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Reply to John Chalmers (Paul E)

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

3/12/1997 9:04:40 AM
From: PAULE

>As for your more complex psychoacoustic model, what meantone-like
>tuning(s) does it generate?

If you are referring to the normal curve model I proposed, the answer is,
the same ones! It's a monotonic decreasing function of the sum of the
squared errors, so minimizing the latter is the same as maximizing the
former. If nothing else, the function is nice because is gives all tunings a
rating between zero and one.

>As for "expressive unisons," they can be very poignant. I believe
>Wyschnegradsky made considerable use of them.

What is this referring to?

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

3/12/1997 3:02:45 PM
From: PAULE

John C. wrote,

>Somewhat to my surprise, using the ABS function did not generate
>any tunings different from the ones I got by the lst squares
>technique or simply setting the errors equal or opposite. The
>exact solution depended upon whether I minimized the abs(sum)
>or sum of ABS (errors) as expected, but the results were still
>tunings I already had.

Uh, abs(sum) seems like a bad idea, since pos. and neg. errors cancel out in
the sum. As for sum of abs(errors), you certinaly get different results from
least squares. Using the three intervals in the 5-limit triad, the best
meantone tunings according to absolute error have fifths of 696.57842466
cents (unweighted) and 696.57841680 cents (limit-weighted). I normally don't
use so many decimal places but you need to to see the difference between
these two! Maybe you only thought you got tunings you already had!

-Paul E.

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