RE: Partch Limit vs Prime Limit (Paul E)

From: "Paul H. Erlich"

Let those of us who don't understand Partch's work not accuse him of
muddy thinking!!! As for priority, I seriously doubt the claim that the
prime-limit concept predated Partch's work; in fact, it is often merely
the result of misunderstanding Partch.

I think we have at least four schools of thought here; add in the
question of dualism and there are eight philosophies about the
mathematical characterization of chords. All these, by the way, leave
aside the issues of temperament and octave equivalence, so so will I for
now.

Let us sidestep the issue of odd vs. prime limits for now, and consider
another dichotomy within the context of 5-limit lattices. There are
those who subscribe to a rectangular matrix/ROHS/LCM/Tenney harmonic
distance idea. According to any of these philosophies, 15:1 is "the
same" as 5:3, i.e., a major seventh is just as dissonant/complex as a
major sixth. (Note that this conclusion is independent of the odd/prime
question; it is important to keep that question from entering the
present dichotomy.) In the rectangular matrix point of view, both
intervals involve moving one unit along the 3-axis and one unit along
the 5-axis. The relevant ROHS is this a 1X1 rectangle. The LCM of both
intervals is 15.

Since I think the major sixth is as basic a consonance as the major
third (for many reasons including some recently discussed experiments on
tuning major triads), I prefer a triangular lattice. Here 5:3 has an
axis unto itself, so the lattice is filled with triangles, which
represent major or minor triads depending on their orienetation. Now 5:3
is just one step, while 15:1 is two. Considering that actual occurences
where a 15:1 is emphasized in music are typically in the form 15:5:1 or
15:3:1 (or 15:5:3:1), while 5:3 is happy all by itself, the triangular
lattice seems more appropriate. In fact, the major seventh in equal
temperament is 11 times closer to 17:9 than to 15:8; the reason that
doesn't matter is that the individual steps in the triangular lattice
are indeed well-represented in equal temperament, and 15:8 is merely a
by-product of linking consonant intervals. 15:8 heard alone is indeed a
dissonance and so its exact tuning is not crucial.

Finally, note that the smallest triangles in the rectangular lattice are
the major triad, the minor triad, the 15:5:1, and the 15:3:1. Are these
equivalent in consonance? I think it is far easier to believe that the
major and minor triads are more consonant that the other two, and the
triangular lattice expresses this by the more compact geometric
representation of the major and minor triads.

Now, sticking to the 5-limit, we can address the odd vs. prime issue.
Clearly, _any_ two points in the lattice, no matter how distant, form a
5-limit interval, if the prime definition is used. If the limit is
supposed to be related to dissonance/complexity, that should mean that
any two points on the lattice form a more consonant interval than an
8:7. But this is absurd. In fact, I would consider 9:8 slighlty more
dissonant than 8:7, if heard in isolation. The Partch theory calls the
former a 9-limit interval, correctly expressing its more complex nature.
The claim that 27:16, heard in isolation, has a certain 3-limit quality
to it, is nonsense to me, and is probably nothing more than a result of
familiarity with the effects of Pythagorean tuning, and of listening to
chords like 27:9:3:1. Relative primality is all that matters to the ear;
absolute primality of the terms in a ratio can have no conceivable
importance to the auditory phenomenon, and 27:16 differs materially from
23:16 or 29:16 only in the simpler ratios they approximate (i.e., we are
essentially getting into issues of temperament).

It's hard to try to discuss these issues seperately since they are all
>so intertwined . . .

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Paul Erlich said:
>Graham, I am puzzled by your statement that these two four note chords
>are complete chords of a nature which would seem to require them to be
>six-note chords. Somehow, you hid two notes under your sleeve. Therefore
>I will reserve judgment on your analysis until you clarify.

I used the phrase "complete chords" in the literal English sense of
"chords from which no notes are missing" rather than specifically
Euler complete chords. I also used the word "saturated" in a
different sense to you. Sorry for any confusion this caused.

>I should add to this post that Erv Wilson's CPS idea is a reflection of
>the triangular lattice philosophy, and deserves more attention, while
>Euler genera and LCMs are a reflection of the rectangular lattice
>philosophy.

Symmetrical saturated chords are indeed Wilson CPSs (e.g. 2)1.3.3.5).
They are also Euler genera without the first and last notes. I can't
make of these definitions work for the non-symmetrical, non-o/utonal
chords mentioned in my more recent post.

I haven't worked out the implications of triangular lattices yet.
Are they what I would call Hexagonal lattices -- that is, is every
point directly connected to 6 others? How do you measure distances
on them? How do you extend them to multiple dimensions? Do they
work without octave invariance?

As I haven't received Digest 1122 yet, I can't be sure my message
was in it. If it wasn't, I'll re-post: it contains the conclusions
I was working towards through looking at octave reduced ROHSs.
However, there is a mistake in it; non-square powers of primes cause
the same problems as rhubarb numbers. The only such number small
enough to be musically relevant is 27. With n', m, for example,
the interval 33/1 will arise.

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