RE: Positive Linear Temperaments (Paul E)

From: PAULE

Finally, optimizing the dominant ninth chord (10 intervals) of the diatonic
scale with respect to a 4:5:6:7:9 tuning yields the fifth sizes

(10+(2*log(3)+3*log(5)-3*log(7))/log(2))/20 octaves or
702.8187 cents (equal-weighted)

(2642+(810*log(3)+631*log(5)-863*log(7))/log(2))/5060 octaves or
703.9231 cents (limit-weighted, 9:3 treated as 9-limit)

(2714+(369*log(3)+631*log(5)-863*log(7))/log(2))/4988 octaves or
703.9515 cents (limit-weighted, 9:3 treated as 3-limit)

So these are good tunings for diatonic music that uses the dominant ninth
chord as a tonic. The subharmonic version of the diatonic dominant ninth
chord is a diatonic dominant ninth chord, so making it approximate 4:5:6:7:9
automatically makes it approximate 1/9:1/7:1/6:1/5:1/4 just as well. I think
it will tend to be heard as 4:5:6:7:9, though.

Note that our optimized tunings have fifths spanning the range from just
over 696 cents to just under 704 cents. It appears that for the diatonic
scale, 12-equal is a nice central point from which other tunings can be
regarded as small deviations.

-Paul Erlich

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On Fri, 14 Mar 1997, Aline Surman wrote:
> Haverstick here...Paul Hahn; in reference to your quip regarding Sagan's
> statement about one's birth, and the admonition to not get too
> "mystical"; Sagan is a dead mofo now, and the gravitational force of the
> Angel of Death was enough to kick his butt into those "mystical" regions
> which you warned me to be so careful of...Hstick

Yes, and your point is . . . ? You'll be dead someday too, Neil. How
does this in any way bear on whether or not scales/notes/timbres based
on chemical elements, astronomical objects, the magnetic field of the
Earth, etc. have any psychoacoustical significance?

--pH http://library.wustl.edu/~manynote
O
/\ "'Jever take'n try to give an ironclad leave to
-\-\-- o yourself from a three-rail billiard shot?"


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