -1|] | | Map: [<1 1 0 -1 2 4|, <0 2 8 13 5 -1|] |
| | |
| EDOs: 17c, 24, 41c, 65c | | EDOs: 17c, 24, 41c, 65c |
| | |
| Badness: 0.0287 | | Badness: 0.0287 |
|
| |
|
| =Orphic= | | =Orphic= |
| Commas: 81/80, 5898240/5764801 | | Commas: 81/80, 5898240/5764801 |
|
| |
|
| POTE generator: ~7/6 = 275.794 | | POTE generator: ~7/6 = 275.794 |
| | |
| Mapping generator: ~343/288 | | Mapping generator: ~343/288 |
|
| |
|
| Map: [<2 1 -4 4|, <0 4 16 3|] | | Map: [<2 1 -4 4|, <0 4 16 3|] |
| | |
| Wedgie: <<8 32 6 32 -13 -76|| | | Wedgie: <<8 32 6 32 -13 -76|| |
| | |
| EDOs: 26, 74, 174bd, 248bd | | EDOs: 26, 74, 174bd, 248bd |
| | |
| Badness: 0.2588 | | Badness: 0.2588 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 81/80, 99/98, 73728/73205 | | Commas: 81/80, 99/98, 73728/73205 |
|
| |
|
| POTE generator: ~7/6 = 275.762 | | POTE generator: ~7/6 = 275.762 |
| | |
| Mapping generator: ~77/64 | | Mapping generator: ~77/64 |
|
| |
|
| Map: [<2 1 -4 4 8|, <0 4 16 3 -2|] | | Map: [<2 1 -4 4 8|, <0 4 16 3 -2|] |
| | |
| EDOs: 26, 48c, 74, 248bd, 322bd | | EDOs: 26, 48c, 74, 248bd, 322bd |
| | |
| Badness: 0.1015 | | Badness: 0.1015 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 81/80, 99/98, 144/143, 2200/2197 | | Commas: 81/80, 99/98, 144/143, 2200/2197 |
|
| |
|
| POTE generator: ~7/6 = 275.774 | | POTE generator: ~7/6 = 275.774 |
| | |
| Mapping generator: ~63/52 | | Mapping generator: ~63/52 |
|
| |
|
| Map: [<2 1 -4 4 8 2|, <0 4 16 3 -2 10|] | | Map: [<2 1 -4 4 8 2|, <0 4 16 3 -2 10|] |
| | |
| EDOs: 26, 48c, 74, 174bd, 248bd, 322bd | | EDOs: 26, 48c, 74, 174bd, 248bd, 322bd |
| | |
| Badness: 0.0535 | | Badness: 0.0535 |
|
| |
|
| =Mothra= | | =Mothra= |
| [[Comma]]s: 81/80, 1029/1024 | | [[Comma|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|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. In the 2.3.7-limit, mothra is identical to [[Slendric|slendric]]. |
|
| |
|
| 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. In the 2.3.7-limit, mothra is identical to [[Slendric|slendric]].
| |
| Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes. | | Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes. |
|
| |
|
| [[7-limit|7]] and [[9-limit]] minimax 1/4 comma | | [[7-limit|7]] and [[9-limit|9-limit]] minimax 1/4 comma |
| | |
| [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>] | | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>] |
| [[Eigenmonzo]]s: 2, 5
| |
|
| |
|
| [[POTE tuning|POTE generator]]: ~8/7 = 232.193 | | [[Eigenmonzo|Eigenmonzo]]s: 2, 5 |
| | |
| | [[POTE_tuning|POTE generator]]: ~8/7 = 232.193 |
| | |
| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
|
| |
|
| Line 844: |
Line 1,098: |
|
| |
|
| 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|| | | [[generator|Generator]]s: 2, 8/7 |
| | |
| | [[wedgie|Wedgie]]: <<3 12 -1 12 -10 -36|| |
| | |
| EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]] | | EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]] |
| [[Badness]]: 0.0371
| |
|
| |
|
| ==11-limit== | | [[Badness|Badness]]: 0.0371 |
| [[Comma]]s: 81/80, 99/98, 385/384 | | |
| | ==11-limit== |
| | [[Comma|Comma]]s: 81/80, 99/98, 385/384 |
|
| |
|
| POTE generator: ~8/7 = 232.031 | | POTE generator: ~8/7 = 232.031 |
| | |
| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
|
| |
|
| Map: [<1 1 0 3 5|, <0 3 12 -1 -8|] | | Map: [<1 1 0 3 5|, <0 3 12 -1 -8|] |
| | |
| EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]] | | EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]] |
| [[Badness]]: 0.0256
| |
|
| |
|
| ==13-limit== | | [[Badness|Badness]]: 0.0256 |
| | |
| | ==13-limit== |
| Commas: 81/80, 99/98, 105/104, 144/143 | | Commas: 81/80, 99/98, 105/104, 144/143 |
|
| |
|
| POTE generator: ~8/7 = 231.811 | | POTE generator: ~8/7 = 231.811 |
| | |
| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
|
| |
|
| Map: [<1 1 0 3 5 1|, <0 3 12 -1 -8 14|] | | Map: [<1 1 0 3 5 1|, <0 3 12 -1 -8 14|] |
| | |
| EDOs: 5, 26, 31, 57, 88 | | EDOs: 5, 26, 31, 57, 88 |
| | |
| Badness: 0.0240 | | Badness: 0.0240 |
|
| |
|
| ==Cynder== | | ==Cynder== |
| Commas: 45/44, 81/80, 1029/1024 | | Commas: 45/44, 81/80, 1029/1024 |
|
| |
|
| POTE generator: ~8/7 = 231.317 | | POTE generator: ~8/7 = 231.317 |
| | |
| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
|
| |
|
| Map: [<1 1 0 3 0|, <0 3 12 -1 18|] | | Map: [<1 1 0 3 0|, <0 3 12 -1 18|] |
| | |
| EDOs: 26, 57e, 83bce | | EDOs: 26, 57e, 83bce |
| | |
| Badness: 0.0557 | | Badness: 0.0557 |
|
| |
|
| ===13-limit=== | | ===13-limit=== |
| Commas: 45/44, 78/77, 81/80, 640/637 | | Commas: 45/44, 78/77, 81/80, 640/637 |
|
| |
|
| POTE generator: ~8/7 = 231.293 | | POTE generator: ~8/7 = 231.293 |
| | |
| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
|
| |
|
| Map: [<1 1 0 3 0 1|, <0 3 12 -1 18 14|] | | Map: [<1 1 0 3 0 1|, <0 3 12 -1 18 14|] |
| | |
| EDOs: 26, 57e, 83bce | | EDOs: 26, 57e, 83bce |
| | |
| Badness: 0.0341 | | Badness: 0.0341 |
|
| |
|
| ==Mosura== | | ==Mosura== |
| Commas: 81/80, 176/175, 1029/1024 | | Commas: 81/80, 176/175, 1029/1024 |
|
| |
|
| POTE generator: ~8/7 = 232.419 | | POTE generator: ~8/7 = 232.419 |
| | |
| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
|
| |
|
| Map: [<1 1 0 3 -1|, <0 3 12 -1 23|] | | Map: [<1 1 0 3 -1|, <0 3 12 -1 23|] |
| | |
| EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce | | EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce |
| | |
| Badness: 0.0313 | | Badness: 0.0313 |
|
| |
|
| ===13-limit=== | | ===13-limit=== |
| Commas: 81/80, 144/143, 176/175, 1029/1024 | | Commas: 81/80, 144/143, 176/175, 1029/1024 |
|
| |
|
| POTE generator: ~8/7 = 232.640 | | POTE generator: ~8/7 = 232.640 |
| | |
| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
|
| |
|
| Map: [<1 1 0 3 -1 7|, <0 3 12 -1 23 -17|] | | Map: [<1 1 0 3 -1 7|, <0 3 12 -1 23 -17|] |
| | |
| EDOs: 31, 67, 98 | | EDOs: 31, 67, 98 |
| | |
| Badness: 0.0369 | | Badness: 0.0369 |
|
| |
|
| =Squares= | | =Squares= |
| [[Comma]]s: 81/80, 2401/2400 | | [[Comma|Comma]]s: 81/80, 2401/2400 |
|
| |
|
| Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third ([[9_7|9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a 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|31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401. |
|
| |
|
| 7 and 9 limit minimax 1/4 comma | | 7 and 9 limit minimax 1/4 comma |
| | |
| [|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]]: ~9/7 = 425.942 | | [[Eigenmonzo|Eigenmonzo]]s: 2, 5 |
| | |
| | [[POTE_tuning|POTE generator]]: ~9/7 = 425.942 |
| | |
| Mapping generator: ~9/7 | | Mapping generator: ~9/7 |
|
| |
|
| Line 924: |
Line 1,203: |
|
| |
|
| Map: [<1 3 8 6|, <0 -4 -16 -9|] | | Map: [<1 3 8 6|, <0 -4 -16 -9|] |
| [[Generator]]s: 2, 9/7 | | |
| | [[generator|Generator]]s: 2, 9/7 |
| | |
| EDOs: [[14edo|14]], [[31edo|31]], [[262edo|262]], [[293edo|293]] | | EDOs: [[14edo|14]], [[31edo|31]], [[262edo|262]], [[293edo|293]] |
| [[Badness]]: 0.0460 | | |
| | [[Badness|Badness]]: 0.0460 |
|
| |
|
| Music: | | Music: |
| By [[Chris Vaisvil]]
| |
| [[http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3|Square 8]]
| |
|
| |
|
| ==11-limit== | | By [[Chris_Vaisvil|Chris Vaisvil]] |
| | |
| | [http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3 Square 8] |
| | |
| | ==11-limit== |
| Commas: 81/80, 99/98, 121/120 | | Commas: 81/80, 99/98, 121/120 |
|
| |
|
| POTE generator: ~9/7 = 425.957 | | POTE generator: ~9/7 = 425.957 |
| | |
| Mapping generator: ~9/7 | | Mapping generator: ~9/7 |
|
| |
|
| Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|] | | Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|] |
| | |
| EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]] | | EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]] |
| [[Badness]]: 0.0216
| |
|
| |
|
| ==13-limit== | | [[Badness|Badness]]: 0.0216 |
| | |
| | ==13-limit== |
| Commas: 81/80, 99/98, 121/120, 66/65 | | Commas: 81/80, 99/98, 121/120, 66/65 |
|
| |
|
| POTE generator: ~9/7 = 425.550 | | POTE generator: ~9/7 = 425.550 |
| | |
| Mapping generator: ~9/7 | | Mapping generator: ~9/7 |
|
| |
|
| Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|] | | Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|] |
| | |
| EDOs: 17c, 31, 79cf, 110cef, 141cef | | EDOs: 17c, 31, 79cf, 110cef, 141cef |
| [[Badness]]: 0.0255
| |
|
| |
|
| ==Agora== | | [[Badness|Badness]]: 0.0255 |
| | |
| | ==Agora== |
| Commas: 81/80, 99/98, 105/104, 121/120 | | Commas: 81/80, 99/98, 105/104, 121/120 |
|
| |
|
| POTE generator: ~9/7 = 426.276 | | POTE generator: ~9/7 = 426.276 |
| | |
| Mapping generator: ~9/7 | | Mapping generator: ~9/7 |
|
| |
|
| Map: [<1 3 8 6 7 14|, <0 -4 -16 -9 -10 -29|] | | Map: [<1 3 8 6 7 14|, <0 -4 -16 -9 -10 -29|] |
| | |
| EDOs: 31, 45ef, 76e | | EDOs: 31, 45ef, 76e |
| | |
| Badness: 0.0245 | | Badness: 0.0245 |
|
| |
|
| =Cuboctahedra= | | =Cuboctahedra= |
| ==11-limit== | | |
| [[Comma]]s: 81/80, 385/384, 1375/1372 | | ==11-limit== |
| | [[Comma|Comma]]s: 81/80, 385/384, 1375/1372 |
| | |
| | [[POTE_tuning|POTE generator]]: ~9/7 = 425.993 |
|
| |
|
| [[POTE tuning|POTE generator]]: ~9/7 = 425.993
| |
| Mapping generator: ~9/7 | | Mapping generator: ~9/7 |
|
| |
|
| Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|] | | Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|] |
| | |
| EDOs: [[14edo|14]], [[31edo|31]], [[45edo|45]], [[200edo|200]] | | EDOs: [[14edo|14]], [[31edo|31]], [[45edo|45]], [[200edo|200]] |
| [[Badness]]: 0.0568
| |
|
| |
|
| =Liese= | | [[Badness|Badness]]: 0.0568 |
| <span style="display: block; text-align: right;">[[xenharmonie/Liese|Deutsch]]
| | |
| </span>
| | =Liese= |
| [[Comma]]s: 81/80, 686/675 | | <span style="display: block; text-align: right;">[[:de:Liese Deutsch]]</span> |
| | |
| | [[Comma|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|74edo]] makes for a good liese tuning, though [[19edo|19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55. |
|
| |
|
| 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]]: ~10/7 = 632.406 | | [[Eigenmonzo|Eigenmonzo]]s: 2, 5 |
| | |
| | [[POTE_tuning|POTE generator]]: ~10/7 = 632.406 |
| | |
| Mapping generator: ~10/7 | | Mapping generator: ~10/7 |
|
| |
|
| Line 990: |
Line 1,290: |
|
| |
|
| Map: [<1 0 -4 -3|, <0 3 12 11|] | | Map: [<1 0 -4 -3|, <0 3 12 11|] |
| [[Generator]]s: 2, 10/7 | | |
| | [[generator|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
| |
|
| |
|
| ==Liesel== | | [[Badness|Badness]]: 0.0467 |
| | |
| | ==Liesel== |
| Commas: 56/55, 81/80, 540/539 | | Commas: 56/55, 81/80, 540/539 |
|
| |
|
| POTE generator: ~10/7 = 633.073 | | POTE generator: ~10/7 = 633.073 |
| | |
| Mapping generator: ~10/7 | | Mapping generator: ~10/7 |
|
| |
|
| Map: [<1 0 -4 -3 4|, <0 3 12 11 -1|] | | Map: [<1 0 -4 -3 4|, <0 3 12 11 -1|] |
| | |
| EDOs: 17c, 19, 36, 91ce | | EDOs: 17c, 19, 36, 91ce |
| | |
| Badness: 0.0407 | | Badness: 0.0407 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Liesel is a very natural 13-limit tuning, given the generator is so near 13/9. | | Liesel is a very natural 13-limit tuning, given the generator is so near 13/9. |
|
| |
|
| Line 1,010: |
Line 1,316: |
|
| |
|
| POTE generator: ~10/7 = ~13/9 = 633.042 | | POTE generator: ~10/7 = ~13/9 = 633.042 |
| | |
| Mapping generator: ~10/7 | | Mapping generator: ~10/7 |
|
| |
|
| Map: [<1 0 -4 -3 4 0|, <0 3 12 11 -1 7|] | | Map: [<1 0 -4 -3 4 0|, <0 3 12 11 -1 7|] |
| | |
| EDOs: 17c, 19, 36, 91cef | | EDOs: 17c, 19, 36, 91cef |
| | |
| Badness: 0.0273 | | Badness: 0.0273 |
|
| |
|
| ==Elisa== | | ==Elisa== |
| Commas: 77/75, 81/80, 99/98 | | Commas: 77/75, 81/80, 99/98 |
|
| |
|
| POTE generator: ~10/7 = 633.061 | | POTE generator: ~10/7 = 633.061 |
| | |
| Mapping generator: ~10/7 | | Mapping generator: ~10/7 |
|
| |
|
| Map: [<1 0 -4 -3 -5|, <0 3 12 11 16|] | | Map: [<1 0 -4 -3 -5|, <0 3 12 11 16|] |
| | |
| EDOs: 19e, 36e | | EDOs: 19e, 36e |
| | |
| Badness: 0.0416 | | Badness: 0.0416 |
|
| |
|
| ==Lisa== | | ==Lisa== |
| Commas: 45/44, 81/80, 343/330 | | Commas: 45/44, 81/80, 343/330 |
|
| |
|
| POTE generator: ~10/7 = 631.370 | | POTE generator: ~10/7 = 631.370 |
| | |
| Mapping generator: ~10/7 | | Mapping generator: ~10/7 |
|
| |
|
| Map: [<1 0 -4 -3 -6|, <0 3 12 11 18|] | | Map: [<1 0 -4 -3 -6|, <0 3 12 11 18|] |
| | |
| EDOs: 19 | | EDOs: 19 |
| | |
| Badness: 0.0548 | | Badness: 0.0548 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 45/44, 81/80, 91/88, 147/143 | | Commas: 45/44, 81/80, 91/88, 147/143 |
|
| |
|
| POTE generator: ~10/7 = 631.221 | | POTE generator: ~10/7 = 631.221 |
| | |
| Mapping generator: ~10/7 | | Mapping generator: ~10/7 |
|
| |
|
| Map: [<1 0 -4 -3 -6 0|, <0 3 12 11 18 7|] | | Map: [<1 0 -4 -3 -6 0|, <0 3 12 11 18 7|] |
| | |
| EDOs: 19 | | EDOs: 19 |
| | |
| Badness: 0.0361 | | Badness: 0.0361 |
|
| |
|
| =Jerome= | | =Jerome= |
| Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size. | | Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size. |
|
| |
|
| Line 1,052: |
Line 1,370: |
|
| |
|
| POTE generator: ~54/49 = 139.343 | | POTE generator: ~54/49 = 139.343 |
| | |
| Mapping generator: ~54/49 | | Mapping generator: ~54/49 |
|
| |
|
| Map: [<1 1 0 2|, <0 5 20 7|] | | Map: [<1 1 0 2|, <0 5 20 7|] |
| | |
| Wedgie: <<5 30 7 20 -3 -40|| | | Wedgie: <<5 30 7 20 -3 -40|| |
| | |
| EDOs: 8, 9, 17, 26, 43, 112 | | EDOs: 8, 9, 17, 26, 43, 112 |
| | |
| Badness: 0.1087 | | Badness: 0.1087 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 81/80, 99/98, 864/847 | | Commas: 81/80, 99/98, 864/847 |
|
| |
|
| POTE generator: ~12/11 = 139.428 | | POTE generator: ~12/11 = 139.428 |
| | |
| Mapping generator: ~12/11 | | Mapping generator: ~12/11 |
|
| |
|
| Map: [<1 1 0 2 3|, <0 5 20 7 4|] | | Map: [<1 1 0 2 3|, <0 5 20 7 4|] |
| | |
| EDOs: 8, 9, 17, 26, 43, 241 | | EDOs: 8, 9, 17, 26, 43, 241 |
| | |
| Badness: 0.0479 | | Badness: 0.0479 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 77/78, 81/80, 99/98, 144/143 | | Commas: 77/78, 81/80, 99/98, 144/143 |
|
| |
|
| POTE generator: ~13/12 = 139.387 | | POTE generator: ~13/12 = 139.387 |
| | |
| Mapping generator: ~12/11 | | Mapping generator: ~12/11 |
|
| |
|
| Map: [<1 1 0 2 3 3|, <0 5 20 7 4 6|] | | Map: [<1 1 0 2 3 3|, <0 5 20 7 4 6|] |
| | |
| EDOs: 8, 9, 17, 26, 43, 155, 198 | | EDOs: 8, 9, 17, 26, 43, 155, 198 |
| | |
| Badness: 0.0293 | | Badness: 0.0293 |
|
| |
|
| ==17-limit== | | ==17-limit== |
| Commas: 78/77, 81/80, 99/98, 144/143, 189/187 | | Commas: 78/77, 81/80, 99/98, 144/143, 189/187 |
|
| |
|
| POTE generator: ~13/12 = 139.362 | | POTE generator: ~13/12 = 139.362 |
| | |
| Mapping generator: ~12/11 | | Mapping generator: ~12/11 |
|
| |
|
| Map: [<1 1 0 2 3 3 2|, <0 5 20 7 4 6 18|] | | Map: [<1 1 0 2 3 3 2|, <0 5 20 7 4 6 18|] |
| | |
| EDOs: 8, 9, 17, 26, 43, 155 | | EDOs: 8, 9, 17, 26, 43, 155 |
| | |
| Badness: 0.0209 | | Badness: 0.0209 |
|
| |
|
| =Meanmag= | | =Meanmag= |
| Commas: 81/80, 3125/3072 | | Commas: 81/80, 3125/3072 |
|
| |
|
| POTE generator: ~8/7 = 238.396 | | POTE generator: ~8/7 = 238.396 |
| | |
| Mapping generator: ~7 | | Mapping generator: ~7 |
|
| |
|
| Map: [<19 30 44 0|, <0 0 0 1|] | | Map: [<19 30 44 0|, <0 0 0 1|] |
| | |
| Wedgie: <<0 0 19 0 30 44|| | | Wedgie: <<0 0 19 0 30 44|| |
| | |
| EDOs: 19, 57, 76, 171bcd | | EDOs: 19, 57, 76, 171bcd |
| | |
| Badness: 0.0770 | | Badness: 0.0770 |
|
| |
|
| =Undevigintone= | | =Undevigintone= |
| Commas: 49/48, 81/80, 126/125 | | Commas: 49/48, 81/80, 126/125 |
|
| |
|
| POTE generator: ~11/8 = 538.047 | | POTE generator: ~11/8 = 538.047 |
| | |
| Mapping generator: ~11 | | Mapping generator: ~11 |
|
| |
|
| Map: [<19 30 44 53 0|, <0 0 0 0 1|] | | Map: [<19 30 44 53 0|, <0 0 0 0 1|] |
| | |
| EDOs: 19, 38d | | EDOs: 19, 38d |
| | |
| Badness: 0.0364 | | Badness: 0.0364 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 49/48, 65/64, 81/80, 126/125 | | Commas: 49/48, 65/64, 81/80, 126/125 |
|
| |
|
| Line 1,116: |
Line 1,454: |
|
| |
|
| Map: [<19 30 44 53 0 70|, <0 0 0 0 1 0|] | | Map: [<19 30 44 53 0 70|, <0 0 0 0 1 0|] |
| | |
| EDOs: 19, 38d | | EDOs: 19, 38d |
| Badness: 0.0229</pre></div> | | |
