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

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This page is a guide, based on RTT, but with a relatively minor approach.

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?

Here are a lot of just a opinion, just a trend on this wiki, etc. please add your point of view!

  • 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.
      • In fact, Graham Breed's temperament finder has different TE tuning results depending on the basis selection.
      • However, it is important to note that this does not mean "2.5-eigenmonzo" or "2.5/3-eigenmonzo".
      • As a reminder, the 2.3.5/3 subgroup is the same abelian group as 2.3.5, as opposed to 2.9.5/3 and 2.9.5 being different abelian groups.
  • Such small errors contribute to describing the relationship between n-EDO and n-ET:

Guide to manipulating a subgroup basis

Here are a lot of incomplete mathematic description, incomplete algorithms. Other approaches are welcome.

  • Usual properties of Column HNF (antitransposed Hermite form):
    • First basis element is a p1-limit interval, second basis element is a p2-limit interval, … where p1 < p2 < … and all pi are minimized.
      • (I think it's a good idea to always take this into consideration.)
    • In practice, each element of the basis is written as a positive interval. (positive ratio form)
      • (This article may not adhere to this.)
  • 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)
    • 5/3.11/3.13/5 = 5/3.11/3.13/3 = 5/3.11/5.13/5 = 11/3.11/5.13/11 = …. Will be 5/3.11/3.13/3 if normalized in column HNF.
  • 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.

Systematic methods

  • 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: (* no explanation for the case such as k = 4 and 3^2 appears)
    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 (or remove it), 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 with 76 edosteps.
      4. 2.5.7.99.117. Oh no, can't represent 11/3 and 13/3.
      5. 2.5.7.11/3.13/3. (non-over-1 for 3, 11 and 13)
      6. 2.3.5.7.11/3.13/3. (superficial exclusion style)
    • Example with 14edo "Hi, I want 5:7:9:11." Okay, …
      1. How to define k, or its candidates, that's the question.
      2. k=2: 2*14 subgroup is 2.3.25.35.55.13. remove 25 and fractionalize, 2.3.7/5.11/5.13. add 5…? It's failing to organize 3 and 5 and you have to calculate the number of steps in 5 to get 5:7:9:11 to the desired number of steps.
      3. k=4: 4*14 subgroup is 2.81.45.189.99.39. remove 81 and fractionalize, 2.5/9.7/3.11/9.39. (39? 13/27?) It's a pity, if it's 7/9, you can assemble 5:7:9:11. I can't help it, so I'll add 3.
      4. k=3 and k=5 both maintain the wall of ±50% error on 14edo, so these cases, 5 is likely to be left out (maybe 2.5/27.7/9.11/9.39 @ k=5). I can't help it, so I'll add 3.
      5. It's inconvenient that the threshold can't be changed continuously. (But for such, see #Handpicking methods.)
      6. 2*14 subgroup on which tempers out the same commas as the patent val for 2.9.5.7.11.13 subgroup 28edo: 2.81.45.63.99.13. remove 81 and fractionalize, 2.5/9.7/9.11/9.13. Can assemble 5:7:9:11. Whether or not to add 3 (or add 9) is up to you. In this case, no problem to add 3 because desired mapping of 9 is even edosteps of 14edo.
      7. If no-3s-but-9s and the order of formal prime numbers is 2, 5, 7, 9, 11, …, 2*14 subgroup is 2.25.35.45.55.13. remove 25 and fractionalize, 2.7/5.9/5.11/5.13. Can assemble 5:7:9:11. But when add 5, If you use the nearest neighbor of 5, then 7, 9, and 11 are not nearest neighbors. After all, the mapping of p to be added needs to be confirmed.
      8. Similarly, the "desired mapping of 9" is not just a nearest approximation, but rather a sharp/flat tendency.
      9. I want the algorithm to judge whether it's a 3s or a 9s if you give it k. Hmm
    • So, from here, I would like to consider an algorithm that determines the recommended value of k and the conditions for adopting 9 instead of 3.
      • In addition, we will not consider adopting 3^3 or more.
      • As a criterion, we will mainly use the nearest neighbor approximation of 2*N-edo.
      • abs(error(3)) < 12.5% means that 9 is approximated (in NN) by a multiple of 4 edosteps at 2*N-edo. Therefore, it is good to use 3 * 3 = 9 without complaint. (use 3)
        • Under this condition, there is no need to modify 3 because the k*N subgroup always has an independent 3 in the range of k up to 4. Therefore, use this algorithm to determine the appropriate k (in the range of 4 or less) for the following prime numbers with a large error.
      • abs(error(3)) > 37.5% means that 9 is approximated by a odd multiple of 2 edosteps at 2*N-edo. Therefore, we want to map 9 to an odd number of steps at N-edo, so we can't use 3 as independent basis element. (use 9)
        • (You may decide which one (or none) to adopt based on whether there are other prime numbers near 3 or near 9.)
        • Under this condition, use this algorithm to determine the appropriate k for next prime number with a large error as k=2 or 3 or 4. (if k=3, then "remove 27 and fractionalize by 27 and add 9". This is surprising, but there is nothing wrong with the fact that 3 and 27 do not match up when 27 appears in the k*N subgroup and makes it 3.)
      • 12.5% < abs(error(3)) < 37.5% means that 9 is approximated with an odd number of edosteps at 2*N-edo.
        • If 12.5% < abs(error(3)) < 25%, then 2*N subgroup will always have independent 3. 9 would normally have an error of over 25%, but since it maps to 3*3, it becomes impossible to express connections with other primes that have over 25% error. (so k=4 or 5 or 6 may be more fit)
        • If 25% < abs(error(3)) < 37.5%, you may wonder whether the appearing fractional basis elements should be /3 or /9. (so k=3 or 4 may be more fit)
      • ???(WIP)
    • Example with 16edo:
      1. abs(error(3))=35.9%.
      2. k=3. 3*16 subgroup will have 27, …
    • Example with 14edo:
      1. abs(error(3))=18.9%.
      2. k=4 or 5 or 6. k*14 subgroup will have 3^k, …

Handpicking methods

(WIP)

  • First, let's outline the steps to get a nonpatent val for a prime basis.
    • ~~
    • Grouping (clustering) the factors on the illustraion of ~~
    • (In this section, "group" is that in the everyday sense.)
    • In other words, remove a few factors to get 40% or bigger blank zone on the error circle.
    • ~~
  • Now it's time to get a nonstandard basis.
    • Group the factors more finely. The group that contains 2 (which is always in the 0% error position) is called the over-1 group, and each of the other groups is called the non-over-1 group.
    • ~~
    • If 3 is not grouped, ~~
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