Meaner tones again

Paul E: The weights I used were 3, 5, and 5. My error function
was thus 3*(F-X)^2 + 5*(TM-4*X+2*OCT)^2 + 5*(MS-3*X+0CT)^2.
(F�nts(3/2), TM�nts(5/4), MScents(5/3), OCT1200)
Your weights are 9 25 25, the squares of these. I weighted
the derivatives, you the squared errors themselves. I'm not
sure which is more correct mathematically, but your unweighted
gives a slightly lower sum of sq'ed errors.

I thought our original question was optimising meantone-like
tunings. When one goes beyond the meantone range, other functions
must be used (8 fifths down for TM, 14 down for the Harmonic 7th,
8 d or 9up for the TM for doubly positive systems).

Yes, I agree that JI triads are better than MT.

I agree that the 15/8 is much less important than the 6/5. In
fact if one sets the errors of the third and fifth equal,
one gets the 1/3 comma meantone, if opposite in sign, the 1/5
comma. In the first case the 6/5 is just, the second, the 15/8.
I have analysed the tunings in my post in terms of the abs values
of the errors. In which case the errors of the 5/3 and 15/8 may
or may not be equal in a given tuning. The sum of the F and T
errors may or may not equal the errors in either the 5/3 or 15/8.
I could take the algebraic sum of the errors rather than the abs
value instead, I suppose.

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.

--John


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From: PAULE

John Chalmers has derived some meantone tunings using the augmented sixth as
the approximation to the harmonic seventh. But what if, more relevant to
modern practice, the minor seventh must approximate the harmonic seventh,
the minor third must approximate the 7:6, and the diminished fifth must
approximate the 7:5? Add terms for these to the usual 5-limit terms, and the
following "optimal" tunings are found:

Equal-weighted:
Fifth3+(log(3)+13*log(5)-11*log(7))/log(2))/75 octaves, or
702.2260 cents

Limit-weighted:
Fifth075+(81*log(3)+469*log(5)-539*log(7))/log(2))/3035 octaves, or
703.4736 cents

So if one wants a diatonic system with optimal dominant seventh (and
half-diminished seventh, if you subscribe to some form of duality) chords,
one must use fifths slightly _larger_ than pure.

-Paul

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From: PAULE

The absolute error criterion is minimized with 1/4-comma meantone whether
the errors are weighted or unweighted. The two slightly different answers I
posted were a result of computer approximation error.

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