| <h4>Original HTML content:</h4>
| | Badness: 0.0229 |
| <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><span style="display: block; text-align: right;"><!-- ws:start:WikiTextTocRule:184:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:184 --><!-- ws:start:WikiTextTocRule:185: --><div style="margin-left: 2em;"><a href="#x-Seven limit children">Seven limit children</a></div>
| | [[Category:catalog]] |
| <!-- ws:end:WikiTextTocRule:185 --><!-- ws:start:WikiTextTocRule:186: --><div style="margin-left: 1em;"><a href="#Septimal meantone">Septimal meantone</a></div>
| | [[Category:family]] |
| <!-- ws:end:WikiTextTocRule:186 --><!-- ws:start:WikiTextTocRule:187: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Bimeantone">Bimeantone</a></div>
| | [[Category:listen]] |
| <!-- ws:end:WikiTextTocRule:187 --><!-- ws:start:WikiTextTocRule:188: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Bimeantone-13-limit">13-limit</a></div>
| | [[Category:meantone]] |
| <!-- ws:end:WikiTextTocRule:188 --><!-- ws:start:WikiTextTocRule:189: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens">Unidecimal meantone aka Huygens</a></div>
| | [[Category:rank_2]] |
| <!-- ws:end:WikiTextTocRule:189 --><!-- ws:start:WikiTextTocRule:190: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone">Tridecimal meantone</a></div>
| | [[Category:theory]] |
| <!-- ws:end:WikiTextTocRule:190 --><!-- ws:start:WikiTextTocRule:191: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone">Grosstone</a></div>
| |
| <!-- ws:end:WikiTextTocRule:191 --><!-- ws:start:WikiTextTocRule:192: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Meridetone">Meridetone</a></div>
| |
| <!-- ws:end:WikiTextTocRule:192 --><!-- ws:start:WikiTextTocRule:193: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Hemimeantone">Hemimeantone</a></div>
| |
| <!-- ws:end:WikiTextTocRule:193 --><!-- ws:start:WikiTextTocRule:194: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanpop">Meanpop</a></div>
| |
| <!-- ws:end:WikiTextTocRule:194 --><!-- ws:start:WikiTextTocRule:195: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanpop-13-limit Meanpop">13-limit Meanpop</a></div>
| |
| <!-- ws:end:WikiTextTocRule:195 --><!-- ws:start:WikiTextTocRule:196: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanpop-Meanplop">Meanplop</a></div>
| |
| <!-- ws:end:WikiTextTocRule:196 --><!-- ws:start:WikiTextTocRule:197: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanenneadecal">Meanenneadecal</a></div>
| |
| <!-- ws:end:WikiTextTocRule:197 --><!-- ws:start:WikiTextTocRule:198: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanenneadecal-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:198 --><!-- ws:start:WikiTextTocRule:199: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanenneadecal-Vincenzo">Vincenzo</a></div>
| |
| <!-- ws:end:WikiTextTocRule:199 --><!-- ws:start:WikiTextTocRule:200: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanundeci">Meanundeci</a></div>
| |
| <!-- ws:end:WikiTextTocRule:200 --><!-- ws:start:WikiTextTocRule:201: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanundeci-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:201 --><!-- ws:start:WikiTextTocRule:202: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanundec">Meanundec</a></div>
| |
| <!-- ws:end:WikiTextTocRule:202 --><!-- ws:start:WikiTextTocRule:203: --><div style="margin-left: 1em;"><a href="#Flattone">Flattone</a></div>
| |
| <!-- ws:end:WikiTextTocRule:203 --><!-- ws:start:WikiTextTocRule:204: --><div style="margin-left: 2em;"><a href="#Flattone-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:204 --><!-- ws:start:WikiTextTocRule:205: --><div style="margin-left: 2em;"><a href="#Flattone-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:205 --><!-- ws:start:WikiTextTocRule:206: --><div style="margin-left: 1em;"><a href="#Dominant">Dominant</a></div>
| |
| <!-- ws:end:WikiTextTocRule:206 --><!-- ws:start:WikiTextTocRule:207: --><div style="margin-left: 2em;"><a href="#Dominant-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:207 --><!-- ws:start:WikiTextTocRule:208: --><div style="margin-left: 2em;"><a href="#Dominant-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:208 --><!-- ws:start:WikiTextTocRule:209: --><div style="margin-left: 2em;"><a href="#Dominant-Dominion">Dominion</a></div>
| |
| <!-- ws:end:WikiTextTocRule:209 --><!-- ws:start:WikiTextTocRule:210: --><div style="margin-left: 2em;"><a href="#Dominant-Domineering">Domineering</a></div>
| |
| <!-- ws:end:WikiTextTocRule:210 --><!-- ws:start:WikiTextTocRule:211: --><div style="margin-left: 2em;"><a href="#Dominant-Domination">Domination</a></div>
| |
| <!-- ws:end:WikiTextTocRule:211 --><!-- ws:start:WikiTextTocRule:212: --><div style="margin-left: 3em;"><a href="#Dominant-Domination-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:212 --><!-- ws:start:WikiTextTocRule:213: --><div style="margin-left: 2em;"><a href="#Dominant-Twelve">Twelve</a></div>
| |
| <!-- ws:end:WikiTextTocRule:213 --><!-- ws:start:WikiTextTocRule:214: --><div style="margin-left: 2em;"><a href="#Dominant-Arnold">Arnold</a></div>
| |
| <!-- ws:end:WikiTextTocRule:214 --><!-- ws:start:WikiTextTocRule:215: --><div style="margin-left: 2em;"><a href="#Dominant-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:215 --><!-- ws:start:WikiTextTocRule:216: --><div style="margin-left: 2em;"><a href="#Dominant-Dominatrix">Dominatrix</a></div>
| |
| <!-- ws:end:WikiTextTocRule:216 --><!-- ws:start:WikiTextTocRule:217: --><div style="margin-left: 1em;"><a href="#Sharptone">Sharptone</a></div>
| |
| <!-- ws:end:WikiTextTocRule:217 --><!-- ws:start:WikiTextTocRule:218: --><div style="margin-left: 1em;"><a href="#Meansept">Meansept</a></div>
| |
| <!-- ws:end:WikiTextTocRule:218 --><!-- ws:start:WikiTextTocRule:219: --><div style="margin-left: 2em;"><a href="#Meansept-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:219 --><!-- ws:start:WikiTextTocRule:220: --><div style="margin-left: 1em;"><a href="#Supermean">Supermean</a></div>
| |
| <!-- ws:end:WikiTextTocRule:220 --><!-- ws:start:WikiTextTocRule:221: --><div style="margin-left: 2em;"><a href="#Supermean-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:221 --><!-- ws:start:WikiTextTocRule:222: --><div style="margin-left: 2em;"><a href="#Supermean-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:222 --><!-- ws:start:WikiTextTocRule:223: --><div style="margin-left: 1em;"><a href="#Injera">Injera</a></div>
| |
| <!-- ws:end:WikiTextTocRule:223 --><!-- ws:start:WikiTextTocRule:224: --><div style="margin-left: 2em;"><a href="#Injera-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:224 --><!-- ws:start:WikiTextTocRule:225: --><div style="margin-left: 2em;"><a href="#Injera-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:225 --><!-- ws:start:WikiTextTocRule:226: --><div style="margin-left: 2em;"><a href="#Injera-Enjera">Enjera</a></div>
| |
| <!-- ws:end:WikiTextTocRule:226 --><!-- ws:start:WikiTextTocRule:227: --><div style="margin-left: 2em;"><a href="#Injera-Injerous">Injerous</a></div>
| |
| <!-- ws:end:WikiTextTocRule:227 --><!-- ws:start:WikiTextTocRule:228: --><div style="margin-left: 2em;"><a href="#Injera-Lahoh">Lahoh</a></div>
| |
| <!-- ws:end:WikiTextTocRule:228 --><!-- ws:start:WikiTextTocRule:229: --><div style="margin-left: 1em;"><a href="#Godzilla">Godzilla</a></div>
| |
| <!-- ws:end:WikiTextTocRule:229 --><!-- ws:start:WikiTextTocRule:230: --><div style="margin-left: 2em;"><a href="#Godzilla-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:230 --><!-- ws:start:WikiTextTocRule:231: --><div style="margin-left: 2em;"><a href="#Godzilla-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:231 --><!-- ws:start:WikiTextTocRule:232: --><div style="margin-left: 2em;"><a href="#Godzilla-Semafour">Semafour</a></div>
| |
| <!-- ws:end:WikiTextTocRule:232 --><!-- ws:start:WikiTextTocRule:233: --><div style="margin-left: 2em;"><a href="#Godzilla-Varan">Varan</a></div>
| |
| <!-- ws:end:WikiTextTocRule:233 --><!-- ws:start:WikiTextTocRule:234: --><div style="margin-left: 3em;"><a href="#Godzilla-Varan-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:234 --><!-- ws:start:WikiTextTocRule:235: --><div style="margin-left: 2em;"><a href="#Godzilla-Baragon">Baragon</a></div>
| |
| <!-- ws:end:WikiTextTocRule:235 --><!-- ws:start:WikiTextTocRule:236: --><div style="margin-left: 2em;"><a href="#Godzilla-Music">Music</a></div>
| |
| <!-- ws:end:WikiTextTocRule:236 --><!-- ws:start:WikiTextTocRule:237: --><div style="margin-left: 1em;"><a href="#Mohajira">Mohajira</a></div>
| |
| <!-- ws:end:WikiTextTocRule:237 --><!-- ws:start:WikiTextTocRule:238: --><div style="margin-left: 2em;"><a href="#Mohajira-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:238 --><!-- ws:start:WikiTextTocRule:239: --><div style="margin-left: 2em;"><a href="#Mohajira-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:239 --><!-- ws:start:WikiTextTocRule:240: --><div style="margin-left: 1em;"><a href="#Ptolemy">Ptolemy</a></div>
| |
| <!-- ws:end:WikiTextTocRule:240 --><!-- ws:start:WikiTextTocRule:241: --><div style="margin-left: 2em;"><a href="#Ptolemy-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:241 --><!-- ws:start:WikiTextTocRule:242: --><div style="margin-left: 1em;"><a href="#Maqamic">Maqamic</a></div>
| |
| <!-- ws:end:WikiTextTocRule:242 --><!-- ws:start:WikiTextTocRule:243: --><div style="margin-left: 2em;"><a href="#Maqamic-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:243 --><!-- ws:start:WikiTextTocRule:244: --><div style="margin-left: 1em;"><a href="#Migration">Migration</a></div>
| |
| <!-- ws:end:WikiTextTocRule:244 --><!-- ws:start:WikiTextTocRule:245: --><div style="margin-left: 1em;"><a href="#Mohamaq">Mohamaq</a></div>
| |
| <!-- ws:end:WikiTextTocRule:245 --><!-- ws:start:WikiTextTocRule:246: --><div style="margin-left: 2em;"><a href="#Mohamaq-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:246 --><!-- ws:start:WikiTextTocRule:247: --><div style="margin-left: 2em;"><a href="#Mohamaq-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:247 --><!-- ws:start:WikiTextTocRule:248: --><div style="margin-left: 1em;"><a href="#Orphic">Orphic</a></div>
| |
| <!-- ws:end:WikiTextTocRule:248 --><!-- ws:start:WikiTextTocRule:249: --><div style="margin-left: 2em;"><a href="#Orphic-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:249 --><!-- ws:start:WikiTextTocRule:250: --><div style="margin-left: 2em;"><a href="#Orphic-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:250 --><!-- ws:start:WikiTextTocRule:251: --><div style="margin-left: 1em;"><a href="#Mothra">Mothra</a></div>
| |
| <!-- ws:end:WikiTextTocRule:251 --><!-- ws:start:WikiTextTocRule:252: --><div style="margin-left: 2em;"><a href="#Mothra-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:252 --><!-- ws:start:WikiTextTocRule:253: --><div style="margin-left: 2em;"><a href="#Mothra-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:253 --><!-- ws:start:WikiTextTocRule:254: --><div style="margin-left: 2em;"><a href="#Mothra-Cynder">Cynder</a></div>
| |
| <!-- ws:end:WikiTextTocRule:254 --><!-- ws:start:WikiTextTocRule:255: --><div style="margin-left: 3em;"><a href="#Mothra-Cynder-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:255 --><!-- ws:start:WikiTextTocRule:256: --><div style="margin-left: 2em;"><a href="#Mothra-Mosura">Mosura</a></div>
| |
| <!-- ws:end:WikiTextTocRule:256 --><!-- ws:start:WikiTextTocRule:257: --><div style="margin-left: 3em;"><a href="#Mothra-Mosura-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:257 --><!-- ws:start:WikiTextTocRule:258: --><div style="margin-left: 1em;"><a href="#Squares">Squares</a></div>
| |
| <!-- ws:end:WikiTextTocRule:258 --><!-- ws:start:WikiTextTocRule:259: --><div style="margin-left: 2em;"><a href="#Squares-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:259 --><!-- ws:start:WikiTextTocRule:260: --><div style="margin-left: 2em;"><a href="#Squares-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:260 --><!-- ws:start:WikiTextTocRule:261: --><div style="margin-left: 2em;"><a href="#Squares-Agora">Agora</a></div>
| |
| <!-- ws:end:WikiTextTocRule:261 --><!-- ws:start:WikiTextTocRule:262: --><div style="margin-left: 1em;"><a href="#Cuboctahedra">Cuboctahedra</a></div>
| |
| <!-- ws:end:WikiTextTocRule:262 --><!-- ws:start:WikiTextTocRule:263: --><div style="margin-left: 2em;"><a href="#Cuboctahedra-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:263 --><!-- ws:start:WikiTextTocRule:264: --><div style="margin-left: 1em;"><a href="#Liese">Liese</a></div>
| |
| <!-- ws:end:WikiTextTocRule:264 --><!-- ws:start:WikiTextTocRule:265: --><div style="margin-left: 2em;"><a href="#Liese-Liesel">Liesel</a></div>
| |
| <!-- ws:end:WikiTextTocRule:265 --><!-- ws:start:WikiTextTocRule:266: --><div style="margin-left: 2em;"><a href="#Liese-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:266 --><!-- ws:start:WikiTextTocRule:267: --><div style="margin-left: 2em;"><a href="#Liese-Elisa">Elisa</a></div>
| |
| <!-- ws:end:WikiTextTocRule:267 --><!-- ws:start:WikiTextTocRule:268: --><div style="margin-left: 2em;"><a href="#Liese-Lisa">Lisa</a></div>
| |
| <!-- ws:end:WikiTextTocRule:268 --><!-- ws:start:WikiTextTocRule:269: --><div style="margin-left: 2em;"><a href="#Liese-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:269 --><!-- ws:start:WikiTextTocRule:270: --><div style="margin-left: 1em;"><a href="#Jerome">Jerome</a></div>
| |
| <!-- ws:end:WikiTextTocRule:270 --><!-- ws:start:WikiTextTocRule:271: --><div style="margin-left: 2em;"><a href="#Jerome-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:271 --><!-- ws:start:WikiTextTocRule:272: --><div style="margin-left: 2em;"><a href="#Jerome-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:272 --><!-- ws:start:WikiTextTocRule:273: --><div style="margin-left: 2em;"><a href="#Jerome-17-limit">17-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:273 --><!-- ws:start:WikiTextTocRule:274: --><div style="margin-left: 1em;"><a href="#Meanmag">Meanmag</a></div>
| |
| <!-- ws:end:WikiTextTocRule:274 --><!-- ws:start:WikiTextTocRule:275: --><div style="margin-left: 1em;"><a href="#Undevigintone">Undevigintone</a></div>
| |
| <!-- ws:end:WikiTextTocRule:275 --><!-- ws:start:WikiTextTocRule:276: --><div style="margin-left: 2em;"><a href="#Undevigintone-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:276 --><!-- ws:start:WikiTextTocRule:277: --></div>
| |
| <!-- ws:end:WikiTextTocRule:277 --><br />
| |
| <a class="wiki_link" href="http://xenharmonie.wikispaces.com/mittelt%C3%B6nig">Deutsch</a><br />
| |
| </span><br />
| |
| The <a class="wiki_link" href="/5-limit">5-limit</a> parent <a class="wiki_link" href="/Comma">comma</a> of the <a class="wiki_link" href="/meantone">meantone</a> family is the Didymus or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.239<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Tuning%20Ranges%20of%20Regular%20Temperaments">valid range</a>: [685.714, 720.000] (7 to 5)<br />
| |
| nice range: [694.786, 701.955] (1/3 comma to Pythagorean)<br />
| |
| strict range: [694.786, 701.955]<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Map">Map</a>: [&lt;1 0 -4|, &lt;0 1 4|]<br />
| |
| EDOs (patent val edo list is complete): <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="/24edo">24</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/36edo">36</a>, <a class="wiki_link" href="/38edo">38</a>, <a class="wiki_link" href="/43edo">43</a>, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/55edo">55</a>, <a class="wiki_link" href="/57edo">57</a>, <a class="wiki_link" href="/62edo">62</a>, <a class="wiki_link" href="/67edo">67</a>, <a class="wiki_link" href="/69edo">69</a>, <a class="wiki_link" href="/74edo">74</a>, <a class="wiki_link" href="/76edo">76</a>, <a class="wiki_link" href="/81edo">81</a>, <a class="wiki_link" href="/86edo">86</a>, <a class="wiki_link" href="/88edo">88</a>, <a class="wiki_link" href="/93edo">93</a>, <a class="wiki_link" href="/98edo">98</a>, <a class="wiki_link" href="/100edo">100</a>, <a class="wiki_link" href="/105edo">105</a>, <a class="wiki_link" href="/117edo">117</a>, <a class="wiki_link" href="/129edo">129</a>, <a class="wiki_link" href="/212edo">212b</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.00736<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>
