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Diaschismic family: Difference between revisions

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== Srutal (12&34, aka diaschismic) ==
== Srutal (12&34, aka diaschismic) ==
Subgroup: 2.3.5
Subgroup: 2.3.5


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{{main| Pajara }}
{{main| Pajara }}


Pajara is closely associated with 22 EDO (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22 EDO, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12 EDO and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.


Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
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POTE generator: ~3/2 = 710.240
POTE generator: ~3/2 = 710.240


Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdf }}
Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdff }}


Badness: 0.025176
Badness: 0.025176
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
Mapping: [{{val| 2 0 11 12 -9 1 5 }}, {{val| 0 1 -2 -2 5 2 1 }}]
POTE generator: ~3/2 = 710.221
Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdff }}
Badness: 0.018249


==== Pajaro ====
==== Pajaro ====
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[[Mapping]]: [{{val|2 2 7 6}}, {{val|0 3 -6 -1}}]
[[Mapping]]: [{{val|2 2 7 6}}, {{val|0 3 -6 -1}}]
{{Multival|legend=1| 6 -12 -2 -33 -20 29 }}


[[POTE generator]]: ~8/7 = 234.492
[[POTE generator]]: ~8/7 = 234.492
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Badness: 0.045731
Badness: 0.045731
== Quadrasruta ==
Subgroup: 2.3.5.7
[[Comma list]]: 2048/2025, 2401/2400
[[Mapping]]: [{{val|2 0 11 8}}, {{val|0 4 -8 -3}}]
{{Multival|legend=1| 8 -16 -6 -44 -32 31 }}
[[POTE generator]]: ~15/14 = 123.784
{{Val list|legend=1| 10, 38c, 48c, 58, 68, 126 }}
[[Badness]]: 0.073569
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 176/175, 896/891, 2401/2400
Mapping: [{{val|2 0 11 8 22}}, {{val|0 4 -8 -3 -19}}]
POTE generator: ~15/14 = 123.882
Optimal GPV sequence: {{Val list| 58, 126, 184c, 310bccde }}
Badness: 0.049018
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 196/195, 512/507, 676/675
Mapping: [{{val|2 0 11 8 22 9}}, {{val|0 4 -8 -3 -19 -2}}]
POTE generator: ~14/13 = 123.901
Optimal GPV sequence: {{Val list| 58, 126f, 184cff }}
Badness: 0.028463
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255
Mapping: [{{val|2 0 11 8 22 9 5}}, {{val|0 4 -8 -3 -19 -2 4}}]
POTE generator: ~14/13 = 123.838
Optimal GPV sequence: {{Val list| 58, 126f }}
Badness: 0.023820
=== Quadrafourths ===
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 2048/2025
Mapping: [{{val|2 0 11 8 -1}}, {{val|0 4 -8 -3 10}}]
POTE generator: ~15/14 = 123.983
Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58 }}
Badness: 0.049114
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 196/195, 243/242, 676/675
Mapping: [{{val|2 0 11 8 -1 9}}, {{val|0 4 -8 -3 10 -2}}]
POTE generator: ~14/13 = 123.972
Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58 }}
Badness: 0.026743
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220
Mapping: [{{val|2 0 11 8 -1 9 5}}, {{val|0 4 -8 -3 10 -2 4}}]
POTE generator: ~14/13 = 123.923
Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58, 126eef, 184ceeff }}
Badness: 0.022239


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]
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