Yahoo Tuning Groups Ultimate Backup tuning prime vs. odd limits

dante rosati wrote,

>Im a little confused. Are ratios usually catagorized by prime limit or
odd
>limit?

This is a great question. Partch, who introduced the term limit, usually
meant odd limit, but was not entirely consistent. In most cases he would
speak of ratios of 9 as belonging either to the 9-limit or as
_secondary_ ratios of the 3-limit. Others after Partch started the prime
limit usage.

Here is the definition of limit I wrote for the Southeast Just
Intonation Center's Just Intonation Reference (compiled by Adam
Silverman) and Joe Monzo's tuning dictionary
(http://www.ixpres.com/interval/dict/index.htm):

*1. prime limit: A pitch system in Just Intonation where all ratios are
of integers containing no prime factors higher than n is said to be an
"n-limit" system.

*2. odd limit: A chord in Just Intonation where all ratios are of
integers containing no odd factors higher than n is said to be an
"n-limit" chord. A composition or style where chords of the n-limit are
considered consonant and chords of any higher limit are considered
dissonant is said to be an "n-limit" composition or style. Note that
this may not agree with the "prime limit" definition.

What I like about this definition is that it almost always fits what
people are saying. If people are talking about a JI tuning system, with
a potentially infinite number of notes, that system is usually
characterized by a prime limit. If people (especially Partch) are
talking about consonance and dissonance, the odd limit is usually more
important.

>Ratios with 9 in them are, I thought, 3 limit ratios.

9/7 is a 9-odd-limit and 7-prime-limit ratio.

>Also, if you're going to draw a 5-limit lattice why would it
>only contain six ratios, and not have, say 15/8? Would 15/8 only appear
in
>a 15-limit lattice?

No, lattices are infinite and a 5-limit lattice would have to contain a
15/8, since definition 1 applies to JI pitch systems such as lattices.

What you're probably looking at is the lattice representation of
Partch's 5-limit tonality diamond. This is, of course, a subset of the
infinite lattice; namely, those pitches which form a 5-limit consonance
with 1/1. Since we're talking about consonance, definition 2 applies,
and 15/8 is not part of the 5-limit.

15/8 would appear in a 15-limit tonality diamond.

Many people think prime limit gives additional information about
consonance after odd-limit is already considered. I disagree, and think
that this is a result of confusion caused by looking at chords with more
than two voices. For example, consider the following chords (from the
definition of saturated in the tuning dictionary):

*the complete 9-limit Otonality *the complete 9-limit Utonality
*12:15:18:20, and *14:18:21:24.

All of these chords are composed entirely of 9-odd-limit intervals, no
higher. But the presence of 15 and 21 in the ratio representations of
the chords on the right might confuse people into thinking that the
odd-limit is 15 and 21, respectively. Since these chords are clearly
more consonant than similar chords with 11 or 13 in them, one might be
led to think that the prime limit is sometimes more important than the
odd limit in determining consonance. But again, that is a result of not
seeing what the odd limit really is.

To use your 15/8 example, Partch considered 15:8 a dissonance. Now many
people may object that 8:10:12:15 (the major seventh chord) is more
consonant than any chord with 11 or 13 in it. That may be true, but I
would say that the consonance of the other 5 intervals in the major
seventh chord (10:8=5:4, 12:8=3:2, 12:10=6:5, 15:10=3:2, 15:12=5:4) is
so much greater than the consonance of any ratio of 11 or 13 that the
dissonance of 15:8 is not as important by comparison. As an isolated
interval, I certainly feel that 15:8 is more dissonant than 11:8 or
13:8, if we could divorce ourselves from familiarity with diatonic
intervals. But the context of the major seventh chord allows 15:8 to
participate in many 5-limit consonances in a way that 11:8 or 13:8 could
never do.

>Wouldn't 3 limit ratios include 9/8 and 16/9, as well as anything else
>containing no prime factor greater than 3?

Again, this depends on the definition, which usually depends on context.

>Since there are an infinite
>number of ratios of each prime limit, what other factor accounts for
when
>to stop filling the octave with them?

One possibility: odd-limit. Then you don't even need the prime limit
anymore, you still get a finite number of pitches per octave.

>On my webpage I suggest an octave
>limit, where intevals of a given prime limit are included up to an
>arbitrary octave limit of the harmonic series. Are there other schemes?