| |
| The <a class="wiki_link" href="/7-limit">7-limit</a> children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1&gt;, |-13 10 0 -1&gt;], flattone, with normal list [|-4 4 -1&gt;, |-17 9 0 1&gt;], dominant, with normal list [|-4 4 -1&gt;, |6 -2 0 -1&gt;], sharptone, with normal list [|-4 4 -1&gt;, |2 -3 0 1&gt;], injera, with normal list [|-4 4 -1&gt;, |-7 8 0 -2&gt;], mohajira, with normal list [|-4 4 -1&gt;, |-23 11 0 2&gt;], godzilla, with normal list [|-4 4 -1&gt;, |-4 -1 0 2&gt;], mothra, with normal list [|-4 4 -1&gt;, |-10 1 0 3&gt;], squares, with normal list [|-4 4 -1&gt;, |-3 9 0 -4&gt;], and liese, with normal list [|-4 4 -1&gt;, |-9 11 0 -3&gt;].<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Septimal meantone"></a><!-- ws:end:WikiTextHeadingRule:2 -->Septimal meantone</h1>
| |
| <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/septimal-mittelt%C3%B6nig">Deutsch</a><br />
| |
| </span><br />
| |
| <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Septimal_meantone_temperament" rel="nofollow">Wikipedia article</a><br />
| |
| <br />
| |
| The comma |-13 10 0 -1&gt; for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the <a class="wiki_link" href="/7_4">7/4</a> of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and <a class="wiki_link" href="/7_5">7/5</a>, C-F#, the tritone. The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for septimal meantone is &lt;&lt;1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and <a class="wiki_link" href="/31edo">31edo</a> is a good tuning for it.<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125<br />
| |
| <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 />
| |
| <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Tuning%20Ranges%20of%20Regular%20Temperaments">valid range</a>: [694.737, 700.000] (19 to 12)<br />
| |
| nice range: [694.786, 701.955]<br />
| |
| strict range: [694.786, 700.000]<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.495<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Map">Map</a>: [&lt;1 0 -4 -13|, &lt;0 1 4 10|]<br />
| |
| <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br />
| |
| <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 10 4 13 12||<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>, <a class="wiki_link" href="/143edo">143b</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0137<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Septimal meantone-Bimeantone"></a><!-- ws:end:WikiTextHeadingRule:4 -->Bimeantone</h2>
| |
| Commas: 81/80, 126/125, 245/242<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.016<br />
| |
| <br />
| |
| Map: [&lt;2 0 -8 -26 -31|, &lt;0 1 4 10 12|]<br />
| |
| EDOs: 12, 38d, 50<br />
| |
| Badness: 0.0381<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Septimal meantone-Bimeantone-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h3>
| |
| Commas: 81/80, 105/104, 126/125, 245/242<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 695.836<br />
| |
| <br />
| |
| Map: [&lt;2 0 -8 -26 -31 -40|, &lt;0 1 4 10 12 15|]<br />
| |
| EDOs: 12f, 50<br />
| |
| Badness: 0.0288<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Septimal meantone-Unidecimal meantone aka Huygens"></a><!-- ws:end:WikiTextHeadingRule:8 -->Unidecimal meantone aka Huygens</h2>
| |
| See also <a class="wiki_link" href="/Meantone%20vs%20meanpop">Meantone vs meanpop</a><br />
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 99/98<br />
| |
| <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 />
| |
| |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 />
| |
| valid range: [696.774, 700.000] (31 to 12)<br />
| |
| nice range: [691.202, 701.955]<br />
| |
| strict range: [696.774, 700.000]<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.967<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Algebraic%20generator">Algebraic generator</a>: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Map">Map</a>: [&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: <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">198be</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0170<br />
| |
| <br />
| |
| <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3" rel="nofollow">Twinkle canon – 74 edo</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a><br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone"></a><!-- ws:end:WikiTextHeadingRule:10 -->Tridecimal meantone</h3>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 66/65, 81/80, 99/98, 105/104<br />
| |
| <br />
| |
| valid range: 697.674 (43)<br />
| |
| nice range: [691.202, 701.955]<br />
| |
| strict range: 697.674<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.642<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|]<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="/267edo">267</a>, <a class="wiki_link" href="/298edo">298</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0180<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Grosstone"></a><!-- ws:end:WikiTextHeadingRule:12 -->Grosstone</h3>
| |
| Commas: 81/80, 99/98, 126/125, 144/143<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 697.264<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 -25 29|, &lt;0 1 4 10 18 -16|]<br />
| |
| EDOs: 12, 31, 43, 74<br />
| |
| Badness: 0.0259<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Meridetone"></a><!-- ws:end:WikiTextHeadingRule:14 -->Meridetone</h3>
| |
| Commas: 78/77, 81/80, 99/98, 126/125<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 697.529<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 -25 -39|, &lt;0 1 4 10 18 27|]<br />
| |
| EDOs: 43, 117df, 160bdf, 203bcdef<br />
| |
| Badness: 0.0264<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Hemimeantone"></a><!-- ws:end:WikiTextHeadingRule:16 -->Hemimeantone</h3>
| |
| Commas: 81/80, 99/98, 126/125, 169/168<br />
| |
| <br />
| |
| POTE generator: ~52/45 = 250.304<br />
| |
| Mapping generator: ~26/15<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 -25 -5|, &lt;0 2 8 20 36 11|]<br />
| |
| EDOs: 43, 62, 167bef, 229bef<br />
| |
| Badness: 0.0314<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Septimal meantone-Meanpop"></a><!-- ws:end:WikiTextHeadingRule:18 -->Meanpop</h2>
| |
| See also <a class="wiki_link" href="/Meantone%20vs%20meanpop">Meantone vs meanpop</a><br />
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 385/384<br />
| |
| <br />
| |
| <a class="wiki_link" href="/11-limit">11-limit</a> <a class="wiki_link" href="/minimax">minimax</a> 1/4 comma<br />
| |
| [|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,<br />
| |
| |-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 />
| |
| valid range: [694.737, 696.774] (19 to 31)<br />
| |
| nice range: [691.202, 701.955]<br />
| |
| strict range: [694.737, 696.774]<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.434<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| <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.<br />
| |
| <br />
| |
| <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/607" rel="nofollow" target="_blank">Scott Joplin's &quot;The Entertainer&quot; tuned into meanpop</a><br />
| |
| <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: <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="/112edo">112</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0215<br />
| |
| <br />
| |
| <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3" rel="nofollow">Twinkle canon – 50 edo</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a><br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Septimal meantone-Meanpop-13-limit Meanpop"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit Meanpop</h3>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 105/104, 144/143, 196/195<br />
| |
| <br />
| |
| valid range: [694.737, 696.774] (19 to 31)<br />
| |
| nice range: [691.202, 701.955]<br />
| |
| strict range: [694.737, 696.774]<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 696.211<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]<br />
| |
| EDOS: <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">131bd</a>, <a class="wiki_link" href="/212edo">212bdf</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0209<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Septimal meantone-Meanpop-Meanplop"></a><!-- ws:end:WikiTextHeadingRule:22 -->Meanplop</h3>
| |
| Commas: 65/64, 78/77, 81/80, 91/90<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 696.202<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 24 10|, &lt;0 1 4 10 -13 -4|]<br />
| |
| EDOs: 12e, 19, 31f, 50f<br />
| |
| Badness: 0.0277<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Septimal meantone-Meanenneadecal"></a><!-- ws:end:WikiTextHeadingRule:24 -->Meanenneadecal</h2>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 81/80<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.250<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]<br />
| |
| 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">31e</a>, <a class="wiki_link" href="/50edo">50e</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0214<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Septimal meantone-Meanenneadecal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h3>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 78/77, 81/80<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.146<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]<br />
| |
| EDOs: <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31e</a>, <a class="wiki_link" href="/50edo">50e</a>]<br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0212<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Septimal meantone-Meanenneadecal-Vincenzo"></a><!-- ws:end:WikiTextHeadingRule:28 -->Vincenzo</h3>
| |
| Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22<br />
| |
| <br />
| |
| POTE generator: ~3/2<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 ... |, &lt;0 1 4 10 6 -4 -5 -3 -6|]<br />
| |
| EDOs: 12<br />
| |
| Badness:<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Septimal meantone-Meanundeci"></a><!-- ws:end:WikiTextHeadingRule:30 -->Meanundeci</h2>
| |
| Commas: 33/32, 55/54, 77/75<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 694.689<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 5|, &lt;0 1 4 10 -1|]<br />
| |
| EDOs: 12e, 19e<br />
| |
| Badness: 0.0315<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Septimal meantone-Meanundeci-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h3>
| |
| Commas: 33/32, 55/54, 77/75, 729/728<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 694.764<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 5 10|, &lt;0 1 4 10 -1 -4|]<br />
| |
| EDOs: 12e, 19e<br />
| |
| Badness: 0.0263<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Septimal meantone-Meanundec"></a><!-- ws:end:WikiTextHeadingRule:34 -->Meanundec</h2>
| |
| Commas: 27/26, 40/39, 45/44, 56/55<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 697.254<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -13 -6 -1|, &lt;0 1 4 10 6 3|]<br />
| |
| EDOS: 12f, 19f, 31ef<br />
| |
| Badness: 0.0242<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:36:&lt;h1&gt; --><h1 id="toc18"><a name="Flattone"></a><!-- ws:end:WikiTextHeadingRule:36 -->Flattone</h1>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 525/512<br />
| |
| <br />
| |
| The <a class="wiki_link" href="/wedgie">wedgie</a> for flattone is &lt;&lt;1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that <a class="wiki_link" href="/7_4">7/4</a> is a diminished seventh interval. Other intervals are <a class="wiki_link" href="/7_6">7/6</a>, a diminished third, and <a class="wiki_link" href="/7_5">7/5</a>, a doubly diminshed fifth. Good tunings for flattone are <a class="wiki_link" href="/26edo">26edo</a>, <a class="wiki_link" href="/45edo">45edo</a> and <a class="wiki_link" href="/64edo">64edo</a>.<br />
| |
| <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 />
| |
| |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 />
| |
| <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 />
| |
| |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 />
| |
| valid range: [692.308, 694.737] (26 to 19)<br />
| |
| nice range: [692.353, 701.955]<br />
| |
| strict range: [692.353, 694.737]<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 693.779<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 17|, &lt;0 1 4 -9|]<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: <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><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0386<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Flattone-11-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->11-limit</h2>
| |
| Commas: 45/44, 81/80, 385/384<br />
| |
| <br />
| |
| valid range: [692.308, 694.737] (26 to 19)<br />
| |
| nice range: [682.502, 701.955]<br />
| |
| strict range: [692.308, 694.737]<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 693.126<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 17 -6|, &lt;0 1 4 -9 6|]<br />
| |
| EDOs: 7, 19, 26, 45, 71bc, 116bcde<br />
| |
| Badness: 0.0338<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Flattone-13-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->13-limit</h2>
| |
| 45/44, 65/64, 78/77, 81/80<br />
| |
| <br />
| |
| valid range: [692.308, 694.737] (26 to 19)<br />
| |
| nice range: [682.502, 701.955]<br />
| |
| strict range: [692.308, 694.737]<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 693.058<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 17 -6 10|, &lt;0 1 4 -9 6 -4|]<br />
| |
| EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef<br />
| |
| Badness: 0.0223<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc21"><a name="Dominant"></a><!-- ws:end:WikiTextHeadingRule:42 -->Dominant</h1>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 36/35, 64/63<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 <a class="wiki_link" href="/3_2">3/2</a> fifths, and with <a class="wiki_link" href="/29edo">29edo</a>, <a class="wiki_link" href="/41edo">41edo</a>, or <a class="wiki_link" href="/53edo">53edo</a>.<br />
| |
| <br />
| |
| valid range: [700.000, 720.000] (12 to 5)<br />
| |
| nice range: [694.786, 715.587]<br />
| |
| strict range: [700.000, 715.587]<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.573<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]<br />
| |
| <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 -2 4 -6 -16||<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="/53edo">53</a>, <a class="wiki_link" href="/65edo">65</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0207<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc22"><a name="Dominant-11-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->11-limit</h2>
| |
| Commas: 36/35, 64/63, 56/55<br />
| |
| <br />
| |
| valid range: [700.000, 705.882] (12 to 17)<br />
| |
| nice range: [691.202, 715.587]<br />
| |
| strict range: [700.000, 705.882]<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 703.254<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 13|, &lt;0 1 4 -2 -6|]<br />
| |
| EDOs: 5, 12, 17c, 29cde<br />
| |
| Badness: 0.0242<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Dominant-13-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->13-limit</h2>
| |
| Commas: 36/35, 56/55, 64/63, 66/65<br />
| |
| <br />
| |
| valid range: 705.882 (17)<br />
| |
| nice range: [691.202, 715.587]<br />
| |
| strict range:705.882<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 703.636<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 13 18|, &lt;0 1 4 -2 -6 -9|]<br />
| |
| EDOs: 12f, 17c, 29cdef<br />
| |
| Badness: 0.0241<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc24"><a name="Dominant-Dominion"></a><!-- ws:end:WikiTextHeadingRule:48 -->Dominion</h2>
