User:Dummy index/Heuristics for picking a basis of JI subgroup

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Domain basis of just intonation subgroup will be normalized in order to detect same subgroup and same temperament. But in exploration, It can be transformed and used as a clue to thinking.

Just intonation subgroup #Normalization

Full-rank

Domain basis

Why not normalize a subgroup basis?

• It may just be a list of candidates of basis elements.
• Nonoctave gang:
  • One says it should be a strict no-twos subgroup for tritave temperament. Another says it might be useful to include 4 for avoiding interval become near-octave (get help from odd-numbered ed4s).
    • Combined with the convention of putting equave at the first base, it is 3.4.….
• There is also a style in which the bases are arranged in ascending order without distinction between prime and composite numbers.
• The desire to choose the basis so that the error of basis is as small as possible.
  • e.g. 2.3.5 12&19 [⟨1 0 -4], ⟨0 1 4]] is intended to 1/4-comma meantone, whereas 2.3.5/3 12&19 [⟨1 0 -4], ⟨0 1 3]] is intended to 1/3-comma meantone.

Guide to manipulating a subgroup basis

• Usual properties of Column HNF:
• Treating the unregularized basis:
  • If different powers of three are included at the same time, it will be replaced with a power of 3 with exponent GCD, or it will be treated as a dual-fifth system.
  • If entangled bases such as 4.14.18.30 seems to be full-rank on related primes subgroup (2.3.5.7 in this case), it is convertible between composite subgroup bases and fractional subgroup bases (4.9/2.15/2.7/2). But it's not always easy. (e.g. 9/2.25/3.5/4)
• Non-over-1 temperament shows some examples of attempts to exclude (octave-reduced) harmonics from the basis because they do not appear on the desired scale.