More beating Intervals

While trying to determine the source of the discrepancies between
my numbers and Manuel's, I recomputed my earlier table and got the
same results. However, it occurred to me that my approach does not
distinguish between intervals that are sharp from those that are
flat, or in other words, between intervals beating because they are to
narrow or too large. Hence, there are two solutions to each of the
relations; one where the beat rates are due to intervals both sharp
or flat, and one where they are dissimilar. I have thus added a
column of "Equal and Opposite" tunings for these new results.


Beating Intervals Equal Equal & Opposite

1. 4th = min 6th 695.8096 697.1769
2. 4th = Maj 6th 696.2958 692.1612, too flat?
3. min 6th= Maj 6th 696.0063 698.1741
4. 5th = Maj 3rd 695.6304 697.2785
5. 5th = min 3rd 695.8096 693.3588, too flat?
6. Maj 3rd = min 3rd 695.7294 (629.9748, unacceptable!)
7. min 3rd = min 6th 698.8781 695.9340
8. 4th = Maj 3rd 697.4747 695.2284
9. 5th = min 6th 697.0390 696.0237

My approach is based on Rasch's beat rate formulae where x is the
temperer and S=81/80. The tempered fifths thus equal 3x/2. The
"register-free" beat rates of the various intervals are as follows:

Fifth 3x - 3
Fourth 4/x - 4
Major 3rd 5Sx^4 - 5
Minor 3rd 6/Sx^3 - 6
Major 6th 5Sx^3 - 5
Minor 6th 8/Sx^4 - 8

Tunings are defined by taking pairs of these relations and setting
one equal to the other or to the negative of the other. Thus #4
where the Fifth beats at the same rate as the Major third is defined
by the formulae 3x - 3 = 5Sx^4 - 5 and 3x - 3 = 5 - 5Sx^4. Solve these
equations for x, then multiply by 3/2 to get the decimal fraction for
the tempered fifth and then multiply this value by some convenient
pitch base such as C261.625 hz. From this frequency, one may compute
the difference between the 3rd partial of the tonic and 2nd of the
tempered fifth. This value is 0 for x=1.

Beat rates of the other intervals are calculated by generating them
from tempered fifths by the traditional Pythagorean cycles. Beat
rates are then determined from the lowest (nearly) coincident partials
as above.

--John



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