RE: [tuning] Re: retuning Schoenberg (was: definition of COFT)

John deLaubenfels wrote,

>OK, I'm confused here. 5-limit music includes 9/8 ratios, yes? 7-limit
>music can include 9/7, no? So what is "9-limit" music?

If you look in the index of Partch's _Genesis_, the book in which "limit"
was first used, you'll see three references to "9-limit". Basically, in
5-limit music, 9:8 is considered dissonant (so its tuning shouldn't be a
factor in your program), and in 7-limit music, 9:7 is considered dissonant
(ditto), but in 9-limit music, both are considered consonant.

> John deLaubenfels wrote,
>
> > OK, I'm confused here. 5-limit music includes 9/8 ratios,
> > yes? 7-limit music can include 9/7, no? So what is "9-limit"
> > music?

Hi John. The explanation Paul Erlich provided at
http://www.egroups.com/message/tuning/12803

is an excellent answer to your question. I just wanted to
supplement it with the comment that you should be very aware
of the two different definitions for 'limit': *prime*- and
*odd*-limit.

When I say 'limit', I almost invariably mean 'prime', and when
Paul says it, he almost always means 'odd'. It just reflects
on what each of us places more importance.

IMO, odd-limit is an important consideration when the object
of study is dyads (i.e., 'bare' intervals), but for any larger
collection of pitches - whether chords, chord progressions, entire
pieces, or entire tuning systems - I'd place much more importance
on the prime-limits that can be observed in action.

My Dictionary definitions of 'prime', 'affect', and the related
links will explain my position in more detail. Paul will probably
respond to this with more details on his views.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> IMO, odd-limit is an important consideration when the object
> of study is dyads (i.e., 'bare' intervals), but for any larger
> collection of pitches - whether chords, chord progressions, entire
> pieces, or entire tuning systems - I'd place much more importance
> on the prime-limits that can be observed in action.

I agree that for entire JI tuning systems, the prime limit is
typically more important and relevant than the odd limit. But for
chords, the issue is a little more intricate. Looking at many JI
chords, one sometimes gets the impression that prime limit is more
important than odd limit. But that is because many chords are
composed of several consonant intervals of a low odd limit, and
perhaps just one interval that has a high odd limit but low prime
limit. For example, the major seventh chord has 5 intervals within
the 5-odd limit and 1 ratio of 15. The chord is pretty consonant
overall, and the tendency is to think that the prime limit (5) is
therefore more relevant than the odd limit (15). But many 5-prime-
limit chords are very dissonant. A better way of looking at the issue
is that the 5 consonant intervals and 1 dissonant interval in the
chord make the chord 5/6 consonant and 1/6 dissonant -- the single
ratio of 15 is simply a splash of dissonance in a rather consonant
cocktail.