Talk:Tenney–Euclidean tuning: Difference between revisions
Cmloegcmluin (talk | contribs) →"Frobenius" tuning: redact out-of-date nonsense, and add'l correction |
→Motivation & "weaknesses": perhaps a better idea |
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== Crazy math theory's dominating the article == | == Crazy math theory's dominating the article == | ||
Anybody can read this article in its current shape and learn how to derive the TE tuning, TE generators, etc.? I can't. I learned it by coming up with the idea of RMS-error tuning myself, posting it on reddit and get told that was actually called TE tuning. | |||
Anybody can read this article in its current shape and learn how to derive the TE tuning, TE generators, etc.? I can't. | |||
That said, TE tuning is an easy problem if you break it down this way. | That said, TE tuning is an easy problem if you break it down this way. | ||
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It's a least squares problem of the following linear equations: | It's a least squares problem of the following linear equations: | ||
<math>( | <math>(VW)^\mathsf{T} \vec{g} = W\vec{p}</math> | ||
where | where ''V'' is the known mapping of the temperament, '''g''' the column vector of each generators in cents, '''p''' the column vector of targeted intervals in cents, usually prime harmonics, and ''W'' the weighting matrix. | ||
This is an overdetermined system saying that the sum of ( | This is an overdetermined system saying that the sum of (''VW'')<sup>T</sup><sub>''ij''</sub> steps of generator '''g'''<sub>''j''</sub> for all ''j'' equals the corresponding interval (''W'''''p''')<sub>''i''</sub>. | ||
'''How to solve it?''' | '''How to solve it?''' | ||
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The only thing that matters is to identify the problem as a least square problem. The rest is nothing but manual labor. | The only thing that matters is to identify the problem as a least square problem. The rest is nothing but manual labor. | ||
I'm gonna try improving the readability of this article by adding my thoughts and probably clear it up. | I'm gonna try improving the readability of this article by adding my thoughts and probably clear it up. | ||
: | [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 18:52, 24 June 2020 (UTC) (updated [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:07, 13 July 2025 (UTC)) | ||
: | : Update: I gave the article some rework to bring its level to my standard. | ||
: | : The conventional way to write the equation is: | ||
: | : <math>GVW = JW</math> | ||
: | : The targeted interval list is known as ''JIP'' and is denoted ''J'' here. The main difference from my previous comment is that the generator list and the JIP are presented as row vectors. It can be further simplified to | ||
: which is pretty clearly | : <math>GV_W = J_W</math> | ||
: which is pretty clearly presented in the article now. | |||
: [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 17:39, 16 December 2021 (UTC) (updated [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:07, 13 July 2025 (UTC)) | |||
== Damage, not error? == | == Damage, not error? == | ||
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::::::: Also, per Sintel's original comment, I have by now realized that the Frobenius tuning is the one which minimizes the Frobenius norm of ''the projection matrix'' (''not'' the mapping matrix), by defining the projection matrix as the mapping matrix left-multiplied by (a generator embedding matrix equal to) its own pseudoinverse. So the name does make sense, but I think it should be clarified where it is mentioned. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:38, 11 December 2022 (UTC) | ::::::: Also, per Sintel's original comment, I have by now realized that the Frobenius tuning is the one which minimizes the Frobenius norm of ''the projection matrix'' (''not'' the mapping matrix), by defining the projection matrix as the mapping matrix left-multiplied by (a generator embedding matrix equal to) its own pseudoinverse. So the name does make sense, but I think it should be clarified where it is mentioned. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:38, 11 December 2022 (UTC) | ||
== Motivation & "weaknesses" == | |||
We'll need to review these sections. It's written way too vague yet still has too many judgements baked in. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:08, 13 July 2025 (UTC) | |||
: Here's some specifics from the "weaknesses" section. | |||
: > TE must give an undue weight to extremely large intervals, as evidenced by the fact that you have to choose a prime limit to get sensible results. It doesn't converge as you keep adding primes. | |||
: This is one of the possible complaints about Tenney weighting in general, and points towards Wilson weighting. I'm not sure this article is the right place to discuss it. | |||
: > The optimization with octaves constrained to be pure (CTE) is controversial, and many believe the implied TE error function being minimized to be incorrect in this case and so generally invalid. Variations to fix this are considered under Constrained tuning. | |||
: This is simply wrong. TE and CTE optimize for the same error function. CTE is controversial not becuz of a problem in the error function but becuz octave tempering is a means of effectively compensating for divisive ratios. | |||
: > Weighting intervals according to their size gives less weight to higher primes than an RMS specifically considering audible ratios within the prime limit. | |||
: This is another possible complaint about Tenney weighting in general, and points towards equal/equilateral weighting, which is an opposite of Wilson weighting. This and the first point should never appear in the same place from the same projected user group. | |||
: > That TE tuning appears to be a limit to infinity of RMS of intervals approaching infinite complexity is meaningless. The human ear can't perceive even moderately complex intervals and the convergence is too slow to be psychoacoustically meaningful. | |||
: This is similar and complementary to the first point, and again points towards Wilson weighting. | |||
: > Optimizing for an average rather than a minimax means intolerably mistuned intervals are balanced by needlessly pure intervals, rather than ensuring all intervals get tempered to within tolerable bounds. | |||
: While I get where this is coming from, minimax gives purely tuned intervals way more frequently than RMS. It's also not guaranteed that minimax can tune all relevant intervals within tolerable bounds. | |||
: > Requiring octaves to be tempered is inconsistent with some electronic intervals with octave-repeating tuning tables. | |||
: This isn't a distinct issue on top of the ubiquity of octave equivalence. Replace "octave" with "fifth" and the logic still holds, so I don't think it needs to be brought up. | |||
: Fixing this section will be difficult as most of the contents are so out of place. <s>I'd say a complete removal is the best way to go. Then we might wanna selectively re-install them to the relevant articles.</s> Moving some of the relevant info to the "otherwise normed tunings" section would make sense, as that's where we compare different norms. | |||
: As for the "motivation" section, most of it is discussing the characteristics of Tenney weighting and Euclidean norms. I'm thinking about condensing it to something similar to the "features" section of Wikipedia's IPA sound articles. | |||
: —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:29, 7 December 2025 (UTC) (updated [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:23, 8 December 2025 (UTC)) | |||