1. How about octave limit without prime limit? This is equivalent to odd
limits of 3, 7, 15, 31 (2^n-1), isn't it?

2. Instead of having a single tonal center, start constructing intervals
from other important intervals, such as 3/2 and 4/3, etc. This is what
Partch did to go from the 29-tone 11-limit tonality diamond (which is
all pitches forming an 11-odd-limit ratio with 1/1) to a 43-tone scale
which allows one to establish tonal centers (in the Partchian sense) on
pitches other than 1/1. His 43-tone scale also approximates 41-tone
equal temperament with two auxiliaries, allowing ideas to be transposed
freely without changing too much (Partch once asked Erv Wilson to
explain what was going on in a particular composition Partch was working
on on the Chromelodeon, which didn't appear to make sense in terms of
ratios. Wilson pointed out to Partch that Partch was in fact transposing
as if it were 41-tET, without regard to the exact ratios produced).

E-mail me and we can go over all kinds of possibilities.

Bram wrote,

>What I'm confused about is why people don't
>more often talk about just plain integer limit - 10/7 being 10 limit
and
>7/5 being 7 limit, that sort of thing.

If you want to get octave-specific, integer limit is important. But
usually it is assumed that any pitch can be transposed to any octave and
any chord can be played in any inversion, and one wants to characterize
all these possibities at the same time. So one ignores even numbers.

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

I was not trying to patronize you dante your question was simple or at least it
seemed simple so i gave a simple answer
I was not listing primes to show my math but to list the ones frequently used in
JI.
Nine is composite Harry Partch or not Bub.
Nine is not a 'prime' limit just a limit or convenient place to stop.
Your overthinking a simple thing maybe you are catching tetitis
and yes you are definitely missing something
Pat

dante rosati wrote:

> From: dante rosati <dante@pop.interport.net>
>
> >From: Patrick Pagano <ppagano@bellsouth.net>
> >
> >its the way they are broken down guys
> >2 times 5 is 10, 10 just being a duple of five.
> >9 a composite of three
> >i think the main limits are 3,5,7,11,13,17,19,31 at least as far as this list
> >has mentioned so far
> >Pat
>
> Gee, thanks, Pat(ronizing), for educating us morons as to what a prime
> number is. However, that wasn't my question. It was: Harry Partch seems to
> treat 9 as a limit. Joe Monzo doesn't seem to distinguish between 9 as a
> limit and actual primes as a limit. Am I missing something here about prime
> vs. odd limits?
>
> dante
>
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🔗Paul Hahn <Paul-Hahn@xxxxxxx.xxxxx.xxxx>

🔗Gary Morrison <mr88cet@xxxxx.xxxx>

> > It depends on who you talk to. What I'm confused about is why people don't
> > more often talk about just plain integer limit - 10/7 being 10 limit and
> > 7/5 being 7 limit, that sort of thing.

The concept behind an odd limit is that octaves give a sense of duplication,
meaning that our ears can reduce one or more octaves from an interval and get an
interval with the same overall effect, and that how many octaves we reduce out of
it doesn't matter much. That then means that each odd number, being not
reduceable by 2, outlines a new musical character.

The concept underlying prime limits is that we can factor out stacks of any
interval (i.e., powers of any number) just as we can any arbitrary number of
octaves. That being the case, powers of a given number become a minor
consideration compared the number being there at all, which in turn means that
the measuring stick has to be primes. The concept then is that you can stack up
as many 3s (for example) as you'd like and the high-level character doesn't
change much from just having one three. That just as a factor of 4 or 8 doesn't
change the character of an interval much from a factor of 2.

I personally am tetering not so much between prime vs. odd, as much whether
the limit concept is really meaningful or not. The best evidence I can see
suggests that it is a valid concept, but I'm not as confident in that conclusion
as I have been in the past.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Paul Hahn wrote,

>This is the crux, I think. 9-ratios used as secondary intervals
>resulting from melodic and harmonic progressions involving 5-limit
>triads (or even 3-limit dyads) are not quite the same as 9-ratios used
>as a basic sonority within a 9-limit pentad. The classification of
>intervals into primary, secondary, tertiary, etc. is essential to the
>definition of a couple of scale parameters I've occasionally promoted
>around here, consistency level and diameter.

Paul H., I know we've argued this before, but let me now ask: since the
secondary and tertiary ratios result from progressions, and not from
being present in a basic, consonant sonority with the 1/1, why worry
about those ratios' agreement with JI? Isn't it enough to require that
the primary ratios be sufficiently accurate and consistent? Then the
secondary and tertiary ratios will still result from stringing together
the primary ratios in progressions -- whether these secondary and
tertiary ratios approximate JI well is irrelevant since they are not
supposed to be consonant. My 22-tone piece on the tape swap uses the
extended 7-limit intervals 25/24, 36/35, 49/48, and 256/243 as melodic
steps resulting from progressions of basic 7-limit sonorities, but
represents them all as 1/22 of an octave -- a fact that contributes
melodic coherence, but no dissonance, to the piece. (Note that I am only
objecting to higher-level consistency, not to diameter).

