Meaner tones

Paul Erlich's and my last posts may have crossed on the Net, but
I will try to answer his question in TD #1005.

He suggested a tuning with Major Thirds (TM) of 372 cents,
Fifths (F) of 720 and minor thirds (tm) of 348 and asks that I
compare this to 15-tet. I can't tune up the triads easily today, but
I did calculate the following:

Fifth Major T minor t (in cents)
PE's tuning 720 372 348
15-tet 720 400 320

720-F 18.05 (720-F)^2 325.62
T-372 14.3137 (T-372)^2 204.88
400-T 13.683 (400-T)^2 187.31
348-tm 32.36 (348-tm)^2 1047.09
320-tm 4.36 (320-tm)^2 19.0

Now let us sum the squared errors of the fifth and the Major
Third for both tunings:
15-tet 325.62 + 187.31 512.93
PE's 325.62 + 204.88 530.5

15-tet is the more consonant tuning by this test, though not by much.

Now let us add the squared error of the minor third to each.
15-tet 512.93 +19531.93
PE's 530.5 + 1047.09 77.59

I will stipulate that this is a very large difference, but we already
knew that 15 tet was the more consonant. Whether the subjective
difference is proportional to the difference in the summed squares
is something to be determined experimentally.

After I did this (to put off doing my taxes, alas), I decided to
look for a tuning that would equalize the errors in major/minor triads
with fifths of 720 cents. I solved these two equations: X + Y 720
(X, Y are the Major and minor thirds) and X-386.3137 Y-315.6413
(errors in TM, tm). By substituting in, one gets 395.3362 for X,
324.6638 for Y, 9.0225 for the absolute differences and 488.43 for
the total squared error as (9.0225^2 x 2 + 325.62), 407.03 without
the tm. This is also the unweighted least squares minimum error solution
as well. [(TM-X)^2 + (tm-720+X)^2, differentiate wrt X, set equal to
zero, solve for X.] As both thirds are on the sharp side, this should
be a pretty good tuning. (Essentially what I did was to take the
error in the Fifth and split it between the two thirds.)

For comparison, the corresponding values for 10-tet with its neutral
third of 360 cents are 1018.03 and 2985.72. Ten fails by both tests
again.

Paul is correct that 15-tet is the more consonant of the two tunings he
suggested and he is certainly correct that adding in the minor third
(or Major Sixth) shows the differences dramatically. My point (in case
someone forgot what we are discussing) is that in this case using just
the
fifth and major third is sufficient to demonstrate the difference in
consonance, on the assumption that squared errors from JI is a good
proxy measure consonance.

In the meantone cases, I think it would be very difficult to hear the
differences between the various tunings we've suggested, but Paul could
be right.

--John



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