| |
| Commas: 26/25, 36/35, 56/55, 64/63<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 704.905<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 13 -9|, &lt;0 1 4 -2 -6 8|]<br />
| |
| EDOs: 5, 12, 17c, 46cde<br />
| |
| Badness: 0.0273<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:50:&lt;h2&gt; --><h2 id="toc25"><a name="Dominant-Domineering"></a><!-- ws:end:WikiTextHeadingRule:50 -->Domineering</h2>
| |
| Commas: 36/35, 45/44, 64/63<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 698.776<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 -6|, &lt;0 1 4 -2 6|]<br />
| |
| EDOs: 7, 12, 43de<br />
| |
| Badness: 0.0220<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc26"><a name="Dominant-Domination"></a><!-- ws:end:WikiTextHeadingRule:52 -->Domination</h2>
| |
| Commas: 36/35, 64/63, 77/75<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 705.004<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 -14|, &lt;0 1 4 -2 11|]<br />
| |
| EDOs: 17c, 46cd<br />
| |
| Badness: 0.0366<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:54:&lt;h3&gt; --><h3 id="toc27"><a name="Dominant-Domination-13-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->13-limit</h3>
| |
| Commas: 26/25, 36/35, 64/63, 66/65<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 705.496<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 -14 -9|, &lt;0 1 4 -2 11 8|]<br />
| |
| EDOs: 17c<br />
| |
| Badness: 0.0274<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><h2 id="toc28"><a name="Dominant-Twelve"></a><!-- ws:end:WikiTextHeadingRule:56 -->Twelve</h2>
| |
| Commas: 81/80 64/63 45/44 65/64 256/255 153/152<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 696.217<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 -6 10 12 9|, &lt;0 1 4 -2 6 -4 -5 -3|]<br />
| |
| EDOs: 7, 12, 19d, 31def<br />
| |
| Badness: 0.0204<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:58:&lt;h2&gt; --><h2 id="toc29"><a name="Dominant-Arnold"></a><!-- ws:end:WikiTextHeadingRule:58 -->Arnold</h2>
| |
| Commas: 22/21, 33/32, 36/35<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 698.491<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 5|, &lt;0 1 4 -2 -1|]<br />
| |
| EDOs: 5, 7, 12e<br />
| |
| Badness: 0.0261<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc30"><a name="Dominant-13-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->13-limit</h2>
| |
| Commas: 22/21, 27/26, 33/32, 40/39<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 696.743<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 5 -1|, &lt;0 1 4 -2 -1 3|]<br />
| |
| EDOs: 5, 7, 12ef, 19def, 31def<br />
| |
| Badness: 0.0233<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc31"><a name="Dominant-Dominatrix"></a><!-- ws:end:WikiTextHeadingRule:62 -->Dominatrix</h2>
| |
| Commas: 27/26 36/35 45/44 64/63<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 698.544<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 6 -6 -1|, &lt;0 1 4 -2 6 3|]<br />
| |
| EDOs: 7, 12f<br />
| |
| Badness: 0.0183<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><h1 id="toc32"><a name="Sharptone"></a><!-- ws:end:WikiTextHeadingRule:64 -->Sharptone</h1>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 21/20, 28/27<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 />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 700.140<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -2|, &lt;0 1 4 3|]<br />
| |
| <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;1 4 3 4 2 -4||<br />
| |
| EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/12edo">12</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0248<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:66:&lt;h1&gt; --><h1 id="toc33"><a name="Meansept"></a><!-- ws:end:WikiTextHeadingRule:66 -->Meansept</h1>
| |
| Commas: 15/14, 81/80<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 682.895<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -5|, &lt;0 1 4 5|]<br />
| |
| Wedgie: &lt;&lt;1 4 5 4 5 0||<br />
| |
| EDOs: 7<br />
| |
| Badness: 0.0453<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:68:&lt;h2&gt; --><h2 id="toc34"><a name="Meansept-11-limit"></a><!-- ws:end:WikiTextHeadingRule:68 -->11-limit</h2>
| |
| Commas: 15/14, 22/21, 125/121<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 685.234<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -5 -6|, &lt;0 1 4 5 6|]<br />
| |
| EDOs: 7<br />
| |
| Badness: 0.0325<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:70:&lt;h1&gt; --><h1 id="toc35"><a name="Supermean"></a><!-- ws:end:WikiTextHeadingRule:70 -->Supermean</h1>
| |
| Commas: 81/80, 672/625<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 704.889<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -21|, &lt;0 1 4 15|]<br />
| |
| EDOs: 17c, 46c<br />
| |
| Badness: 0.1342<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc36"><a name="Supermean-11-limit"></a><!-- ws:end:WikiTextHeadingRule:72 -->11-limit</h2>
| |
| Commas: 56/55, 81/80, 132/125<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 705.096<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -21 -14|, &lt;0 1 4 15 11|]<br />
| |
| EDOs: 17c, 46c<br />
| |
| Badness: 0.0633<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:74:&lt;h2&gt; --><h2 id="toc37"><a name="Supermean-13-limit"></a><!-- ws:end:WikiTextHeadingRule:74 -->13-limit</h2>
| |
| Commas: 26/25, 56/55, 66/65, 81/80<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 705.094<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -21 -14 -9|, &lt;0 1 4 15 11 8|]<br />
| |
| EDOs: 17c, 46c<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:76:&lt;h1&gt; --><h1 id="toc38"><a name="Injera"></a><!-- ws:end:WikiTextHeadingRule:76 -->Injera</h1>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 50/49, 81/80<br />
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| <br />
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| The wedgie for injera is &lt;&lt;2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. <a class="wiki_link" href="/38edo">38edo</a>, which is two parallel <a class="wiki_link" href="/19edo">19edo</a>s, is an excellent tuning for injera.<br />
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| <br />
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| <a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/3091" rel="nofollow">Origin of the name</a><br />
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| <br />
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| valid range: [685.714, 700.000] (14c to 12)<br />
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| nice range: [688.957, 701.955]<br />
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| strict range: [688.957, 700.000]<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 694.375<br />
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| Mapping generator: ~3<br />
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| <br />
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| Map: [&lt;2 0 -8 -7|, &lt;0 1 4 4|]<br />
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| <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;2 8 8 8 7 -4||<br />
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| 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="/102edo">102bcd</a>, <a class="wiki_link" href="/140edo">140bcd</a>, <a class="wiki_link" href="/178edo">178bcd</a><br />
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| <a class="wiki_link" href="/Badness">Badness</a>: 0.0311<br />
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| <br />
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| <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3" rel="nofollow">Two Pairs of Socks</a> (in <a class="wiki_link" href="/26edo">26edo</a>) by <a class="wiki_link" href="/Igliashon%20Jones">Igliashon Calvin Jones-Coolidge</a><br />
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| <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Injera%20Jam.mp3" rel="nofollow">Injera Jam</a> (in <a class="wiki_link" href="/26edo">26edo</a>) by <a class="wiki_link" href="/Zach%20Curley">Zach Curley</a><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:78:&lt;h2&gt; --><h2 id="toc39"><a name="Injera-11-limit"></a><!-- ws:end:WikiTextHeadingRule:78 -->11-limit</h2>
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| Commas: 45/44, 50/49, 81/80<br />
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| <br />
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| valid range: [685.714, 700.000] (14c to 12)<br />
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| nice range: [682.458, 701.955]<br />
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| strict range: [685.714, 700.000]<br />
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| <br />
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| POTE generator: ~3/2 = 692.840<br />
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| Mapping generator: ~3<br />
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| <br />
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| Map: [&lt;2 0 -8 -7 -12|, &lt;0 1 4 4 6|]<br />
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| EDOs: 12, 14c, 26. 90bce, 116bce<br />
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| Badness: 0.0231<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:80:&lt;h2&gt; --><h2 id="toc40"><a name="Injera-13-limit"></a><!-- ws:end:WikiTextHeadingRule:80 -->13-limit</h2>
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| Commas: 45/44, 50/49, 81/80, 78/77<br />
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| <br />
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| valid range: 692.308 (26)<br />
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| nice range: [682.458, 701.955]<br />
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| strict range: 692.308 (26)<br />
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| <br />
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| POTE generator: ~3/2 = 692.673<br />
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| Mapping generator: ~3<br />
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| <br />
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| Map: [&lt;2 0 -8 -7 -12 -21|, &lt;0 1 4 4 6 9|]<br />
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| EDOs: 26, 104bcf<br />
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| Badness: 0.0216<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:82:&lt;h2&gt; --><h2 id="toc41"><a name="Injera-Enjera"></a><!-- ws:end:WikiTextHeadingRule:82 -->Enjera</h2>
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| Commas: 27/26, 40/39, 45/44, 99/98<br />
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| <br />
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| POTE generator: ~3/2 = 694.121<br />
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| Mapping generator: ~3<br />
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| <br />
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| Map: [&lt;2 0 -8 -7 -12 -2|, &lt;0 1 4 4 6 3|]<br />
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| EDOs: 12f, 26f, 38ef<br />
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| Badness: 0.0265<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc42"><a name="Injera-Injerous"></a><!-- ws:end:WikiTextHeadingRule:84 -->Injerous</h2>
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| Commas: 33/32, 50/49, 55/54<br />
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| <br />
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| POTE generator: ~3/2 = 690.548<br />
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| Mapping generator: ~3<br />
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| <br />
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| Map: [&lt;2 0 -8 -7 10|, &lt;0 1 4 4 -1|]<br />
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| EDOs: 12e, 14c, 26e, 40ce<br />
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| Badness: 0.0386<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc43"><a name="Injera-Lahoh"></a><!-- ws:end:WikiTextHeadingRule:86 -->Lahoh</h2>
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| Commas: 50/49, 56/55, 81/77<br />
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| <br />
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| POTE generator: ~3/2 = 699.001<br />
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| Mapping generator: ~3<br />
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| <br />
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| Map: [&lt;2 0 -8 -7 7|, &lt;0 1 4 4 0|]<br />
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| EDOs: 12<br />
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| Badness: 0.0431<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:88:&lt;h1&gt; --><h1 id="toc44"><a name="Godzilla"></a><!-- ws:end:WikiTextHeadingRule:88 -->Godzilla</h1>
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| <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/Semiphor%2C%20Semaphor%2C%20Godzilla">Deutsch</a><br />
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| </span><br />
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| Main article: <a class="wiki_link" href="/Semaphore%20and%20Godzilla">Semaphore and Godzilla</a><br />
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| <a class="wiki_link" href="/Comma">Comma</a>s: 49/48, 81/80<br />
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| <br />
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| Godzilla has wedgie &lt;&lt;2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. <a class="wiki_link" href="/19edo">19edo</a> is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.<br />
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| <br />
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| valid range: [240.000, 257.143] (5 to 14c)<br />
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| nice range: [231.174, 266.871]<br />
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| strict range: [240.000, 257.143]<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~8/7 = 252.635<br />
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| Mapping generator: ~7/4<br />
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| <br />
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| Map: [&lt;1 0 -4 2|, &lt;0 2 8 1|]<br />
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| <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;2 8 1 8 -4 -20||<br />
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| EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/9edo">9c</a>, <a class="wiki_link" href="/14edo">14c</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/62edo">62d</a>, <a class="wiki_link" href="/81edo">81d</a>, 143bd<br />
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| <a class="wiki_link" href="/Badness">Badness</a>: 0.0267<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:90:&lt;h2&gt; --><h2 id="toc45"><a name="Godzilla-11-limit"></a><!-- ws:end:WikiTextHeadingRule:90 -->11-limit</h2>
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| Commas: 45/44, 49/48, 81/80<br />
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| <br />
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| valid range: [252.632, 257.143] (19 to 14c)<br />
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| nice range: [231.174, 266.871]<br />
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| strict range: [252.632, 257.143]<br />
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| <br />
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| POTE generator: ~8/7 = 254.027<br />
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| Mapping generator: ~7/4<br />
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| <br />
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| Map: [&lt;1 0 -4 2 -6|, &lt;0 2 8 1 12|]<br />
