RE: Lattice, LCM, and Aliquot Parts

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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Marion wrote:

''Gosh, I thought Pythagorus started the whole thing (at least in
the west) with his famous 3 limit. I certainly was working with
prime limits before I ever heard of Partch.''


This is a twofold misunderstanding. The tradition is muddy enough, but
evidently Pythagoras (or better: the Pythagoreans) was (were) not workingwith a conscious ''limit'' of three, but with the hypothesis that the
numbers 1 and 2 alone yielded musically insufficient materials, and that
the addition of some number of factors of 3 would yield the rational
resources to construct the known scales. This is subtly but vitally
different from the notion of _limit_.

Although an obvious enough concept, the term _limit_ appears to be Partch's
innovation in musical terminology. His late romantic harmonic view was
based upon a ''stacked thirds'' model, frequently found in American theory
texts of the time, and used by Partch in terms of odd, not prime chordal
components. (Interestingly, Partch's viewpoint leads - to my ears - to a
better explaination of the lowered ninth (as the 17th) than the
conventional treatment as an ''altered tone''.) In view of Partch's
priority and importance, I think it is due respect to use ''limit'' as
''odd limit'' and explicitly say ''prime limit'' when that is the case.

The leading alternative to the limit classification is that of Euler, which
lists the set of generative factors. This has the advantage in that factors
left out in a given system that would otherwise be implied by a limit
labeling are not falsely incorporated. (Example: Sometimes I use the
Eikosany with (1,3,7,9,11,15) as factors in preference to (1,3,5,7,9,11);a
limit notation would not make the distinction clear).

This discussed really repeats a discussion from last fall (and probably
from earlier than that!). Are the archives to the list available and
indexed in such a way that we can easily direct new members to earlier
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>I have not recieved TD 1122, so I write with some tentativeness . . .
Could it be that no-one got that digest?

I've drawn myself some triangular lattices, and I can see how a
lot of nice shapes correspond to a CPS. I've also tried to
visualise a regular tetrahedral lattice. The points of this seem
to be in a hexagonal close packed structure. If I can get a
quantitative formula worked out, I'll change the metric. Making
the 5/3 dimension infinite turns a triangular into a square
lattice. What fun!

The saturated 15-limit chords 3:7:15:21 and 5:7:15:35 both make
the same shape on a tetrahedral lattice: two triangles joining at
one edge, not in the same plane. On a tetrahedral, octave
specific lattice a 4:5:6 chord lies on three corners of the same
shape. 3:4:5 and 5:6:8 don't look as good. 4:5:6:8 doesn't
either, but I still think this is leading somewhere. I don't see
that we need a new dimension for 9.

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Graham Breed wrote:

''I don't see
that we need a new dimension for 9.''

This depends entirely on how 9 is used. If - like Partch or Wilson, or incertain temperaments (the TX81Z resolution comes to mind) - 9 is used as a
distinctive harmonic entity, then it is useful to visualize 9 on its own
axis. If, on the other hand, 9 is only used in terms of 3^2, then by all
means let 9 share the 3 axis. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
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