limits and ??

hi.. i've just learned that i've been using the word "limit" wrong -- so
now i'm looking for a term that means what i *thought* limit meant.

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.

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.

so, if limit isn't the right word to refer to these "dimensions", what is?
is there a term for the infinite 1-dimensional lattice made with powers of 3?
the 2-dimensional lattice with powers of 3 and 5? etc etc...

--
P a u l F l y
http://www.neuron.net/~pfly

>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.

No sweat! -- This subject has surfaced before. If you read my recent post
on CPSs, I touch on the difference between JI tunings (whose pitches are
written as fractions) and JI harmony (where all the intervals in a sonority
appear in a list of consonances). The terms "just intonation" and "limit"
have been confusingly applied to both these things.

For purposes of discussing consonant intervals, many listers have decided
to write the ratios involved with colons (:), distinguishing _intervals_
from _pitches in a tuning_ (which they write with slashes (/), following
Ellis, Partch, the American Gamelan folks, et all). So the JI diatonic
scale may be written...

1/1 9/8 5/4 4/3 3/2 5/3 15/8
C D E F G A B

...and the interval between D and E may be written as 10:9.

For tunings, the "limit" used is often a "prime limit", when the
diatonic scale above is considered 5-limit, since 5 is the largest
prime factor needed to write all the fractions involved.

For harmony, the "limit" used is often "odd", since consonance seems
to change, roughly, with the largest odd factor appearing in any
interval in the chord sounding. There are good psychoacoustic and
practical reasons for thinking in terms of an odd-limit for harmony;
Partch did so. In we play all the notes in the above scale at once,
the odd limit would be 45, since there's a 45:32 between F and B.

Prime limit may have come into use for tunings because many tunings
are designed to provide several copies of simple chords (like the
diatonic scale contains triads), and often if you use a prime limit,
the factorization will reveal the harmonic limit of the simple chords
which have been transposed throughout the scale (the triads in the
diatonic scale are 5-odd-limit).

More examples:

1/1 and 15/8
prime= 5
odd= 15

1/1 and 5/4 and 15/8
prime= 5
odd= 15

1/1 and 8/5 and 16/15
prime= 5
odd= 15

1/1 and 11/8 and 15/8
prime= 11
odd= 15

As you can see from these examples, either "limit" can be somewhat
limiting for harmony, since a "limit" assumes you'll be using every
factor up to the limit before you increase the limit. But you may
not always want to do this, and when you don't, it's best to list the
factors you consider consonant. In the last example, they would be
(1, 11, 15) -- 15-limit and 11-limit are both deceptive here because
7, 5, and 3 are missing.

>so, if limit isn't the right word to refer to these "dimensions", what
>is? is there a term for the infinite 1-dimensional lattice made with
>powers of 3? the 2-dimensional lattice with powers of 3 and 5? etc etc...

On a lattice, each dimension is usually mapped to a single factor. If
you include a dimension for every prime factor up to x, then I'd call it
a x-prime-limit lattice. Same goes for odd-limit lattices, but they do
have a quirk where composite odd numbers will appear in more than once
place -- 9 will be one step on the 9-axis, or two steps on the 3 axis, for
example. Normally, this doesn't cause a problem, since our ordinary 3-D
space can only accomidate the 7-limit (which is the same wether odd or
prime, since the first composite odd number is 9). But on some of Erv
Wilson's diagrams, you can see pitches appearing in multiple places.

-Carl