source file: mills2.txt
Date: Wed, 2 Jul 1997 08:35:05 +0200

Subject: Re: Lattice, LCM, and Aliquot Parts

From: rtomes@kcbbs.gen.nz (Ray Tomes)

On Tue, 1 Jul 1997 Mckyyy@aol.com typed:

>On octave invariance, it has just occurred to me that, in LCM
>terms, the octave has a property that no other ratio shares. 
>That is two notes that differ only by octaves can be mixed
>without making the length of the resulting "interference" pattern
>longer than the waveform of the lowest note.

>If you mix two notes of wavelength 1 and 8, the wavelength of the
>resulting pattern is still 8, but if you mix two notes of
>wavelength 2 and 3, the resulting pattern has a wavelength of 6.

Marion, it is not so.  You can equally well say that 1, 3 and 9 mix
without making the wavelength any longer but that putting a 2 in there
results in it being longer.  It is the mix of primes that does.

I think that the reason that a ratio of 2 is more fundamental than any
other is that 2 is a smaller number than any other (OK 1 is better!).
In my opinion the simplicity of a prime ratio is inversely proportional
to p*ln(p) where p is the prime.  For the first few primes this gives:

p         2      3      5       7      11      13
p*in(p) 1.386  3.296  8.047  13.621  26.376  33.344

Also, I think that a ratio of 1:12 is more concordant than one of 1:16.
So it is not totally correct to just cast out all the 2s and saying that
a D is a D and a G is a G.  The octave is relevant.

>Aliquot parts are simply a list of all the possible numbers that
>can be derived by multiplying subsets of the prime factors of a
>given number.  For example, here is a list of the aliquot parts
>of 2880:

Ah!  My second most favourite number (after 34560).

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
     http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

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