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| EDOs: 14c, 19, 33cd, 52cd<br />
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| Badness: 0.0290<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:92:&lt;h2&gt; --><h2 id="toc46"><a name="Godzilla-13-limit"></a><!-- ws:end:WikiTextHeadingRule:92 -->13-limit</h2>
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| Commas: 45/44, 49/48, 78/77, 81/80<br />
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| <br />
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| valid range: 694.737 (19)<br />
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| nice range: [621.581, 737.652]<br />
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| strict range: 694.737<br />
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| <br />
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| POTE generator: ~8/7 = 253.603<br />
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| Mapping generator: ~7/4<br />
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| <br />
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| Map: [&lt;1 0 -4 2 -6 -5|, &lt;0 2 8 1 12 11|]<br />
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| EDOs: 14cf, 19, 33cdf, 52cdf<br />
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| Badness: 0.0225<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:94:&lt;h2&gt; --><h2 id="toc47"><a name="Godzilla-Semafour"></a><!-- ws:end:WikiTextHeadingRule:94 -->Semafour</h2>
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| Commas: 33/32, 49/48, 55/54<br />
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| <br />
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| POTE generator: ~8/7 = 254.042<br />
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| Mapping generator: ~7/4<br />
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| <br />
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| Map: [&lt;1 0 -4 2 5|, &lt;0 2 8 1 -2|]<br />
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| EDOs: 5, 14c, 19e, 33cde<br />
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| Badness: 0.0285<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:96:&lt;h2&gt; --><h2 id="toc48"><a name="Godzilla-Varan"></a><!-- ws:end:WikiTextHeadingRule:96 -->Varan</h2>
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| Commas: 49/48, 77/75, 81/80<br />
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| <br />
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| POTE generator: ~8/7 = 251.079<br />
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| Mapping generator: ~7/4<br />
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| <br />
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| Map: [&lt;1 0 -4 2 -10|, &lt;0 2 8 1 17|]<br />
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| EDOs: 19e, 24, 43de<br />
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| Badness: 0.0396<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:98:&lt;h3&gt; --><h3 id="toc49"><a name="Godzilla-Varan-13-limit"></a><!-- ws:end:WikiTextHeadingRule:98 -->13-limit</h3>
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| Commas: 49/48, 66/65, 77/75, 81/80<br />
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| <br />
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| POTE generator: ~8/7 = 251.165<br />
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| Mapping generator: ~7/4<br />
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| <br />
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| Map: [&lt;1 0 -4 2 -10 -5|, &lt;0 2 8 1 17 11|]<br />
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| EDOs: 19e, 24, 43de<br />
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| Badness: 0.0257<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:100:&lt;h2&gt; --><h2 id="toc50"><a name="Godzilla-Baragon"></a><!-- ws:end:WikiTextHeadingRule:100 -->Baragon</h2>
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| Commas: 49/48, 56/55, 81/80<br />
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| <br />
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| POTE generator: ~8/7 = 251.173<br />
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| Mapping generator: ~7/4<br />
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| <br />
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| Map: [&lt;1 0 -4 2 9|, &lt;0 2 8 1 -7|]<br />
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| EDOs: 19, 24, 43d<br />
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| Badness: 0.0357<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:102:&lt;h2&gt; --><h2 id="toc51"><a name="Godzilla-Music"></a><!-- ws:end:WikiTextHeadingRule:102 -->Music</h2>
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| <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3" rel="nofollow">Godzilla Example</a> by <a class="wiki_link" href="/Cameron%20Bobro">Cameron Bobro</a><br />
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| <a class="wiki_link_ext" href="http://tinyurl.com/4uyumk9" rel="nofollow">&quot;Change is on the Wind&quot;</a> in Godzilla[9] by <a class="wiki_link" href="/Igliashon%20Jones">Igliashon Jones</a><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:104:&lt;h1&gt; --><h1 id="toc52"><a name="Mohajira"></a><!-- ws:end:WikiTextHeadingRule:104 -->Mohajira</h1>
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| <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/Mohajira">Deutsch</a><br />
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| </span><br />
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| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 6144/6125<br />
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| <br />
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| Mohajira, with wedgie &lt;&lt;2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. <a class="wiki_link" href="/31edo">31edo</a> makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.<br />
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| <br />
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| Mohajira can also be thought of, intuitively, as &quot;meantone with quarter tones&quot;; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.<br />
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| <br />
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| <a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> minimax 1/4 comma<br />
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| [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]<br />
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| <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~128/105 = 348.415<br />
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| Mapping generator: ~128/105<br />
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| <br />
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| Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.<br />
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| <br />
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| Map: [&lt;1 1 0 6|, &lt;0 2 8 -11|]<br />
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| <a class="wiki_link" href="/Generator">Generator</a>s: 2, 128/105<br />
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| <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;2 8 -11 8 -23 -48||<br />
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| 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 />
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| <a class="wiki_link" href="/Badness">Badness</a>: 0.0557<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:106:&lt;h2&gt; --><h2 id="toc53"><a name="Mohajira-11-limit"></a><!-- ws:end:WikiTextHeadingRule:106 -->11-limit</h2>
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| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 121/120, 176/175<br />
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| <br />
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| <a class="wiki_link" href="/11-limit">11-limit</a> minimax 1/4 comma<br />
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| [|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,<br />
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| |6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]<br />
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| <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~11/9 = 348.477<br />
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| Mapping generator: ~11/9<br />
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| <br />
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| Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]<br />
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| <a class="wiki_link" href="/Generator">Generator</a>s: 2, 11/9<br />
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| 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 />
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| <a class="wiki_link" href="/Badness">Badness</a>: 0.0261<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:108:&lt;h2&gt; --><h2 id="toc54"><a name="Mohajira-13-limit"></a><!-- ws:end:WikiTextHeadingRule:108 -->13-limit</h2>
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| Commas: 81/80, 121/120, 105/104, 66/65<br />
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| <br />
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| POTE generator: ~11/9 = 348.558<br />
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| Mapping generator: ~11/9<br />
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| <br />
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| Map: [&lt;1 1 0 6 2 4|, &lt;0 2 8 -11 5 -1|]<br />
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| EDOs: 7, 24, 31, 117ef, 148bef<br />
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| Badness: 0.0234<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:110:&lt;h1&gt; --><h1 id="toc55"><a name="Ptolemy"></a><!-- ws:end:WikiTextHeadingRule:110 -->Ptolemy</h1>
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| Commas: 81/80, 121/120, 525/512<br />
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| <br />
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| POTE generator: ~11/9 = 346.922<br />
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| <br />
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| Map: [&lt;1 1 0 8 2|, &lt;0 2 8 -18 5|]<br />
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| EDOs: 7, 38d, 45e, 83bcde<br />
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| Badness: 0.0588<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:112:&lt;h2&gt; --><h2 id="toc56"><a name="Ptolemy-13-limit"></a><!-- ws:end:WikiTextHeadingRule:112 -->13-limit</h2>
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| Commas: 65/64, 81/80, 105/104, 121/120<br />
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| <br />
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| POTE generator: ~11/9 = 346.910<br />
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| <br />
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| Map: [&lt;1 1 0 8 2 6|, &lt;0 2 8 -18 5 -8|]<br />
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| EDOs: 7, 38df, 45ef, 83bcdef<br />
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| Badness: 0.0343<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:114:&lt;h1&gt; --><h1 id="toc57"><a name="Maqamic"></a><!-- ws:end:WikiTextHeadingRule:114 -->Maqamic</h1>
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| <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/maqamisch">Deutsch</a><br />
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| </span><br />
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| Main article: <a class="wiki_link" href="/Maqamic">Maqamic</a><br />
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| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 36/35, 121/120<br />
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| <br />
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| Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered &quot;middle-of-the-road&quot; intonation.<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~11/9 = 350.934<br />
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| Mapping generator: ~11/9<br />
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| <br />
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| Map: [&lt;1 1 0 4 2|, &lt;0 2 8 -4 5|]<br />
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| <a class="wiki_link" href="/Generator">Generator</a>s: 2, 11/9<br />
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| EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/10edo">10c</a>, <a class="wiki_link" href="/17edo">17c</a>, <a class="wiki_link" href="/24edo">24d</a>, <a class="wiki_link" href="/31edo">31d</a><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:116:&lt;h2&gt; --><h2 id="toc58"><a name="Maqamic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:116 -->13-limit</h2>
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| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 36/35, 121/120, 144/143<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~11/9 = 350.816<br />
| |
| Mapping generator: ~11/9<br />
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| <br />
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| Map: [&lt;1 1 0 4 2 4|, &lt;0 2 8 -4 5 -1|]<br />
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| Generators: 2, 11/9<br />
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| EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/10edo">10c</a>, <a class="wiki_link" href="/17edo">17c</a>, <a class="wiki_link" href="/24edo">24d</a>,<a class="wiki_link" href="/31edo"> 31d</a><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:118:&lt;h1&gt; --><h1 id="toc59"><a name="Migration"></a><!-- ws:end:WikiTextHeadingRule:118 -->Migration</h1>
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| Commas: 81/80, 121/120, 126/125<br />
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| <br />
| |
| POTE generator: ~11/9 = 348.182<br />
| |
| Mapping generator: ~11/9<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 -3 2|, &lt;0 2 8 20 5|]<br />
| |
| EDOs: 31, 100de, 131bde, 162bde<br />
| |
| Badness: 0.0255<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:120:&lt;h1&gt; --><h1 id="toc60"><a name="Mohamaq"></a><!-- ws:end:WikiTextHeadingRule:120 -->Mohamaq</h1>
| |
| Commas: 81/80, 392/375<br />
| |
| <br />
| |
| POTE generator: ~25/21 = 350.586<br />
| |
| Mapping generator: ~25/21<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 -1|, &lt;0 2 8 13|]<br />
| |
| EDOs: 17c, 24, 65c, 89cd<br />
| |
| Badness: 0.0777<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:122:&lt;h2&gt; --><h2 id="toc61"><a name="Mohamaq-11-limit"></a><!-- ws:end:WikiTextHeadingRule:122 -->11-limit</h2>
| |
| Commas: 56/55, 77/75, 243/242<br />
| |
| <br />
| |
| POTE generator: ~11/9 = 350.565<br />
| |
| Mapping generator: ~11/9<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 -1 2|, &lt;0 2 8 13 5|]<br />
| |
| EDOs: 17c, 24, 65c, 89cd<br />
| |
| Badness: 0.0362<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:124:&lt;h2&gt; --><h2 id="toc62"><a name="Mohamaq-13-limit"></a><!-- ws:end:WikiTextHeadingRule:124 -->13-limit</h2>
| |
| Commas: 56/55, 66/65, 77/75, 243/242<br />
| |
| <br />
| |
| POTE generator: ~11/9 = 350.745<br />
| |
| Mapping generator: ~11/9<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 -1 2 4|, &lt;0 2 8 13 5 -1|]<br />
| |
| EDOs: 17c, 24, 41c, 65c<br />
| |
| Badness: 0.0287<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:126:&lt;h1&gt; --><h1 id="toc63"><a name="Orphic"></a><!-- ws:end:WikiTextHeadingRule:126 -->Orphic</h1>
| |
| Commas: 81/80, 5898240/5764801<br />
| |
| <br />
| |
| POTE generator: ~7/6 = 275.794<br />
| |
| Mapping generator: ~343/288<br />
| |
| <br />
| |
| Map: [&lt;2 1 -4 4|, &lt;0 4 16 3|]<br />
