Re: JI question (reply to Greg, Paul H., and Gary)

> 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.