Re: TD 701 -- limits (prime and odd)

> From: Paul Fly <pfly@neuron.net>
> Subject: limits and ??
>
> my confusion seems to come from not making a sharp difference between
> factors and roots. i knew that a ratio like 10/8 was 5-limit because
> it can be factored down to 5/4, but i thought a ratio like 10/9 was
> also 5-limit because 9 can be square-rooted to 3.

Hello, there, and I would say that 10:9 is, of course, a 5-limit interval,
exactly as you explain: it can be constructed from the prime factors of 2,
3, and 5.

More specifically, we can sometimes call this a "5-prime-limit" interval
to distinguish it from "5-odd-limit" intervals such as 5:4, 6:5, 5:3, and
8:5.

In musical terms, a 16th-century just intonation (JI) system of the
classic variety (all integer ratios) uses 5-(prime)-limit intervals
including 9:8, 10:9, 16:15, 15:8, 25:24, and so on -- some of which are
also 5-(odd)-limit intervals such as 3:2, 5:4, and 6:5. Only the latter
intervals, however, are treated as stable concords; the others are
unstable, and in 16th-century terms more specifically "dissonances"
subject to rather cautious restraints.

If you were to say simply "5-limit," I would take it to mean the whole
system of intervals, concordant (5-odd-limit) or otherwise. For some
people, "5-limit" can imply "5-odd-limit," in other words "5-limit
concords only," but I would call this a difference in usage, not an error
on your part (as it happens, I favor your usage).

> in lattice terms, i thought 3-limit referred to the entire dimension of
> ratios that can be made by taking powers of 3; and 5-limit, the whole
> 2-dimensional lattice made by taking powers of 3 and 5.

Apart from the question of whether we also count the octave (2:1) as a
dimension -- and doing so seems to be a minority view (e.g. Carter Scholz)
-- I would very much agree with this statement, not only mathematically
but musically. One of the problems I have as a Pythagorean tuning advocate
with the approach that takes "odd-limit" as the default meaning of limit
as that it maybe tends to filter all the neat _large_-integer ratios that
a 3-limit system can come up with.

Maybe rather than debating who is "right" or "wrong," it might be
interesting to see what musical or cultural perspectives might lead people
to prefer one default usage or the other. When in doubt, explicitly saying
"5-odd-limit" or "5-prime-limit" solves any question of ambiguity.

Most respectfully,

Margo Schulter
mschulter@value.net