Talk:Tenney–Euclidean tuning: Difference between revisions
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It's a least square problem of the following linear equations: | It's a least square problem of the following linear equations: | ||
<math> | <math>(MW)^\mathsf{T} \vec{g} = W\vec{p}</math> | ||
where M 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 M 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 (MW)<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) | |||
Revision as of 18:52, 24 June 2020
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This page also contains archived Wikispaces discussion. |
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. 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.
What's the problem?
It's a least square problem of the following linear equations:
[math]\displaystyle{ (MW)^\mathsf{T} \vec{g} = W\vec{p} }[/math]
where M 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 (MW)Tij steps of generator gj for all j equals the corresponding interval (Wp)i.
How to solve it?
The pseudoinverse is a common means to solve least square problems.
We don't need to document what a pseudoinverse is, at least not in so much amount of detail, cuz it's not a concept specific in tuning, and it's well documented on wikipedia. Nor do we need to document why pseudoinverses solve least square problems. Again, that's not a question specific in tuning.
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.