source file: mills2.txt
Date: Wed, 2 Jul 1997 18:53:05 +0200

Subject: RE: Lattice, LCM, and Aliquot Parts

From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>

Marion wrote,

>Another form of invariance, ratio invariance, is important in LCM
>analysis.  This became clear to me during a discussion I had with
>Paul E. some time ago.  Thinking back over that, it seems to me
>that our differences of opinion on the subject of the length of
>LCM patterns was mostly due to different assumptions about ratio
>invariance.  I was assuming ratio invariance, he was not.  I'm
>still not absolutely clear on all this and would appreciate any
>comments.

I thought we had decided that our "differences of opinion" were simply
an error on your part. You were claiming that the LCM of a chord was
proportional to the pattern length of the chord. I pointed out that this
was only true under certain wild assumptions about the relative register
of the chords. For example, the major triad (4:5:6) and the minor triad
(10:12:15) both have an LCM of 60. For both chords, the pattern length
of the chord as a whole is that of the frequency represented by the
number 1. So in order to give both chords the same wavelength, you have
to play the minor triad an octave plus a major third higher than the
major triad. If you play them in the same register, the wavelength of
the minor triad is 10/4 2.5 times longer than that of the major triad.

The LCM was used by Euler to classify chords. The only physical
significance of the LCM is that it is the ratio of the lowest common
overtone to the highest common fundamental. This fundamental is a note
whose wavelength is the same as the pattern length of the chord as a
whole. So your attempt at equating the LCM with pattern length is
correct in general only if we assume all chords are built below the same
common overtone. In other words, it assumes that there is some note (say
2880Hz) which is the lowest common overtone of the notes within each and
every chord. Needless to say, that is a preposterous assumption (calling
it "ratio invariance" doesn't help). For example, adding the major third
to an open fifth increases the LCM from 6 to 60; in this case the
pattern length increases not by a factor of 10 but only by a factor of
2, since the lowest common overtone goes up by a factor of 5. Meanwhile,
adding a minor third to the fifth again increases the LCM from 6 to 60;
this time the pattern length increases by a factor of 5 and the lowest
common overtone goes up by a factor of 2.

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