| |
| Wedgie: &lt;&lt;8 32 6 32 -13 -76||<br />
| |
| EDOs: 26, 74, 174bd, 248bd<br />
| |
| Badness: 0.2588<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:128:&lt;h2&gt; --><h2 id="toc64"><a name="Orphic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:128 -->11-limit</h2>
| |
| Commas: 81/80, 99/98, 73728/73205<br />
| |
| <br />
| |
| POTE generator: ~7/6 = 275.762<br />
| |
| Mapping generator: ~77/64<br />
| |
| <br />
| |
| Map: [&lt;2 1 -4 4 8|, &lt;0 4 16 3 -2|]<br />
| |
| EDOs: 26, 48c, 74, 248bd, 322bd<br />
| |
| Badness: 0.1015<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:130:&lt;h2&gt; --><h2 id="toc65"><a name="Orphic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:130 -->13-limit</h2>
| |
| Commas: 81/80, 99/98, 144/143, 2200/2197<br />
| |
| <br />
| |
| POTE generator: ~7/6 = 275.774<br />
| |
| Mapping generator: ~63/52<br />
| |
| <br />
| |
| Map: [&lt;2 1 -4 4 8 2|, &lt;0 4 16 3 -2 10|]<br />
| |
| EDOs: 26, 48c, 74, 174bd, 248bd, 322bd<br />
| |
| Badness: 0.0535<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:132:&lt;h1&gt; --><h1 id="toc66"><a name="Mothra"></a><!-- ws:end:WikiTextHeadingRule:132 -->Mothra</h1>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 1029/1024<br />
| |
| <br />
| |
| Mothra, with wedgie &lt;&lt;3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using <a class="wiki_link" href="/31edo">31edo</a> with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to <a class="wiki_link" href="/Slendric">slendric</a>.<br />
| |
| Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.<br />
| |
| <br />
| |
| <a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> minimax 1/4 comma<br />
| |
| [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]<br />
| |
| <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~8/7 = 232.193<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.<br />
| |
| <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 />
| |
| <a class="wiki_link" href="/Wedgie">Wedgie</a>: &lt;&lt;3 12 -1 12 -10 -36||<br />
| |
| 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><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0371<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:134:&lt;h2&gt; --><h2 id="toc67"><a name="Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:134 -->11-limit</h2>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 99/98, 385/384<br />
| |
| <br />
| |
| POTE generator: ~8/7 = 232.031<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]<br />
| |
| 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><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0256<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:136:&lt;h2&gt; --><h2 id="toc68"><a name="Mothra-13-limit"></a><!-- ws:end:WikiTextHeadingRule:136 -->13-limit</h2>
| |
| Commas: 81/80, 99/98, 105/104, 144/143<br />
| |
| <br />
| |
| POTE generator: ~8/7 = 231.811<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 3 5 1|, &lt;0 3 12 -1 -8 14|]<br />
| |
| EDOs: 5, 26, 31, 57, 88<br />
| |
| Badness: 0.0240<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:138:&lt;h2&gt; --><h2 id="toc69"><a name="Mothra-Cynder"></a><!-- ws:end:WikiTextHeadingRule:138 -->Cynder</h2>
| |
| Commas: 45/44, 81/80, 1029/1024<br />
| |
| <br />
| |
| POTE generator: ~8/7 = 231.317<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 3 0|, &lt;0 3 12 -1 18|]<br />
| |
| EDOs: 26, 57e, 83bce<br />
| |
| Badness: 0.0557<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:140:&lt;h3&gt; --><h3 id="toc70"><a name="Mothra-Cynder-13-limit"></a><!-- ws:end:WikiTextHeadingRule:140 -->13-limit</h3>
| |
| Commas: 45/44, 78/77, 81/80, 640/637<br />
| |
| <br />
| |
| POTE generator: ~8/7 = 231.293<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 3 0 1|, &lt;0 3 12 -1 18 14|]<br />
| |
| EDOs: 26, 57e, 83bce<br />
| |
| Badness: 0.0341<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:142:&lt;h2&gt; --><h2 id="toc71"><a name="Mothra-Mosura"></a><!-- ws:end:WikiTextHeadingRule:142 -->Mosura</h2>
| |
| Commas: 81/80, 176/175, 1029/1024<br />
| |
| <br />
| |
| POTE generator: ~8/7 = 232.419<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 3 -1|, &lt;0 3 12 -1 23|]<br />
| |
| EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce<br />
| |
| Badness: 0.0313<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:144:&lt;h3&gt; --><h3 id="toc72"><a name="Mothra-Mosura-13-limit"></a><!-- ws:end:WikiTextHeadingRule:144 -->13-limit</h3>
| |
| Commas: 81/80, 144/143, 176/175, 1029/1024<br />
| |
| <br />
| |
| POTE generator: ~8/7 = 232.640<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 3 -1 7|, &lt;0 3 12 -1 23 -17|]<br />
| |
| EDOs: 31, 67, 98<br />
| |
| Badness: 0.0369<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:146:&lt;h1&gt; --><h1 id="toc73"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:146 -->Squares</h1>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 2401/2400<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 (<a class="wiki_link" href="/9_7">9/7</a>) intervals, and uses it for a generator. <a class="wiki_link" href="/31edo">31edo</a>, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.<br />
| |
| <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 />
| |
| <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~9/7 = 425.942<br />
| |
| Mapping generator: ~9/7<br />
| |
| <br />
| |
| Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.<br />
| |
| <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: <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/262edo">262</a>, <a class="wiki_link" href="/293edo">293</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0460<br />
| |
| <br />
| |
| Music:<br />
| |
| By <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br />
| |
| <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3" rel="nofollow">Square 8</a><br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:148:&lt;h2&gt; --><h2 id="toc74"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:148 -->11-limit</h2>
| |
| Commas: 81/80, 99/98, 121/120<br />
| |
| <br />
| |
| POTE generator: ~9/7 = 425.957<br />
| |
| Mapping generator: ~9/7<br />
| |
| <br />
| |
| Map: [&lt;1 3 8 6 7|, &lt;0 -4 -16 -9 -10|]<br />
| |
| 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><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0216<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:150:&lt;h2&gt; --><h2 id="toc75"><a name="Squares-13-limit"></a><!-- ws:end:WikiTextHeadingRule:150 -->13-limit</h2>
| |
| Commas: 81/80, 99/98, 121/120, 66/65<br />
| |
| <br />
| |
| POTE generator: ~9/7 = 425.550<br />
| |
| Mapping generator: ~9/7<br />
| |
| <br />
| |
| Map: [&lt;1 3 8 6 7 3|, &lt;0 -4 -16 -9 -10 2|]<br />
| |
| EDOs: 17c, 31, 79cf, 110cef, 141cef<br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0255<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:152:&lt;h2&gt; --><h2 id="toc76"><a name="Squares-Agora"></a><!-- ws:end:WikiTextHeadingRule:152 -->Agora</h2>
| |
| Commas: 81/80, 99/98, 105/104, 121/120<br />
| |
| <br />
| |
| POTE generator: ~9/7 = 426.276<br />
| |
| Mapping generator: ~9/7<br />
| |
| <br />
| |
| Map: [&lt;1 3 8 6 7 14|, &lt;0 -4 -16 -9 -10 -29|]<br />
| |
| EDOs: 31, 45ef, 76e<br />
| |
| Badness: 0.0245<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:154:&lt;h1&gt; --><h1 id="toc77"><a name="Cuboctahedra"></a><!-- ws:end:WikiTextHeadingRule:154 -->Cuboctahedra</h1>
| |
| <!-- ws:start:WikiTextHeadingRule:156:&lt;h2&gt; --><h2 id="toc78"><a name="Cuboctahedra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:156 -->11-limit</h2>
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 385/384, 1375/1372<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~9/7 = 425.993<br />
| |
| Mapping generator: ~9/7<br />
| |
| <br />
| |
| Map: [&lt;1 3 8 6 -4|, &lt;0 -4 -16 -9 21|]<br />
| |
| EDOs: <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/200edo">200</a><br />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0568<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:158:&lt;h1&gt; --><h1 id="toc79"><a name="Liese"></a><!-- ws:end:WikiTextHeadingRule:158 -->Liese</h1>
| |
| <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/Liese">Deutsch</a><br />
| |
| </span><br />
| |
| <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 686/675<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 />
| |
| <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 />
| |
| <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~10/7 = 632.406<br />
| |
| Mapping generator: ~10/7<br />
| |
| <br />
| |
| Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.<br />
| |
| <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 />
| |
| <a class="wiki_link" href="/Badness">Badness</a>: 0.0467<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:160:&lt;h2&gt; --><h2 id="toc80"><a name="Liese-Liesel"></a><!-- ws:end:WikiTextHeadingRule:160 -->Liesel</h2>
| |
| Commas: 56/55, 81/80, 540/539<br />
| |
| <br />
| |
| POTE generator: ~10/7 = 633.073<br />
| |
| Mapping generator: ~10/7<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -3 4|, &lt;0 3 12 11 -1|]<br />
| |
| EDOs: 17c, 19, 36, 91ce<br />
| |
| Badness: 0.0407<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:162:&lt;h2&gt; --><h2 id="toc81"><a name="Liese-13-limit"></a><!-- ws:end:WikiTextHeadingRule:162 -->13-limit</h2>
| |
| Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.<br />
| |
| <br />
| |
| Commas: 56/55, 78/77, 81/80, 91/90<br />
| |
| <br />
| |
| POTE generator: ~10/7 = ~13/9 = 633.042<br />
| |
| Mapping generator: ~10/7<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -3 4 0|, &lt;0 3 12 11 -1 7|]<br />
| |
| EDOs: 17c, 19, 36, 91cef<br />
| |
| Badness: 0.0273<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:164:&lt;h2&gt; --><h2 id="toc82"><a name="Liese-Elisa"></a><!-- ws:end:WikiTextHeadingRule:164 -->Elisa</h2>
| |
| Commas: 77/75, 81/80, 99/98<br />
| |
| <br />
| |
| POTE generator: ~10/7 = 633.061<br />
| |
| Mapping generator: ~10/7<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -3 -5|, &lt;0 3 12 11 16|]<br />
| |
| EDOs: 19e, 36e<br />
| |
| Badness: 0.0416<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:166:&lt;h2&gt; --><h2 id="toc83"><a name="Liese-Lisa"></a><!-- ws:end:WikiTextHeadingRule:166 -->Lisa</h2>
| |
| Commas: 45/44, 81/80, 343/330<br />
| |
| <br />
| |
| POTE generator: ~10/7 = 631.370<br />
| |
| Mapping generator: ~10/7<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -3 -6|, &lt;0 3 12 11 18|]<br />
| |
| EDOs: 19<br />
| |
| Badness: 0.0548<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:168:&lt;h2&gt; --><h2 id="toc84"><a name="Liese-13-limit"></a><!-- ws:end:WikiTextHeadingRule:168 -->13-limit</h2>
| |
| Commas: 45/44, 81/80, 91/88, 147/143<br />
| |
| <br />
| |
| POTE generator: ~10/7 = 631.221<br />
| |
| Mapping generator: ~10/7<br />
| |
| <br />
| |
| Map: [&lt;1 0 -4 -3 -6 0|, &lt;0 3 12 11 18 7|]<br />
| |
| EDOs: 19<br />
| |
| Badness: 0.0361<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:170:&lt;h1&gt; --><h1 id="toc85"><a name="Jerome"></a><!-- ws:end:WikiTextHeadingRule:170 -->Jerome</h1>
| |
| Jerome is related to <a class="wiki_link" href="/20ed5">Hieronymus' tuning</a>; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.<br />
| |
| <br />
| |
| Commas: 81/80, 17280/16807<br />
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| <br />
| |
| POTE generator: ~54/49 = 139.343<br />
| |
| Mapping generator: ~54/49<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 2|, &lt;0 5 20 7|]<br />
| |
| Wedgie: &lt;&lt;5 30 7 20 -3 -40||<br />
| |
| EDOs: 8, 9, 17, 26, 43, 112<br />
| |
| Badness: 0.1087<br />
| |
| <br />
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| <!-- ws:start:WikiTextHeadingRule:172:&lt;h2&gt; --><h2 id="toc86"><a name="Jerome-11-limit"></a><!-- ws:end:WikiTextHeadingRule:172 -->11-limit</h2>
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| Commas: 81/80, 99/98, 864/847<br />
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| <br />
| |
| POTE generator: ~12/11 = 139.428<br />
| |
| Mapping generator: ~12/11<br />
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| <br />
| |
| Map: [&lt;1 1 0 2 3|, &lt;0 5 20 7 4|]<br />
| |
| EDOs: 8, 9, 17, 26, 43, 241<br />
| |
| Badness: 0.0479<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:174:&lt;h2&gt; --><h2 id="toc87"><a name="Jerome-13-limit"></a><!-- ws:end:WikiTextHeadingRule:174 -->13-limit</h2>
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| Commas: 77/78, 81/80, 99/98, 144/143<br />
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| <br />
| |
| POTE generator: ~13/12 = 139.387<br />
| |
| Mapping generator: ~12/11<br />
| |
| <br />
| |
| Map: [&lt;1 1 0 2 3 3|, &lt;0 5 20 7 4 6|]<br />
| |
| EDOs: 8, 9, 17, 26, 43, 155, 198<br />
| |
| Badness: 0.0293<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:176:&lt;h2&gt; --><h2 id="toc88"><a name="Jerome-17-limit"></a><!-- ws:end:WikiTextHeadingRule:176 -->17-limit</h2>
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| Commas: 78/77, 81/80, 99/98, 144/143, 189/187<br />
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| <br />
| |
| POTE generator: ~13/12 = 139.362<br />
| |
| Mapping generator: ~12/11<br />
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| <br />
| |
| Map: [&lt;1 1 0 2 3 3 2|, &lt;0 5 20 7 4 6 18|]<br />
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| EDOs: 8, 9, 17, 26, 43, 155<br />
| |
| Badness: 0.0209<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:178:&lt;h1&gt; --><h1 id="toc89"><a name="Meanmag"></a><!-- ws:end:WikiTextHeadingRule:178 -->Meanmag</h1>
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| Commas: 81/80, 3125/3072<br />
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| <br />
| |
| POTE generator: ~8/7 = 238.396<br />
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| Mapping generator: ~7<br />
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| <br />
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| Map: [&lt;19 30 44 0|, &lt;0 0 0 1|]<br />
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| Wedgie: &lt;&lt;0 0 19 0 30 44||<br />
| |
| EDOs: 19, 57, 76, 171bcd<br />
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| Badness: 0.0770<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:180:&lt;h1&gt; --><h1 id="toc90"><a name="Undevigintone"></a><!-- ws:end:WikiTextHeadingRule:180 -->Undevigintone</h1>
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| Commas: 49/48, 81/80, 126/125<br />
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| <br />
| |
| POTE generator: ~11/8 = 538.047<br />
| |
| Mapping generator: ~11<br />
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| <br />
| |
| Map: [&lt;19 30 44 53 0|, &lt;0 0 0 0 1|]<br />
| |
| EDOs: 19, 38d<br />
| |
| Badness: 0.0364<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:182:&lt;h2&gt; --><h2 id="toc91"><a name="Undevigintone-13-limit"></a><!-- ws:end:WikiTextHeadingRule:182 -->13-limit</h2>
| |
| Commas: 49/48, 65/64, 81/80, 126/125<br />
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| <br />
| |
| POTE generator: ~11/8 = 537.061<br />
| |
| <br />
| |
| Map: [&lt;19 30 44 53 0 70|, &lt;0 0 0 0 1 0|]<br />
| |
| EDOs: 19, 38d<br />
| |
| Badness: 0.0229</body></html></pre></div>
| |
de:mitteltönig Deutsch
The 5-limit parent comma of the meantone family is the Didymus or syntonic comma, 81/80. This is the one they all temper out. The monzo for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding 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 generator: ~3/2 = 696.239
Mapping generator: ~3
valid range: [685.714, 720.000] (7 to 5)
nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
strict range: [694.786, 701.955]
Map: [<1 0 -4|, <0 1 4|]
EDOs (patent val edo list is complete): 5, 7, 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, 55, 57, 62, 67, 69, 74, 76, 81, 86, 88, 93, 98, 100, 105, 117, 129, 212b
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>].
