User:Dummy index/Heuristics for picking a nonstandard 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

Subgroup basis matrix

Generator form manipulation – This is not an article about domain basis, but it is helpful to know how to transform it with mental calculation.

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 basis elements 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.
• Such small errors contribute to describing the relationship between EDO and ET: the mapping becomes "patent val for such subgroup basis", reducing the need to describe nonpatent vals.

Guide to manipulating a subgroup basis

• Usual properties of Column HNF:
  • First basis element is a p₁-limit interval, second basis element is a p₂-limit interval, … where p₁ < p₂ < … and all p_i are minimized.
• 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. Similarly, if there are four basis elements within the 5-limit, they are not linearly independent or need to be interpreted as Dual-n.
  • If entangled combination of basis elements such as 4.9/2.15/2.7/2 seems to be full-rank on related primes subgroup (2.3.5.7 in this case), it will be convertible between fractional subgroup basis elements and combinational subgroup basis elements (4.14.18.30). 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.
  • Note that this actually removes harmonics from the subgroup, not a superficial exclusion like in 2.3.5/3 in the above example.
• If you do something similar systematically for a specific N-EDO:
  1. Determine k*N subgroup with applicable k and prime limit.
  2. (p = 3, 5, 7, 11, …):
  3.  If it has p^k and you don't want include it with k-odd edosteps:
  4.   Remove p^k from the basis elements.
  5.   Fractionalize rest elements with p^k if necessary.
  6.   Add p to the basis elements if necessary.
  7. Try to fractionalize other big combinational basis elements, to minimize their taxicab distance but maintain their error.
  • Example with 16edo:
    1. 13-limit 3*16 subgroup is 2.27.5.7.99.117.
    2. There is only one matchable case 3^3.
    3. I don't want include 27.
    4. 2.5.7.99.117. Oh no, can't represent 11/3 and 13/3.
    5. 2.5.7.11/3.13/3.
    6. 2.3.5.7.11/3.13/3.