RE: Partch Limit vs Prime Limit

Marion wrote:

>For me, the whole "lattice" approach is much less useful than
>Aliquot Parts and Least Common Multiple for analyzing scales,
>chords, and intervals.

Aliquot Parts are unknown to me, so I can't comment on them.
I have been a card carrying LCM-ist for quite a while, though,
but I'm confused at this dichotomy between them and the
lattices. I express the LCM in my lattice notation, but
call it a region of harmonic space (ROHS) because an LCM
should really be a number and not a matrix. An LCM of 60,
then, corresponds to a ROHS of (2 1 1)H.

In lattice terms, the ROHS is the size of the smallest n-box
in n-dimensional ratio space that includes, on or within its
boundary, all the notes under consideration. The distance
between opposite corners of this n-box, measured using a
city block metric with each dimension weighted according to
the logarithm of the corresponding prime number, is the
logarithm of the LCM of the same notes. Don't worry if you
didn't get that last bit, folks, it isn't important!

Anyway, my main point in writing is to draw a comparison
between the odd-limit and the ROHS. Referring to my last
message of a few days ago, the 7-limit clearly lies within
the ROHS (x 1 1 1)H where x can be anything, to give octave
equivalence. This is the same as the Euler genus 3.5.7.

Not all the members of the genus 3.5.7 are within the 7-limit.
(3 -1 1 0)H, or 15/8, for a start. For intervals within an
octave, though, the ROHS (2 1 1 1)H contains all 7-limit
intervals and only a few outside it -- (-1 1 1 -1)H or 15/14,
etc. In otonal or utonal form, these exceptions disappear as
odd primes cannot appear in both the numerator and
denominator.

If you reduce your chord so that all the intervals are
within the first octave above the otonal root, the ROHS
(2 1 1 1)H is identical to the 7-limit. The saturated
chord being:

(-2 0 0 1)
(-2 0 1 0)
(-1 1 0 0)H
( 0 0 0 0)

The ROHS occupied by the largest interval within a chord will
always be the same as the ROHS occupied by the whole chord.
This is why the ROHS is superior to the odd limit from a
number theoretic point of view. It is also why, IMHO, the
ROHS is the superior way of classing scales.

The equivalent utonal chord can be found by multiplying the
chord above by -1. This is the same as arranging the notes
in the octave below the utonal root. Where the otonal root
will always be (0 0 0 0)H, the utonal root is equal to the
ROHS with the octave term negated. Therefore, the octave
reduced utonal (2 1 1 1)H chord is:

(0 0 0 0) (-2 1 1 1)
(1 -1 0 0) (-1 0 1 1)
(2 0 -1 0)H (2 -1 -1 -1)H + ( 0 1 0 1)H
(2 0 0 -1) ( 0 1 1 0)

For consistency with the odd-limit idea, only otonal and
utonal reductions are allowed. This rules out octave reduced
(2 1 1 1)H, non 7-limit chords like the following:

(-2 0 1 1)
(-1 1 0 1)H
( 0 1 1 0)

Hence o/utonal 7-limit chords are the same thing as o/utonal
octave reduced ROHSs. Beyond the 7-limit, though, this
equivalence no longer holds, just as the two kinds of
odd-limit diverge. As the odd-limit idea is itself an
extrapolation from conventional harmony, I would suggest
that the ROHS method is the best way of extending it to
higher primes.

The ROHS is also a useful way of categorising chord changes.
This is a very important subject, and one that is rarely
mentioned on the list. However, I will not go into it today.

I suggest that the ROHS should also be emancipated from
octave reduction. Then the interval 15/1 comes out as
less dissonant than 15/8. My ears tell me that there is
some justification for this. I don't go with octave
invariance, because different inversons of major triads
definitely sound different. All the 'nice' versions of 4
note major triads fall within the ROHS (3 1 1)H. The ones
that stay within (1 1 1)H also sound particularly nice.
However, the (2 1 1)H ones are not neccessarily better than
the (3 1 1)H ones. There seems to be a premium on having the
root doubled. The idea of root doubling, though, implies the
idea of octave invariance. I might some time get round to
extending Kameoka and Kuriyagawa's dissonance algorithm to 4
note chords to see if it sheds any light on this.

Graham

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>Second, some of us evidently don't believe that octave equivalence (2:1, give
>or take a few cents :-) should be taken for granted ... Holy cow! ...
>one of which has a frequency at a power-of-two of the other, the lower one
>will cross the zero point only at points at which the higher one is crossing
>as well.

Hmmm... Well, certainly you can phase them so that the zero-crossings
never correspond, or microscopically detune them, by 1/5Hz for example,
they will not correspond very closely for very long, and yet we recognize
both as octaves.

Here's another curious observation to throw into this line of inquiry.
I suppose it could be just me, but I'd be surprised if so:

I have found there to be something interesting about ratios of the form
(N+1):N. I've noticed that if I listen to a narrow range of small,
adjacent Ns for long enough, they start masquerading as one another.

An easy example to hear occurs when I practice singing the distinction
between 6:5 and 7:6 thirds. After I sing and think about nothing but those
two intervals for five minutes or so, I find that the 6:5 starts having the
sensation I normally associate with 5:4. "Sensation" in the sense that I'm
inclined to momentarily misidentify 6:5 as 5:4 - an auditory illusion in
essence. And further, when I suddenly toss into the mix a 5:4, it starts
seeming like a 4:3, and 3:2 seems like an octave.

I'd be curious as to whether anybody else has noticed this effect too.

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Gary Morrison wrote:

'' An easy example to hear occurs when I practice singing the distinction
between 6:5 and 7:6 thirds. After I sing and think about nothing but those
two intervals for five minutes or so, I find that the 6:5 starts having the
sensation I normally associate with 5:4. "Sensation" in the sense that I'm
inclined to momentarily misidentify 6:5 as 5:4 - an auditory illusion in
essence. And further, when I suddenly toss into the mix a 5:4, it startsseeming like a 4:3, and 3:2 seems like an octave. ''

I have heard similar stories from La Monte Young, for whom the 9:8 has come
to function as a boundary interval analogous to the 4:3 in classical
tetrachords, and from Morton Feldman, who claimed that his use of tightlypacked chromatic clusters had so changed his perception of intervals thatthe minor second was, for him, as perceptually wide as a minor third had
previously been.

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>I have heard similar stories from La Monte Young, for whom the 9:8 has come
>to function as a boundary interval analogous to the 4:3 in classical
>tetrachords, and from Morton Feldman, who claimed that his use of tightly
>packed chromatic clusters had so changed his perception of intervals that>the minor second was, for him, as perceptually wide as a minor third had
>previously been.

Interesting. Thanks for the perspective.

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