primes vs. odds in chords, feedback requested

[Lumma, TD 130:]

> It seems that the prime-limit ...<snip>
> ... reflects the odd-limits ...<snip>
> ... better than the odd-limit ...
>
> This may be source of confusion and/or argument over
> prime and odd limits.

I admit I'm a little confused by what you're saying.
If I've understood you correctly, this is obvious to me.

If you're examining the interval structure of a chord,
the prime factors are the only ones that can't be reduced,
while the same odd factors may describe more than one interval
in the chord.

The recognition of the odd-factorization gives no new
information about the interval structure of the chord than
that which can already be seen in its prime-factorization.
The primes are the furthest simplification.

-monzo
http://www.ixpres.com/interval/monzo/homepage.html

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Carl Lumma wrote,

>It seems that the prime-limit of all the lowest-term frequency ratios of
>notes in a chord reflects the odd-limits of all the intervalsin the chord
>better than the odd-limit of the lowest-term ratios for the notes.

>This may be source of confussion and/or argument over prime and odd limits.
>If it is right...

>3:5:7:15 may be heard as a 7-odd-limit chord in many of its voicings.
>1:3:5:15 may be heard as a 5-limit chord in many contexts.

>Actually, this only works if the odds considered share factors with at
>least one other note in the chord. So maybe it should be called
>"coprime-limit"...

Coprime and relatively prime are synonymous. So Carl and I were really
saying the same thing.

>In most musical contexts the odds will share factors
>with other notes, so this can still explain prime numbers as being useful
>when looking at music, and why people like Partch, Doty, Canright, and
>others have used them.

In the context of this discussion (chords), I don't think Partch belongs in
this list (though many have misunderstood him, including Gilmore). Partch
looked at intervals and chords in terms of odd numbers. It is only the pitch
systems of just intonation which are usefully characterized (e. g., by
Partch) by their prime limit. One only needs to know how to tune the prime
intervals in order to construct any interval in a just intonation tuning
system; the former can be stacked in various ways to create the latter.

Joe Monzo wrote,

>I admit I'm a little confused by what you're saying.
>If I've understood you correctly, this is obvious to me.

in response to what Carl wrote, which was

>This may be source of confussion and/or argument over prime and odd limits.
>If it is right...

>It seems that the prime-limit of all the lowest-term frequency ratios of
>notes in a chord reflects the odd-limits of all the intervals in the chord
>better than the odd-limit of the lowest-term ratios for the notes.

Judging from the rest of your post, Joe, I don't think you understood what
Carl was saying. I think I do understand because it goes back to a
conversation Carl and I were having at the time the tuning list changed
servers. Carl was thinking, much as you are now, that while odd-limit may be
appropriate for dyads, prime-limit may be appropriate for larger chords. I
demonstrated to him with some examples (check the archives) that I thought
this was an illusion. He appears to have bought my argument, as he is now
using it "against" you.

What Carl is saying:

1. The odd-limits of _all_ intervals in the chord must be considered to
understand the dissonance of the chord as a whole. The total number of
intervals in a chord of n notes is n*(n-1)/2.

2. Most people, unfortunately, will only look at an n-term "ratio" when
trying to understand the dissonance of a chord. This is a set of n integers
which gives the correct proportions for the chord, and the overall set of n
integers is in lowest terms.

3. In many cases, the true dissonance of the chord (as in #1) is more
closely paralleled by the highest prime factor of the integers in the n-term
"ratio" than by the highest odd factor in the n-term "ratio".

4. Most people are therefore deceived into thinking that prime-limit is more
important than odd-limit for understanding the dissonance of a chord.

Carl and I also mentioned "coprimality" and "relative primality." Two (or
more) numbers are coprime or relatively prime if they share no common
factors. A set of prime numbers is relatively prime but the converse is not
necessarily true. What really distinguishes two chords with similar-sized
odd numbers in the n-term "ratio" representation, is not primality itself
but the "relative primality" or "coprimality" of its terms. If there are
terms in the ratio that share common factors, then some of the intervals are
in fact simpler than the n-term "ratio" would indicate. E. g., in the chord
8:10:12:15, although the overall 4-term "ratio" is in lowest terms, all
pairs except 8:15 are not coprime and therefore can be reduced; thus the
chord is more consonant than the size of the numbers 8, 10, 12, and 15 (or
the odds 1, 3, 5, and 15) would indicate. Attributing this consonance to the
low prime-limit of the chord is incorrect, albeit a common error.