Aliquot parts, lattices, etc.

re "9-Limit" I believe Ervin Wilson was the first to construct lattices in
which odd composite factors had their own axes back in the late 60's when
he invented or discovered the CPS and similar types of scales. As far as I
know, he was also the first to use polygonal lattices higher than the
triangular (the tetrahedral-octahedral lattice Paul Erlich mentioned may be
drawn as a centered triangular lattice). Personally, I find both
representations useful.

As for "aliquot parts," Marion's usage is clear enough, but I prefer to use
the term only in reference to the linear division of a string into equal
parts (what aliquot parts mean) to generate utonal scales such as
Schlesinger's harmoniai. Thus the division of 1/2 of a string into 7
aliquot parts (or the whole into 14) generates KS's Mixolydian harmonia (14
13 12 11 10 9 8 7 or 1/1 14/13 7/6 14/11 7/5 14/9 7/4 2/1).

In the logarithmic domain (as opposed to the "acoustic," the result of
dividing an interval into aliquot parts is an equal temperament, unless the
interval is a product of equal just intervals (9/4 into 3/2 x 3/2, etc), in
which case one has an equal division, but not a temperament. Geometric
division or equal division would seem to cover these concepts adequately.

I usually use the term "submultiple" to refer to the notes other than the
"Guiding Tone" (Fokker's term) and the 1/1 of an Euler-Fokker genus
musicum. The logic is that each note is both a multiple of the 1/1 and a
divisor of the guiding tone, the whole genus being simultaneously an
otonality of 1/1 and a utonality of the GT (at the GT limit, though it
usually is not complete). Since E-F genera are classified by their GT's,
the tones are conveniently described as submultiples of them.

Marion's LCM when expressed as a product of factors is equivalent to the
GT,
though one may disagree about the necessity of explicitely writing the
powers of 2 (Euler did, as I recall, Fokker usually did not).


--John

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