More Meantones

Least Square Tunings: With somewhat simpler assumptions than Paul
Erlich, I generated some linear temperaments designed to minimize
the errors of the Fifth, Major Third, and Harmonic 7th (C-A#).
It seems to me that the 5/3 and 6/5 are not independent of the 5/4
and 3/2 and not really necessary, though I believe Thomas Young
included all the simple 5 limit consonances in 3 or 4 octaves to
generate his temperaments (5/18, 11/40 comma, approximately 50-tet).

Therefore, I have used only functions of the cent values of the
intervals and X (as the temperament fifth) to construct error functions
Thus (F-X) is the error of Fifth, (T-4*X-2*oct)the error of the Major
Third, and (S-10*X+5*oct) that of the natural 7th. These error functions
may be squared, summed, differentiated and set equal to zero as Paul
described.

Analogous functions may be written for positive systems where X
is greater than 700 cents: (F-X), (T+8*X-5*oct), and (S+14*X-9*oct).

Finally, the terms may be weighted inversely as the magnitude of
their primes. Thus the (F-X) term may be divided by 3, the Major
Third term by 5, etc. after squaring.

Negative Systems:

1/4-comma Meantone, 5/4 Just696.5784

LST35696.8947

LST37696.9328

LST57 696.8406

LST357 696.8843

Just 7/4 NEG696.8826

Weighted LST35 697.0857

Weighted LST37 696.9982

Weighted LST57 696.8269

Weighted LST357 696.9229

LucyTuning 695.4930

Wilson's Meta-Meantone 695.6304

Wilson's Meta-1/5-Comma Meantone 697.0676

3 1/3-Comma, 6/5 Just 694.7862

3op5, 1/5-Comma, 15/8 Just 697.6537

Positive Systems:

Pythagorean , 3/2 Just 701.9550

6/5 Just, 1/9-SKHISMA, 3 Sabat-Garibaldi 701.7379

15/8 Just, 1/7-SKHISMA, 3op5 701.6759

5/4 Just, 1/8-SKHISMA, Helmholtzian 701.7108

7/4 Just 702.2267

LST35 701.71454

LST357 702.0992

The unweighted negative Lst Square systems and those yielding certain
ratios appeared in Xenharmonikon 2, 1974 along with a number of
other tunings at the 11 limit not shown here.

--John


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