Understanding tuning optimizations geometrically

I'm trying to figure out how to visualize TE and TOP optimization. I
get that on some level we're minimizing the L[p] distance from the JIP
to -something- related to the "tuning map" of the temperament, but I'm
having trouble picturing exactly what it ought to look like.

It seems to me, as I think about it, that the tuning map for a rank-n
temperament can take as possible values anything on an n-dimensional
flat which passes through the origin, and that we're trying to find
the point on this flat which is closest to the line passing through
the JIP. If we're using TE, we'll end up trying to minimize the L2
distance to this point, and if we're using TOP, we'll end up trying to
minimize the Linfty distance.

What I don't get, in the case of L2, is that because we're using the
pseudoinverse, this whole thing must be somehow equivalent to a least
squares optimization. But, it's backwards from the sort of least
squares optimizations I know. The kind I know typically involve
fitting a line or a curve to a scatter plot of data, whereas in this
case, we're trying to find the point on a flat closest to some other
point. Is there some way to transform the question of TE optimization
into the other kind of least squares problem, the one that's like
regression?

-Mike

Distance in euclidean space is a sum of squares. Minimizing it is a least squares optimization.
The problem is conventional but the errors are backwards. JI intervals reflect physical reality. You can think of them as inexact measurements of tempered intervals if you like

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------Original message------
From: Mike Battaglia <battaglia01@gmail.com>
To: <tuning-math@yahoogroups.com>
Date: Friday, January 20, 2012 2:19:09 AM GMT-0500
Subject: [tuning-math] Understanding tuning optimizations geometrically

I'm trying to figure out how to visualize TE and TOP optimization. I
get that on some level we're minimizing the L[p] distance from the JIP
to -something- related to the "tuning map" of the temperament, but I'm
having trouble picturing exactly what it ought to look like.

It seems to me, as I think about it, that the tuning map for a rank-n
temperament can take as possible values anything on an n-dimensional
flat which passes through the origin, and that we're trying to find
the point on this flat which is closest to the line passing through
the JIP. If we're using TE, we'll end up trying to minimize the L2
distance to this point, and if we're using TOP, we'll end up trying to
minimize the Linfty distance.

What I don't get, in the case of L2, is that because we're using the
pseudoinverse, this whole thing must be somehow equivalent to a least
squares optimization. But, it's backwards from the sort of least
squares optimizations I know. The kind I know typically involve
fitting a line or a curve to a scatter plot of data, whereas in this
case, we're trying to find the point on a flat closest to some other
point. Is there some way to transform the question of TE optimization
into the other kind of least squares problem, the one that's like
regression?

-Mike

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