> Greg Schiemer wrote:

>

>> >> How many n-limit intervals must there be in a scale/chord for it

to

>> be called

>> >> n-limit ?

>

Paul Erlich wrote:

>> All of the intervals must be n-limit for the scale or chord to be

called

>> n-limit. Note that by the very definition of the term "limit", the

>> 7-limit includes the 5-limit within it, and the 5-limit includes the

>> 3-limit within it, etc.

>> Kraig Grady wrote,

>>

>> >ONE!

>>

>> In a sense, this is correct, although the language here is tricky so

it

>> is very important to be careful. For a scale or chord to be called

>> n-limit, there must be at least one interval that is outside all

limits

>> smaller than n, and I suspect that is what Kraig meant.

>Paul, I would have thought the answer would be that some scales and

chords are more n-limit >than others ranging from as few as one n-limit

interval and as many as all n-limit intervals. >Would it always have to

include lower limits ?

I'm not sure I understand your question. But in the odd-limit

definition, I can't think of any set of three or more notes where all

intervals are at the same limit. In the prime-limit definition, though,

you can endlessly stack one interval of a given prime limit and all

resulting intervals will be of the same prime limit.

Paul Erlich wrote:

>> . . . there can

be

>> no interval in the scale or chord that is outside the n-limit, if we

>> want to call the scale or chord 'n-limit'.

Paul Hahn wrote:

>Buh? For chords I agree, but for scales I strongly disagree. Consider

>the standard JI version of the major scale:

>1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

>Most would consider this a 5-limit scale. But it contains intervals up

>to the odd limit of 45 (the 45/32 between the 4/3 and the 15/8).

Surely

>no one would call this a 45-limit scale.

Most would consider it a 5-limit scale, but would either be referring to

the prime limit, or to the odd limit of the harmonies they intend to use

as consonances. If one did use the entire scale as a chord, you would

agree that the odd limit would be 45. So I guess a scale's odd limit can

only be defined when a description of how the composer intends to

construct consonant harmonies is provided.

>'Course, it's only 'cos I (and Paul E.) advocate the odd-limit

>interpretation over the prime-limit that this is a problem.

It was Partch who coined the term "limit", and though his usage was not

entirely consistent, it was far closer to the odd-limit definition than

to the prime one now in common currency.

Gary Morrison wrote,

>> How many n-limit intervals must there be in a scale/chord for it to

be called

>> n-limit ?

> My impression of the term is that there needs to only be one.

OK, let's get this one straight. Harry Partch defined a "ratio of N" as

an interval where the largest odd factor of the numerator and

denominator is N. The "N-limit" is defined as all ratios of N, ratios of

N-2, . . . ratios of 1. (The N-limit tonality diamond is the set of all

pitches that make N-limit ratios with a fixed pitch, usually denoted

1/1).

So, an N-limit chord only needs to have one ratio of N, but may not have

any ratios of odd numbers higher than N.

>It's probably

>safe to say that in any scale or chord with more than two pitches, if

you have

>one N-limit interval you pretty much have to have more than one. If

for

>example, you have pitches X, Y, and Z in a chord, and the interval from

X to Y

>is N-limit, and the interval from X to Z is not, then the interval from

Y to Z

>is.

> Actually, that's true as long as X to Z has a lower limit than N.

If it has

>a higher limit, then that is the chord's limit.

This is all right if you replace "N-limit" with "ratio of N" and

rephrase the first two instances of "limit" in the last sentence so that

"ratio" is used instead.