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Reply to John Chalmers/Lst Sqr Tun (Paul E)

🔗Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul)

3/5/1997 3:25:08 PM
From: PAULE

>My Variant of Paul's 696.0743748037858
>TF(6*F+40*T+30*MS+110*OCT)/256

Could you explain how this is derived? Did you also include absolute-error
versions of my optimizations in your list? Is this one?

>However I'm still not completely convinced
>that the intonation of the 5/3 and 6/5 are as important as the
>that of 3/2 and 5/4, but even if it is, there are not independent
>intervals.

Ok, let's assume the intonation of the 6/5 is as important as that of 3/2
and 5/4. They are not independent intervals. So what? Yes, optimizing on 3/2
and 5/4 guarantees a pretty good 5/3. But optimizing on 3/2 and 6/5
guarantees a pretty good 5/4. However, these optimizations lead to different
tunings, neither of which is as optimal (under our assumption) as using all
three. Who cares about independence?

John, under your optimization, the tuning of 6/5 (cents(3/2)-cents(5/4)) is
as important as that of 15/8 (cents(3/2)+cents(5/4)). I would argue that the
tuning of the 15/8 is unimportant, while the tuning of the 6/5 is very
important. At the very least, 6/5 should be included in the optimization
with a smaller weight than 3/2 and 5/4, but I prefer to give it full weight.

>In triads, one has both a major third and a minor third and
>adjusting either the fifth or the major third adjusts the minor third
>as well. There are only two degrees of freedom, not three, which reduces
>to one in meantone-like tunings or their positive analogs.

This is what made it possible to make this a one-dimensional minimization
problem. But again, who cares about degrees of freedom? This is not
statistical inference here!

>I thoroughly agree that the triads of 15-tet are more consonant than
>those of 10-tet, but this is not relevant.

It sure is relevant! See below.

>We were both talking about
>meantone-like tunings which form their major thirds as 4 fifths up,
>their major sixths as 3 fifths up and their minor thirds as the difference
>between the fifth and the major third, which is equivalent to taking
>the minor third as 3 fifths down. If the fifth is greater than 700 cents,
>these relations no longer hold in any meaningful sense.

True, but the original objective ("consonance") function is expressed
independently of these relations. In the space of meantone tunings, these
relations can be used to make the problem into a one-dimensional problem.
But in general, it can only be reduced to a two-dimensional problem, since,
as you point out, there are two independent intervals (any two can be chosen
without affecting the result.) For example, would you not agree that just
intonation gives better triads than any meantone tuning?

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