EB again

I have derived equally-beating tunings similar to Manuel's by
the methods described earlier.

E.B. Intervals Equal Opposite
5/4 = twice 3/2 694.2728 697.8167
3/2 = twice 5/4 696.143 696.9529
6/5 = 3/2 695.8096 693.3588
6/5 = twice 3/2 696.5784 691.2348
3/2 = twice 6/5 695.337 694.1365
3/2 = 7/4 696.6556 697.0906

Manuel and I have posted a number of E.B. tunings recently. As I
don't have Rasch's paper handy, I'm not sure which correspond to
his (See Rudolf Rasch, Music Perception 1(3):308-322 (1984)).

While experimenting with EB tunings, I came across a septimal
komma that is apparently not well-known. It is the difference between
10 Fifths up - 5 octaves and the 7/4 and is 59049/57344 or 3^10/(7*2^13)
or 50.72+ cents.It plays the same role the syntonic comma 81/80 does
in defining meantone tunings -- the tempered fifth which yields a
beatless 7/4 is the 10th root of 56 (compare the 4th root of 5 for
1/4-comma meantone or the cube root of 10/3 for the 1/3 comma). This
fifth is tempered flat by 10th root of 59049/5734 or 696.88259 cents.
Whether this komma is useful notationally remains to be seen.

Other septimally tempered fifths are the 6th root of 56/5
(beatless 7/5) and the 9th root of 112/3 (beatless 7/6). The
corresponding kommas (-ta) are 3645/3584 and 59049/57344.

Manuel said in a personal communication that the latter komma is called
"Harrison's comma" and is on his interval list. He also pointed out
that one can compute the beat rates directly rather than as functions
of kommata. I agree, but at least in the case of the syntonic comma, it
is interesting to see the relations between the historical
fractional-comma tunings and the beat rates of the consonant intervals. I
also agree
with him that these tunings are extremely subtle and would require very
precisely tunable and stable instruments to be used and appreciated.


--John



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