Prime/Odd-limits; omega functions

John Chalmers wrote:
>
> RE Limits: I agree with Paul Erlich about limit terminology.
> As generally used on this List, "N-Prime limit" implicitly means
> that powers of N and multiples of N with smaller primes are
> included in the tuning. Partch's usage of the term N Limit
> means that N is the largest ODD number appearing in either the
> numerator or denominator of the ratios defining the tuning.
> Thus 33/32 would be said to be at the 11-prime limit by conventional
> usage, but at the 33-limit in Partch's terms. I think these
> nomenclatures are clear enough, but one could add 'odd' to be
> unambiguous. Statements such as at the "33 odd-number limit" or
> "33(odd) limit" should be clear.

This is related to the number-theoretic functions small-omega(n) and
large-omega(n). The small-omega function gives the number of different
prime factors of n (i.e. each prime factor is counted once, regardless
of the power to which it may be raised), while the large-omega function
gives us the total number of prime factors (i.e. if p^n is a factor,for
any prime p, it counts as n factors). Our limit terminology in tuning
theory can be seen as an extension of these number-theoretic functions
into the rationals. The prime-limit of a given ratio is thus related to
small-omega(n_1/n_2), where n_1 and n_2 are relatively prime, while the
odd-number limit of the same ratio is related to large-omega(n_1/n_2).

Partch's original odd-number limit terminology seems to be a product of
its times, and of its creators compositional preferences: one of
Schoenberg's strategies for justifying his move into atonality concerned
the historical acceptance of successively larger accumulations of
thirds, first triads, then 7th chords, 9th chords and 13th chords;
Schoenberg tried to argue further that such chords invoked successive
odd-numbered members of the harmonic series, albeit approximated within
equal temperament (an argument which Partch mercilessly lambasts). Even
without the harmonic series accretion, the argument is still highly
dubious, but it had a certain persuasive power in its day (Schoenberg
wasn't its only advocate). Partch himself was concerned, as a composer,
in using intervallic structures moving up through successive
odd-numbered harmonics, although unlike Schoenberg he was determined to
do the job properly (and as everyone knows, for purely pragmatic reasons
he chose not to go beyond the 11-limit).

However, the prime-limit modification of Partch's terminology, related
to the small-omega function seems to have won greater favour, possibly
because it is more open to generalisation. For instance, how do we
specify the limit of a fully chromatic 14th-century monochord, or even
worse, a schismatic 15th-century monochord (i.e. Fb serving for E etc.)
Both are constructed entirely from Pythagorean ratios, and thus 3-limit
by the prime terminology; but if we use the odd-limit terminology we
must say, for example, 531441-limit, even though all odd numbers up to
this limit which are not powers of 3 are excluded. I notice that John,
above, suggests adding "odd" for the Partchian limit terminology,
leaving the prime limit as a default. Have I construed you correctly,
John?

Perhaps it would be better to use the Euler-Fokker notation, and simply
list all the primes [p_1,p_2,...p_n] (according to the small-omega
criterion) that are used in a given composition. Even the prime limit
terminology can become awkward when, as with some of Ben Johnston's
works of the 70s, some prime smaller than the limit are not used as
factors, whereas the Euler-Fokker notation is unambiguous.

OK, now I'd like someone to do me a small favour: could you give me the
source in Euler for the specification of prime factors in musical
contexts. Is there a brief English-language account of the Euler-Fokker
notation? (I'm not aware of anything in Xenharmonikon that fits this
description.) Or perhaps someone would be kind enough to post a few
words on the topic to the list. For instance, is the convention to omit
2 from the list of primes, or to include it? Thanks.

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/


Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Thu, 20 Mar 1997 17:41 +0100
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07084; Thu, 20 Mar 1997 17:41:04 +0100
Received: from ella.mills.edu by ns (smtpxd); id XA07078
Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
id IAA24982; Thu, 20 Mar 1997 08:39:18 -0800
Date: Thu, 20 Mar 1997 08:39:18 -0800
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@ella.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@ella.mills.edu