Septimal meantone
de:septimal-mitteltönig Deutsch
Wikipedia article
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 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.
Commas: 81/80, 126/125
7 and 9-limit minimax
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>]
Eigenmonzos: 2, 5
valid range: [694.737, 700.000] (19 to 12)
nice range: [694.786, 701.955]
strict range: [694.786, 700.000]
POTE generator: 696.495
Mapping generator: ~3
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|]
Generators: 2, 3
Wedgie: <<1 4 10 4 13 12||
EDOs: 12, 19, 31, 81, 143b
Badness: 0.0137
Bimeantone
Commas: 81/80, 126/125, 245/242
POTE generator: ~3/2 = 696.016
Map: [<2 0 -8 -26 -31|, <0 1 4 10 12|]
EDOs: 12, 38d, 50
Badness: 0.0381
13-limit
Commas: 81/80, 105/104, 126/125, 245/242
POTE generator: ~3/2 = 695.836
Map: [<2 0 -8 -26 -31 -40|, <0 1 4 10 12 15|]
EDOs: 12f, 50
Badness: 0.0288
Unidecimal meantone aka Huygens
See also Meantone vs meanpop
Commas: 81/80, 126/125, 99/98
11-limit minimax
[|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>]
Eigenmonzos: 2, 11/9
valid range: [696.774, 700.000] (31 to 12)
nice range: [691.202, 701.955]
strict range: [696.774, 700.000]
POTE generator: 696.967
Mapping generator: ~3
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|]
Generators: 2, 3
EDOs: 7, 12, 31, 105, 198be
Badness: 0.0170
Twinkle canon – 74 edo by Claudi Meneghin
Tridecimal meantone
Commas: 66/65, 81/80, 99/98, 105/104
valid range: 697.674 (43)
nice range: [691.202, 701.955]
strict range: 697.674
POTE generator: ~3/2 = 696.642
Mapping generator: ~3
Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]
EDOs: 12, 19, 31, 267, 298
Badness: 0.0180
Grosstone
Commas: 81/80, 99/98, 126/125, 144/143
POTE generator: ~3/2 = 697.264
Mapping generator: ~3
Map: [<1 0 -4 -13 -25 29|, <0 1 4 10 18 -16|]
EDOs: 12, 31, 43, 74
Badness: 0.0259
Meridetone
Commas: 78/77, 81/80, 99/98, 126/125
POTE generator: ~3/2 = 697.529
Mapping generator: ~3
Map: [<1 0 -4 -13 -25 -39|, <0 1 4 10 18 27|]
EDOs: 43, 117df, 160bdf, 203bcdef
Badness: 0.0264
Hemimeantone
Commas: 81/80, 99/98, 126/125, 169/168
POTE generator: ~52/45 = 250.304
Mapping generator: ~26/15
Map: [<1 0 -4 -13 -25 -5|, <0 2 8 20 36 11|]
EDOs: 43, 62, 167bef, 229bef
Badness: 0.0314
Meanpop
See also Meantone vs meanpop
Commas: 81/80, 126/125, 385/384
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>]
Eigenmonzos: 2, 5
valid range: [694.737, 696.774] (19 to 31)
nice range: [691.202, 701.955]
strict range: [694.737, 696.774]
POTE generator: 696.434
Mapping generator: ~3
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Scott Joplin's "The Entertainer" tuned into meanpop
Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]
Generators: 2, 3
EDOs: 12, 19, 31, 81, 112
Badness: 0.0215
Twinkle canon – 50 edo by Claudi Meneghin
13-limit Meanpop
Commas: 81/80, 105/104, 144/143, 196/195
valid range: [694.737, 696.774] (19 to 31)
nice range: [691.202, 701.955]
strict range: [694.737, 696.774]
POTE generator: ~3/2 = 696.211
Mapping generator: ~3
Map: [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|]
EDOS: 19, 31, 50, 81, 131bd, 212bdf
Badness: 0.0209
Meanplop
Commas: 65/64, 78/77, 81/80, 91/90
POTE generator: ~3/2 = 696.202
Mapping generator: ~3
Map: [<1 0 -4 -13 24 10|, <0 1 4 10 -13 -4|]
EDOs: 12e, 19, 31f, 50f
Badness: 0.0277
Meanenneadecal
Commas: 45/44, 56/55, 81/80
POTE generator: ~3/2 = 696.250
Mapping generator: ~3
Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]
EDOs: 7, 12, 19, 31e, 50e
Badness: 0.0214
13-limit
Commas: 45/44, 56/55, 78/77, 81/80
POTE generator: ~3/2 = 696.146
Mapping generator: ~3
Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]
EDOs: 19, 31e, 50e]
Badness: 0.0212
Vincenzo
Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22
POTE generator: ~3/2
Mapping generator: ~3
Map: [<1 0 -4 -13 ... |, <0 1 4 10 6 -4 -5 -3 -6|]
EDOs: 12
Badness:
Meanundeci
Commas: 33/32, 55/54, 77/75
POTE generator: ~3/2 = 694.689
Mapping generator: ~3
Map: [<1 0 -4 -13 5|, <0 1 4 10 -1|]
EDOs: 12e, 19e
Badness: 0.0315
13-limit
Commas: 33/32, 55/54, 77/75, 729/728
POTE generator: ~3/2 = 694.764
Mapping generator: ~3
Map: [<1 0 -4 -13 5 10|, <0 1 4 10 -1 -4|]
EDOs: 12e, 19e
Badness: 0.0263
Meanundec
Commas: 27/26, 40/39, 45/44, 56/55
POTE generator: ~3/2 = 697.254
Mapping generator: ~3
Map: [<1 0 -4 -13 -6 -1|, <0 1 4 10 6 3|]
EDOS: 12f, 19f, 31ef
Badness: 0.0242
Flattone
Commas: 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 seventh interval. Other intervals are 7/6, a diminished third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are 26edo, 45edo and 64edo.
7-limit minimax
[|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>]
Eigenmonzos: 2, 7/5
9-limit minimax
[|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>]
Eigenmonzos: 2, 9/7
valid range: [692.308, 694.737] (26 to 19)
nice range: [692.353, 701.955]
strict range: [692.353, 694.737]
POTE generator: 693.779
Mapping generator: ~3
Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.
Map: [<1 0 -4 17|, <0 1 4 -9|]
Wedgie: <<1 4 -9 4 -17 -32||
Generators: 2, 3
EDOs: 7, 19, 45, 64
Badness: 0.0386
11-limit
Commas: 45/44, 81/80, 385/384
valid range: [692.308, 694.737] (26 to 19)
nice range: [682.502, 701.955]
strict range: [692.308, 694.737]
POTE generator: ~3/2 = 693.126
Mapping generator: ~3
Map: [<1 0 -4 17 -6|, <0 1 4 -9 6|]
EDOs: 7, 19, 26, 45, 71bc, 116bcde
Badness: 0.0338
13-limit
45/44, 65/64, 78/77, 81/80
valid range: [692.308, 694.737] (26 to 19)
nice range: [682.502, 701.955]
strict range: [692.308, 694.737]
POTE generator: ~3/2 = 693.058
Mapping generator: ~3
Map: [<1 0 -4 17 -6 10|, <0 1 4 -9 6 -4|]
EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef
Badness: 0.0223
Dominant
Commas: 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.
valid range: [700.000, 720.000] (12 to 5)
nice range: [694.786, 715.587]
strict range: [700.000, 715.587]
POTE generator: 701.573
Mapping generator: ~3
Map: [<1 0 -4 6|, <0 1 4 -2|]
Wedgie: <<1 4 -2 4 -6 -16||
EDOs: 5, 7, 12, 53, 65
Badness: 0.0207
11-limit
Commas: 36/35, 64/63, 56/55
valid range: [700.000, 705.882] (12 to 17)
nice range: [691.202, 715.587]
strict range: [700.000, 705.882]
POTE generator: ~3/2 = 703.254
Mapping generator: ~3
Map: [<1 0 -4 6 13|, <0 1 4 -2 -6|]
EDOs: 5, 12, 17c, 29cde
Badness: 0.0242
13-limit
Commas: 36/35, 56/55, 64/63, 66/65
valid range: 705.882 (17)
nice range: [691.202, 715.587]
strict range:705.882
POTE generator: ~3/2 = 703.636
Map: [<1 0 -4 6 13 18|, <0 1 4 -2 -6 -9|]
EDOs: 12f, 17c, 29cdef
Badness: 0.0241
Dominion
Commas: 26/25, 36/35, 56/55, 64/63
POTE generator: ~3/2 = 704.905
Map: [<1 0 -4 6 13 -9|, <0 1 4 -2 -6 8|]
EDOs: 5, 12, 17c, 46cde
Badness: 0.0273
Domineering
Commas: 36/35, 45/44, 64/63
POTE generator: ~3/2 = 698.776
Mapping generator: ~3
Map: [<1 0 -4 6 -6|, <0 1 4 -2 6|]
EDOs: 7, 12, 43de
Badness: 0.0220
Domination
Commas: 36/35, 64/63, 77/75
POTE generator: ~3/2 = 705.004
Mapping generator: ~3
Map: [<1 0 -4 6 -14|, <0 1 4 -2 11|]
EDOs: 17c, 46cd
Badness: 0.0366
13-limit
Commas: 26/25, 36/35, 64/63, 66/65
POTE generator: ~3/2 = 705.496
Mapping generator: ~3
Map: [<1 0 -4 6 -14 -9|, <0 1 4 -2 11 8|]
EDOs: 17c
Badness: 0.0274
Twelve
Commas: 81/80 64/63 45/44 65/64 256/255 153/152
POTE generator: ~3/2 = 696.217
Mapping generator: ~3
Map: [<1 0 -4 6 -6 10 12 9|, <0 1 4 -2 6 -4 -5 -3|]
EDOs: 7, 12, 19d, 31def
Badness: 0.0204
Arnold
Commas: 22/21, 33/32, 36/35
POTE generator: ~3/2 = 698.491
Mapping generator: ~3
Map: [<1 0 -4 6 5|, <0 1 4 -2 -1|]
EDOs: 5, 7, 12e
Badness: 0.0261
13-limit
Commas: 22/21, 27/26, 33/32, 40/39
POTE generator: ~3/2 = 696.743
Mapping generator: ~3
Map: [<1 0 -4 6 5 -1|, <0 1 4 -2 -1 3|]
EDOs: 5, 7, 12ef, 19def, 31def
Badness: 0.0233
Dominatrix
Commas: 27/26 36/35 45/44 64/63
POTE generator: ~3/2 = 698.544
Mapping generator: ~3
Map: [<1 0 -4 6 -6 -1|, <0 1 4 -2 6 3|]
EDOs: 7, 12f
Badness: 0.0183
Sharptone
Commas: 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.
POTE generator: 700.140
Mapping generator: ~3
Map: [<1 0 -4 -2|, <0 1 4 3|]
Wedgie: <<1 4 3 4 2 -4||
EDOs: 5, 12
Badness: 0.0248
Meansept
Commas: 15/14, 81/80
POTE generator: ~3/2 = 682.895
Mapping generator: ~3
Map: [<1 0 -4 -5|, <0 1 4 5|]
Wedgie: <<1 4 5 4 5 0||
EDOs: 7
Badness: 0.0453
11-limit
Commas: 15/14, 22/21, 125/121
POTE generator: ~3/2 = 685.234
Mapping generator: ~3
Map: [<1 0 -4 -5 -6|, <0 1 4 5 6|]
EDOs: 7
Badness: 0.0325
Supermean
Commas: 81/80, 672/625
POTE generator: ~3/2 = 704.889
Map: [<1 0 -4 -21|, <0 1 4 15|]
EDOs: 17c, 46c
Badness: 0.1342
11-limit
Commas: 56/55, 81/80, 132/125
POTE generator: ~3/2 = 705.096
Map: [<1 0 -4 -21 -14|, <0 1 4 15 11|]
EDOs: 17c, 46c
Badness: 0.0633
13-limit
Commas: 26/25, 56/55, 66/65, 81/80
POTE generator: ~3/2 = 705.094
Map: [<1 0 -4 -21 -14 -9|, <0 1 4 15 11 8|]
EDOs: 17c, 46c
Injera
Commas: 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.
