Diaschismic family: Difference between revisions
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→Five-limit srutal (aka diaschismic): update keys; +srutal archagall |
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The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities. | The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities. | ||
== | == Srutal aka diaschismic == | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 2048/2025 | [[Comma list]]: 2048/2025 | ||
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[[Mapping]]: [{{val| 2 0 11 }}, {{val| 0 1 -2 }}] | [[Mapping]]: [{{val| 2 0 11 }}, {{val| 0 1 -2 }}] | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.898 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
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[[Badness]]: 0.019915 | [[Badness]]: 0.019915 | ||
=== | === Overview to extensions === | ||
==== 7-limit extensions ==== | |||
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. | The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. | ||
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Pajara, diaschismic, srutal and keen keep the same half-octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Bidia has a quarter-octave period and a fifth generator. | Pajara, diaschismic, srutal and keen keep the same half-octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Bidia has a quarter-octave period and a fifth generator. | ||
==== Subgroup extensions ==== | |||
Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup, resulting in ''srutal archagall''. | |||
=== Srutal archagall === | |||
Subgroup: 2.3.5.17 | |||
Comma list: 256/255, 289/288 | |||
Sval mapping: [{{val| 2 0 11 5 }}], {{val| 0 1 -2 1 }}] | |||
Sval mapping generators: ~17/12, ~3 | |||
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272 | |||
Optimal GPV sequence: {{Val list| 10, 12, 22, 34, 80, 114, 194bc }} | |||
Badness: 0.00575 | |||
== Srutal == | == Srutal == | ||