>Hmm, I haven't yet written
>up definitions for the online glossary yet--in the meantime, I refer
you
>to the Appendix of Patrick Ozzard-Low's paper on consistency.

That is, the Appendix on consistency of Patrick Ozzard-Low's paper on
21st Century Orchestral Instruments.

🔗Joseph L Monzo <monz@xxxx.xxxx>

Gary Morrison wrote:

> The concept underlying prime limits is that we
> can factor out stacks of any interval (i.e., powers
> of any number) just as we can any arbitrary
> number of octaves. That being the case, powers
> of a given number become a minor consideration
> compared the number being there at all, which in
> turn means that the measuring stick has to be
> primes. The concept then is that you can stack up
> as many 3s (for example) as you'd like and the
> high-level character doesn't change much from just
> having one three. That just as a factor of 4 or 8
> doesn't change the character of an interval much
> from a factor of 2.

I think that mostly this is one of the clearest descriptions
of this idea that I've read. However, I can't say that
I completely agree with the wording of "powers of a
given number become a minor consideration compared
the number being there at all". I think most people
working in JI would agree that increasing prime-bases
*and* increasing powers both create increasing
dissonance. I know that your point is to emphasize
the importance of the primes (in this explanation,
anyway), but it sounds to me like you slighted the
importance of the powers a bit.

- Monzo
http://www.ixpres.com/interval/monzo/homepage.html
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🔗Gary Morrison <mr88cet@xxxxx.xxxx>

> I think most people
> working in JI would agree that increasing prime-bases
> *and* increasing powers both create increasing
> dissonance.

I'll buy that. Notice that I didn't suggest that prime-limit advocates
find the power of a prime completely unimportant, but just that they find
it less important than the presence of a given prime at all, when it comes
to major, broad classes of harmonic effect.

I'll go ahead and say in greater detail what I personally have found to
be the case regarding limits. I'm a little reluctant to say this because I
haven't studied the idea in fair but not excruciating detail, and because I
suspect that a lot of people will disagree.

I personally have found that the prime-limit idea is a better
characterization than odd limit, but that the limit concept is strongly
limited (haha) by two more basic considerations: First, in almost context
with the possible exception of large dense chords, the overall size of an
interval has a more profound effect upon its character than its constituent
numbers. By that I mean that *part* of the sense that 3s have a "steely
cold logical simplicity" to them, is just simply because the first
intervals they form are the P4 and P5, and that "steely cold..." character
is more that of P4s and P5s than of 3s. But there is certainly more to it
than that, as evidenced by a comparison of 27:16 to 5:3 - 5:3 sounds to my
ears to have a sweetness that 27:16 lacks.

The second limiting factor for the concept of limits, in my view, is
that once the numbers in a ratio get bigger than a few tens, it really
starts to lose unique character pretty quickly. 2:1, 3:2, 4:3, 5:4, and
7:6, for example have very clearly recognizable musical character. But
once you start talking about 81:64, 243:128, and such, the pitch
relationship becomes complicated enough that it's much more difficult to
attribute a clearly apparent musical character to in typical-speed music
with harmony of typical density, on typical timbres. So ratios involving
any power of, say, 13 (other than 13 itself) are more prone to sound like a
muddled, nondescript, out-of-tune version of a nearby simpler ratio, than
any distinct in their own right. That much less having distinct a 13ish
character.

Now I'm not suggesting that the size of the numbers in a ratio is the
*only* factor determining how audibly intuitive it is, but it's a sig-
nificant factor, I've found.

prime vs. odd limits

🔗dante rosati <dante@pop.interport.net>

🔗bram <bram@gawth.com>

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

🔗dante rosati <dante@xxx.xxxxxxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Patrick Pagano <ppagano@xxxxxxxxx.xxxx>

🔗Paul Hahn <Paul-Hahn@xxxxxxx.xxxxx.xxxx>

🔗Paul Hahn <Paul-Hahn@xxxxxxx.xxxxx.xxxx>

🔗Gary Morrison <mr88cet@xxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Joseph L Monzo <monz@xxxx.xxxx>

🔗Gary Morrison <mr88cet@xxxxx.xxxx>