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Talk:Tenney–Euclidean tuning: Difference between revisions

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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. How I learned it was by coming up with the idea of RMS-error tuning, posting it on reddit and get told that was actually called TE tuning.  


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>(AW)^\mathsf{T} \vec{g} = W\vec{p}</math>
<math>(VW)^\mathsf{T} \vec{g} = W\vec{p}</math>


where A 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.  
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 (AW)<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>.  
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. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 18:52, 24 June 2020 (UTC)
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.


: Update: the page is clear enough now.
: The conventional way to write the equation is:


: The standard way to write the equation is:
: <math>GVW = JW</math>


: <math>G(AW) = J_0 W</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


: The targeted interval list is known as ''JIP'' and is denoted J<sub>0</sub> 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
: <math>GV_W = J_W</math>


: <math>GV = J</math>
: which is pretty clearly presented in the article now.


: which is pretty clearly displayed in the article. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 17:39, 16 December 2021 (UTC)
: [[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)
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