Diasem: Difference between revisions
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'''Diasem''' (also denoted 2s in [[groundfault]]'s [[aberrismic theory]]) is a 9-note [[Maximum variety|max-variety-3]], [[generator-offset]] scale with [[step signature]] 5L 2m 2s, equivalent to the [[semiquartal]] ([[5L 4s]]) mos with two of the small steps made larger and the other two made smaller. Diasem is [[chiral]], with two rotationally non-equivalent variants: ''right-handed (RH) diasem'' LmLsLmLsL and ''left-handed (LH) diasem'' LsLmLsLmL; these [[step pattern]]s are mirror images. The fact that the small step of diatonic is made smaller results in [[26edo]] and [[31edo]] diasem having better melodic properties than the respective diatonic scales. [[21edo]] is the smallest edo to support a non-degenerate diasem. | '''Diasem''' (also denoted 2s in [[groundfault]]'s [[aberrismic theory]]) is a 9-note [[Maximum variety|max-variety-3]], [[generator-offset]] scale with [[step signature]] 5L 2m 2s, equivalent to the [[semiquartal]] ([[5L 4s]]) mos with two of the small steps made larger and the other two made smaller. Diasem is [[chiral]], with two rotationally non-equivalent variants: ''right-handed (RH) diasem'' LmLsLmLsL and ''left-handed (LH) diasem'' LsLmLsLmL; these [[step pattern]]s are mirror images. The fact that the small step of diatonic is made smaller results in [[26edo]] and [[31edo]] diasem having better melodic properties than the respective diatonic scales. [[21edo]] is the smallest edo to support a non-degenerate diasem. | ||
Diasem can be tuned as a [[Just intonation subgroup|2.3.7 subgroup]] JI scale or a tempered version thereof, where L represents [[9/8]], m represents [[28/27]], and s represents [[64/63]]. This interpretation, or more generally the series of [[generator sequence]] scales generated by GS(7/6, 8/7) or GS(8/7, 7/6), has been named | Diasem can be tuned as a [[Just intonation subgroup|2.3.7 subgroup]] JI scale or a tempered version thereof, where L represents [[9/8]], m represents [[28/27]], and s represents [[64/63]]. This interpretation, or more generally the series of [[generator sequence]] scales generated by GS(7/6, 8/7) or GS(8/7, 7/6), has been named [[Tas]]. | ||
"Diasem" is a name given by [[groundfault]] (though others have discussed the scale before her). The name is a portmanteau of "diatonic" and "semiquartal" (or "[[Semaphore]]") since its step sizes are intermediate between that of [[diatonic]] (5L 2s) and [[semiquartal]] (5L 4s); it is also a pun based on the [[diesis]], which appears as the small step in the scale in the [[31edo]] and [[36edo]] tunings. | "Diasem" is a name given by [[groundfault]] (though others have discussed the scale before her). The name is a portmanteau of "diatonic" and "semiquartal" (or "[[Semaphore]]") since its step sizes are intermediate between that of [[diatonic]] (5L 2s) and [[semiquartal]] (5L 4s); it is also a pun based on the [[diesis]], which appears as the small step in the scale in the [[31edo]] and [[36edo]] tunings. | ||
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The notes of diasem form the {49/48, 567/512} Fokker block, which is a fundamental domain of the 2.3.7 pitch class lattice; it is possible to tile the entire infinite lattice with copies of right-hand diasem transLated by (49/48)<sup>''m''</sup>(567/512)<sup>''n''</sup> for integer ''m'' and ''n''. Including any one of the other three points on the boundary (28/27, 147/128, or 64/63) instead of 9/8 aLso yields Fokker blocks, more specifically, modes of three of the other [[dome]]s of diasem, and transLates of the parallelogram that do not have lattice points on the boundary lead to other domes of this Fokker block. However, only one other choice, 28/27, yields a diasem scale, and it yields the left-handed diasem mode mLLsLmLsL. | The notes of diasem form the {49/48, 567/512} Fokker block, which is a fundamental domain of the 2.3.7 pitch class lattice; it is possible to tile the entire infinite lattice with copies of right-hand diasem transLated by (49/48)<sup>''m''</sup>(567/512)<sup>''n''</sup> for integer ''m'' and ''n''. Including any one of the other three points on the boundary (28/27, 147/128, or 64/63) instead of 9/8 aLso yields Fokker blocks, more specifically, modes of three of the other [[dome]]s of diasem, and transLates of the parallelogram that do not have lattice points on the boundary lead to other domes of this Fokker block. However, only one other choice, 28/27, yields a diasem scale, and it yields the left-handed diasem mode mLLsLmLsL. | ||
As a Fokker block, 2.3.7 JI diasem is aLso | As a Fokker block, 2.3.7 JI diasem is aLso a product of the tempered 2.3.7 mosses Semaphore[9] (LsLsLsLsL) and septimal Mavila[9] (LLLsLLLsL). | ||
== Tunings == | == Tunings == | ||
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2.3.7 JI diasem also has the following generator-offset, SV3 supersets: | 2.3.7 JI diasem also has the following generator-offset, SV3 supersets: | ||
* a 19-note superset: mLsmsLmsmLsmsLmsmLs (5L 7m 7s), with L = 2187/2048, m = 28/27, and s = 64/63, | * a 19-note superset: mLsmsLmsmLsmsLmsmLs (5L 7m 7s), with L = 2187/2048, m = 28/27, and s = 64/63, | ||
* a 29-note superset: mLsmLmLsmLmLmsLmLmLsmLmLmsLmL (12L 12m 5s), with L = 28/27, m = 64/63, and s = 531441/ | * a 29-note superset: mLsmLmLsmLmLmsLmLmLsmLmLmsLmL (12L 12m 5s), with L = 28/27, m = 64/63, and s = 531441/524288. | ||
== See also == | == See also == | ||