2.3.17.19 subgroup: Difference between revisions
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The 2.3.17.19 subgroup is arguably quite significant to [[12edo]]. Its fifth, minor second, and minor third almost perfectly correspond to the octave-reduced harmonics of 3, 17, and 19, respectively. 12edo's patent val for this subgroup is {{val|12 19 49 51}}. | The '''2.3.17.19 subgroup''' (a.k.a. ''sana'' in [[color notation]]) consists of [[just intonation]] [[Interval|intervals]] such that the only [[prime factor]]<nowiki/>s in all [[Ratio|ratios]] are 2, 3, 17 and 19. This subgroup is a [[Rank and codimension|rank-4]] system, and can be modeled in a 3-dimensional [[lattice]], with the primes 2, 3, 17 and [[19/1|19]] represented by each dimension. The prime [[2/1|2]] does not appear in typical lattices because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a seventh dimension is needed. | ||
The 2.3.17.19 subgroup is arguably quite significant to [[12edo]], as it is the subgroup it does best with difference. Its fifth, minor second, and minor third almost perfectly correspond to the octave-reduced harmonics of 3, 17, and 19, respectively. 12edo's patent val for this subgroup is {{val|12 19 49 51}}. | |||
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Latest revision as of 11:53, 6 August 2026
The 2.3.17.19 subgroup (a.k.a. sana in color notation) consists of just intonation intervals such that the only prime factors in all ratios are 2, 3, 17 and 19. This subgroup is a rank-4 system, and can be modeled in a 3-dimensional lattice, with the primes 2, 3, 17 and 19 represented by each dimension. The prime 2 does not appear in typical lattices because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is needed.
The 2.3.17.19 subgroup is arguably quite significant to 12edo, as it is the subgroup it does best with difference. Its fifth, minor second, and minor third almost perfectly correspond to the octave-reduced harmonics of 3, 17, and 19, respectively. 12edo's patent val for this subgroup is ⟨12 19 49 51].
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