Hi yall-

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

limit? Ratios with 9 in them are, I thought, 3 limit ratios. I was looking

at Joe Monzo's page and he speaks of prime limits and then treats 9 as a

limit also. 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?

Wouldn't 3 limit ratios include 9/8 and 16/9, as well as anything else

containing no prime factor greater than 3? 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? 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?

thanks,

dante

On Wed, 24 Feb 1999, dante rosati wrote:

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

> limit?

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.

-Bram

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

bram wrote:

> From: bram <bram@gawth.com>

>

> On Wed, 24 Feb 1999, dante rosati wrote:

>

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

> > limit?

>

> 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.

>

> -Bram

>

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>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

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.

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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On Wed, 24 Feb 1999, Paul H. Erlich wrote:

> 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.

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. 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. (Not you,

Paul E., 'cos you know all this stuff; I mean the newer folks.)

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote

O

/\ "Well, so far, every time I break he runs out.

-\-\-- o But he's gotta slip up sometime . . . "

NOTE: dehyphenate node to remove spamblock. <*>

On Thu, 25 Feb 1999, Paul Hahn wrote:

> --in the meantime, I refer you

> to the Appendix of Patrick Ozzard-Low's paper on consistency.

Arrgh! Misplaced modifier. Patricks paper is on 21st century

orchestral instruments. The _appendix_ is on consistency. Sorry.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote

O

/\ "Well, so far, every time I break he runs out.

-\-\-- o But he's gotta slip up sometime . . . "

NOTE: dehyphenate node to remove spamblock. <*>

> > 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 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.

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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> 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.