Origin of the name
valid range: [685.714, 700.000] (14c to 12)
nice range: [688.957, 701.955]
strict range: [688.957, 700.000]
POTE generator: 694.375
Mapping generator: ~3
Map: [<2 0 -8 -7|, <0 1 4 4|]
Wedgie: <<2 8 8 8 7 -4||
EDOs: 12, 26, 38, 102bcd, 140bcd, 178bcd
Badness: 0.0311
Two Pairs of Socks (in 26edo) by Igliashon Calvin Jones-Coolidge
Injera Jam (in 26edo) by Zach Curley
11-limit
Commas: 45/44, 50/49, 81/80
valid range: [685.714, 700.000] (14c to 12)
nice range: [682.458, 701.955]
strict range: [685.714, 700.000]
POTE generator: ~3/2 = 692.840
Mapping generator: ~3
Map: [<2 0 -8 -7 -12|, <0 1 4 4 6|]
EDOs: 12, 14c, 26. 90bce, 116bce
Badness: 0.0231
13-limit
Commas: 45/44, 50/49, 81/80, 78/77
valid range: 692.308 (26)
nice range: [682.458, 701.955]
strict range: 692.308 (26)
POTE generator: ~3/2 = 692.673
Mapping generator: ~3
Map: [<2 0 -8 -7 -12 -21|, <0 1 4 4 6 9|]
EDOs: 26, 104bcf
Badness: 0.0216
Enjera
Commas: 27/26, 40/39, 45/44, 99/98
POTE generator: ~3/2 = 694.121
Mapping generator: ~3
Map: [<2 0 -8 -7 -12 -2|, <0 1 4 4 6 3|]
EDOs: 12f, 26f, 38ef
Badness: 0.0265
Injerous
Commas: 33/32, 50/49, 55/54
POTE generator: ~3/2 = 690.548
Mapping generator: ~3
Map: [<2 0 -8 -7 10|, <0 1 4 4 -1|]
EDOs: 12e, 14c, 26e, 40ce
Badness: 0.0386
Lahoh
Commas: 50/49, 56/55, 81/77
POTE generator: ~3/2 = 699.001
Mapping generator: ~3
Map: [<2 0 -8 -7 7|, <0 1 4 4 0|]
EDOs: 12
Badness: 0.0431
Godzilla
de:Semiphor,_Semaphor,_Godzilla Deutsch
Main article: Semaphore and Godzilla
Commas: 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-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. 19edo is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
valid range: [240.000, 257.143] (5 to 14c)
nice range: [231.174, 266.871]
strict range: [240.000, 257.143]
POTE generator: ~8/7 = 252.635
Mapping generator: ~7/4
Map: [<1 0 -4 2|, <0 2 8 1|]
Wedgie: <<2 8 1 8 -4 -20||
EDOs: 5, 9c, 14c, 19, 62d, 81d, 143bd
Badness: 0.0267
11-limit
Commas: 45/44, 49/48, 81/80
valid range: [252.632, 257.143] (19 to 14c)
nice range: [231.174, 266.871]
strict range: [252.632, 257.143]
POTE generator: ~8/7 = 254.027
Mapping generator: ~7/4
Map: [<1 0 -4 2 -6|, <0 2 8 1 12|]
EDOs: 14c, 19, 33cd, 52cd
Badness: 0.0290
13-limit
Commas: 45/44, 49/48, 78/77, 81/80
valid range: 694.737 (19)
nice range: [621.581, 737.652]
strict range: 694.737
POTE generator: ~8/7 = 253.603
Mapping generator: ~7/4
Map: [<1 0 -4 2 -6 -5|, <0 2 8 1 12 11|]
EDOs: 14cf, 19, 33cdf, 52cdf
Badness: 0.0225
Semafour
Commas: 33/32, 49/48, 55/54
POTE generator: ~8/7 = 254.042
Mapping generator: ~7/4
Map: [<1 0 -4 2 5|, <0 2 8 1 -2|]
EDOs: 5, 14c, 19e, 33cde
Badness: 0.0285
Varan
Commas: 49/48, 77/75, 81/80
POTE generator: ~8/7 = 251.079
Mapping generator: ~7/4
Map: [<1 0 -4 2 -10|, <0 2 8 1 17|]
EDOs: 19e, 24, 43de
Badness: 0.0396
13-limit
Commas: 49/48, 66/65, 77/75, 81/80
POTE generator: ~8/7 = 251.165
Mapping generator: ~7/4
Map: [<1 0 -4 2 -10 -5|, <0 2 8 1 17 11|]
EDOs: 19e, 24, 43de
Badness: 0.0257
Baragon
Commas: 49/48, 56/55, 81/80
POTE generator: ~8/7 = 251.173
Mapping generator: ~7/4
Map: [<1 0 -4 2 9|, <0 2 8 1 -7|]
EDOs: 19, 24, 43d
Badness: 0.0357
Music
Godzilla Example by Cameron Bobro
"Change is on the Wind" in Godzilla[9] by Igliashon Jones
Mohajira
de:Mohajira Deutsch
Commas: 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 can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.
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>]
Eigenmonzos: 2, 5
POTE generator: ~128/105 = 348.415
Mapping generator: ~128/105
Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.
Map: [<1 1 0 6|, <0 2 8 -11|]
Generators: 2, 128/105
Wedgie: <<2 8 -11 8 -23 -48||
EDOs: 7, 24, 31
Badness: 0.0557
11-limit
Commas: 81/80, 121/120, 176/175
11-limit minimax 1/4 comma
[|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>]
Eigenmonzos: 2, 5
POTE generator: ~11/9 = 348.477
Mapping generator: ~11/9
Map: [<1 1 0 6 2|, <0 2 8 -11 5|]
Generators: 2, 11/9
EDOs: 7, 24, 31
Badness: 0.0261
13-limit
Commas: 81/80, 121/120, 105/104, 66/65
POTE generator: ~11/9 = 348.558
Mapping generator: ~11/9
Map: [<1 1 0 6 2 4|, <0 2 8 -11 5 -1|]
EDOs: 7, 24, 31, 117ef, 148bef
Badness: 0.0234
Ptolemy
Commas: 81/80, 121/120, 525/512
POTE generator: ~11/9 = 346.922
Map: [<1 1 0 8 2|, <0 2 8 -18 5|]
EDOs: 7, 38d, 45e, 83bcde
Badness: 0.0588
13-limit
Commas: 65/64, 81/80, 105/104, 121/120
POTE generator: ~11/9 = 346.910
Map: [<1 1 0 8 2 6|, <0 2 8 -18 5 -8|]
EDOs: 7, 38df, 45ef, 83bcdef
Badness: 0.0343
Maqamic
de:maqamisch Deutsch
Main article: Maqamic
Commas: 81/80, 36/35, 121/120
Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.
POTE generator: ~11/9 = 350.934
Mapping generator: ~11/9
Map: [<1 1 0 4 2|, <0 2 8 -4 5|]
Generators: 2, 11/9
EDOs: 7, 10c, 17c, 24d, 31d
13-limit
Commas: 81/80, 36/35, 121/120, 144/143
POTE generator: ~11/9 = 350.816
Mapping generator: ~11/9
Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|]
Generators: 2, 11/9
EDOs: 7, 10c, 17c, 24d, 31d
Migration
Commas: 81/80, 121/120, 126/125
POTE generator: ~11/9 = 348.182
Mapping generator: ~11/9
Map: [<1 1 0 -3 2|, <0 2 8 20 5|]
EDOs: 31, 100de, 131bde, 162bde
Badness: 0.0255
Mohamaq
Commas: 81/80, 392/375
POTE generator: ~25/21 = 350.586
Mapping generator: ~25/21
Map: [<1 1 0 -1|, <0 2 8 13|]
EDOs: 17c, 24, 65c, 89cd
Badness: 0.0777
11-limit
Commas: 56/55, 77/75, 243/242
POTE generator: ~11/9 = 350.565
Mapping generator: ~11/9
Map: [<1 1 0 -1 2|, <0 2 8 13 5|]
EDOs: 17c, 24, 65c, 89cd
Badness: 0.0362
13-limit
Commas: 56/55, 66/65, 77/75, 243/242
POTE generator: ~11/9 = 350.745
Mapping generator: ~11/9
Map: [<1 1 0 -1 2 4|, <0 2 8 13 5 -1|]
EDOs: 17c, 24, 41c, 65c
Badness: 0.0287
Orphic
Commas: 81/80, 5898240/5764801
POTE generator: ~7/6 = 275.794
Mapping generator: ~343/288
Map: [<2 1 -4 4|, <0 4 16 3|]
Wedgie: <<8 32 6 32 -13 -76||
EDOs: 26, 74, 174bd, 248bd
Badness: 0.2588
11-limit
Commas: 81/80, 99/98, 73728/73205
POTE generator: ~7/6 = 275.762
Mapping generator: ~77/64
Map: [<2 1 -4 4 8|, <0 4 16 3 -2|]
EDOs: 26, 48c, 74, 248bd, 322bd
Badness: 0.1015
13-limit
Commas: 81/80, 99/98, 144/143, 2200/2197
POTE generator: ~7/6 = 275.774
Mapping generator: ~63/52
Map: [<2 1 -4 4 8 2|, <0 4 16 3 -2 10|]
EDOs: 26, 48c, 74, 174bd, 248bd, 322bd
Badness: 0.0535
Mothra
Commas: 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. In the 2.3.7-limit, mothra is identical to slendric.
Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.
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
POTE generator: ~8/7 = 232.193
Mapping generator: ~8/7
Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.
Map: [<1 1 0 3|, <0 3 12 -1|]
Generators: 2, 8/7
Wedgie: <<3 12 -1 12 -10 -36||
EDOs: 5, 26, 31
Badness: 0.0371
11-limit
Commas: 81/80, 99/98, 385/384
POTE generator: ~8/7 = 232.031
Mapping generator: ~8/7
Map: [<1 1 0 3 5|, <0 3 12 -1 -8|]
EDOs: 5, 26, 31, 88, 150, 181
Badness: 0.0256
13-limit
Commas: 81/80, 99/98, 105/104, 144/143
POTE generator: ~8/7 = 231.811
Mapping generator: ~8/7
Map: [<1 1 0 3 5 1|, <0 3 12 -1 -8 14|]
EDOs: 5, 26, 31, 57, 88
Badness: 0.0240
Cynder
Commas: 45/44, 81/80, 1029/1024
POTE generator: ~8/7 = 231.317
Mapping generator: ~8/7
Map: [<1 1 0 3 0|, <0 3 12 -1 18|]
EDOs: 26, 57e, 83bce
Badness: 0.0557
13-limit
Commas: 45/44, 78/77, 81/80, 640/637
POTE generator: ~8/7 = 231.293
Mapping generator: ~8/7
Map: [<1 1 0 3 0 1|, <0 3 12 -1 18 14|]
EDOs: 26, 57e, 83bce
Badness: 0.0341
Mosura
Commas: 81/80, 176/175, 1029/1024
POTE generator: ~8/7 = 232.419
Mapping generator: ~8/7
Map: [<1 1 0 3 -1|, <0 3 12 -1 23|]
EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce
Badness: 0.0313
13-limit
Commas: 81/80, 144/143, 176/175, 1029/1024
POTE generator: ~8/7 = 232.640
Mapping generator: ~8/7
Map: [<1 1 0 3 -1 7|, <0 3 12 -1 23 -17|]
EDOs: 31, 67, 98
Badness: 0.0369
Squares
Commas: 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.
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>]
Eigenmonzos: 2, 5
POTE generator: ~9/7 = 425.942
Mapping generator: ~9/7
Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.
Map: [<1 3 8 6|, <0 -4 -16 -9|]
Generators: 2, 9/7
EDOs: 14, 31, 262, 293
Badness: 0.0460
Music:
By Chris Vaisvil
Square 8
11-limit
Commas: 81/80, 99/98, 121/120
POTE generator: ~9/7 = 425.957
Mapping generator: ~9/7
Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|]
EDOs: 5, 8, 11, 14, 17, 31
Badness: 0.0216
13-limit
Commas: 81/80, 99/98, 121/120, 66/65
POTE generator: ~9/7 = 425.550
Mapping generator: ~9/7
Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|]
EDOs: 17c, 31, 79cf, 110cef, 141cef
Badness: 0.0255
Agora
Commas: 81/80, 99/98, 105/104, 121/120
POTE generator: ~9/7 = 426.276
Mapping generator: ~9/7
Map: [<1 3 8 6 7 14|, <0 -4 -16 -9 -10 -29|]
EDOs: 31, 45ef, 76e
Badness: 0.0245
Cuboctahedra
11-limit
Commas: 81/80, 385/384, 1375/1372
POTE generator: ~9/7 = 425.993
Mapping generator: ~9/7
Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|]
EDOs: 14, 31, 45, 200
Badness: 0.0568
Liese
de:Liese Deutsch
Commas: 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.
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>]
Eigenmonzos: 2, 5
POTE generator: ~10/7 = 632.406
Mapping generator: ~10/7
Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.
Map: [<1 0 -4 -3|, <0 3 12 11|]
Generators: 2, 10/7
EDOs: 17, 19, 55, 74
Badness: 0.0467
Liesel
Commas: 56/55, 81/80, 540/539
POTE generator: ~10/7 = 633.073
Mapping generator: ~10/7
Map: [<1 0 -4 -3 4|, <0 3 12 11 -1|]
EDOs: 17c, 19, 36, 91ce
Badness: 0.0407
13-limit
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
Commas: 56/55, 78/77, 81/80, 91/90
POTE generator: ~10/7 = ~13/9 = 633.042
Mapping generator: ~10/7
Map: [<1 0 -4 -3 4 0|, <0 3 12 11 -1 7|]
EDOs: 17c, 19, 36, 91cef
Badness: 0.0273
Elisa
Commas: 77/75, 81/80, 99/98
POTE generator: ~10/7 = 633.061
Mapping generator: ~10/7
Map: [<1 0 -4 -3 -5|, <0 3 12 11 16|]
EDOs: 19e, 36e
Badness: 0.0416
Lisa
Commas: 45/44, 81/80, 343/330
POTE generator: ~10/7 = 631.370
Mapping generator: ~10/7
Map: [<1 0 -4 -3 -6|, <0 3 12 11 18|]
EDOs: 19
Badness: 0.0548
13-limit
Commas: 45/44, 81/80, 91/88, 147/143
POTE generator: ~10/7 = 631.221
Mapping generator: ~10/7
Map: [<1 0 -4 -3 -6 0|, <0 3 12 11 18 7|]
EDOs: 19
Badness: 0.0361
Jerome
Jerome is related to Hieronymus' tuning; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
Commas: 81/80, 17280/16807
POTE generator: ~54/49 = 139.343
Mapping generator: ~54/49
Map: [<1 1 0 2|, <0 5 20 7|]
Wedgie: <<5 30 7 20 -3 -40||
EDOs: 8, 9, 17, 26, 43, 112
Badness: 0.1087
11-limit
Commas: 81/80, 99/98, 864/847
POTE generator: ~12/11 = 139.428
Mapping generator: ~12/11
Map: [<1 1 0 2 3|, <0 5 20 7 4|]
EDOs: 8, 9, 17, 26, 43, 241
Badness: 0.0479
13-limit
Commas: 77/78, 81/80, 99/98, 144/143
POTE generator: ~13/12 = 139.387
Mapping generator: ~12/11
Map: [<1 1 0 2 3 3|, <0 5 20 7 4 6|]
EDOs: 8, 9, 17, 26, 43, 155, 198
Badness: 0.0293
17-limit
Commas: 78/77, 81/80, 99/98, 144/143, 189/187
POTE generator: ~13/12 = 139.362
Mapping generator: ~12/11
Map: [<1 1 0 2 3 3 2|, <0 5 20 7 4 6 18|]
EDOs: 8, 9, 17, 26, 43, 155
Badness: 0.0209
Meanmag
Commas: 81/80, 3125/3072
POTE generator: ~8/7 = 238.396
Mapping generator: ~7
Map: [<19 30 44 0|, <0 0 0 1|]
Wedgie: <<0 0 19 0 30 44||
EDOs: 19, 57, 76, 171bcd
Badness: 0.0770
Undevigintone
Commas: 49/48, 81/80, 126/125
POTE generator: ~11/8 = 538.047
Mapping generator: ~11
Map: [<19 30 44 53 0|, <0 0 0 0 1|]
EDOs: 19, 38d
Badness: 0.0364
13-limit
Commas: 49/48, 65/64, 81/80, 126/125
POTE generator: ~11/8 = 537.061
Map: [<19 30 44 53 0 70|, <0 0 0 0 1 0|]
EDOs: 19, 38d
Badness: 0.0229