notation, 20-tET, etc.

Paul Ehrlich wrote:

Well, in Paul Rapoport's point of view, 22 would fall into the same
category as 27, 41, 46, and 53, where syntotic comma (81:80) displacements
are necessary to get good thirds, and if 27, 41, 46, or 53 is traditional,
then so is 22. Paul Rapoport's notation probably deals with 15 and 34 in
the same way. Is this correct?

-- The assumption is close (some options and subtleties aren't captured by
it). If "the same way" means it is possible to use the syntonic komma,
then fine. In 22 the Pythagorean komma is 2, which makes a difference in
the circle of fifths, of course. 46 is the closest in that regard, being
22+24.

However, I'm not aware of much of a "tradition" of notating comma
displacements in actual musical practice, so I'd have to say the only
traditional tunings are 12, 19, 31, 43, 50, 55, etc.

-- This would agree with Eric Regener, who rejected positive tunings.
However, notating and using them is not a problem, and there's a decent
tradition for it, mainly because of the use and study of 53.

I have a feeling I would have a bigger problem with Paul Rapoport's
system, since it seems the minor third and its purity are being ignored.
It's that consistency issue again. If the best approximation of the
perfect fifth minus the best approximation of the major third is not the
best approximation of the minor third, a notation based on just 5-limit
intervals is not appropriate for the tuning. In 20tet, for example, the
perfect fifth is 12 steps, and the closest approximation to the major
third is 6 steps, so should the minor third be 6 steps too? No, 5 steps is
a far better minor third. This seems to be a fatal blow to notating 20tet
with Pythagorean scales and syntotic commas.

-- The minor third and its purity are not in fact ignored. The method works,
I believe, for tunings such as 20. If the best M3 and m3 don't add up to
a P5 (as in this case), that doesn't make the notation a problem.
Similarly, if there's no good P5 in an ET (as in 13, 16, 23, others), the
notation isn't a problem; in fact, it reveals the problem.

-- In 25, there are two minor thirds, both of them fairly awful but
usable. No problem there either.

-- However, it is desirable to use the 7th and other primes in the
notation system to reflect their use in the tuning. I look forward to
hearing about recent work in this connection.

Paul Rapoport

Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Sun, 21 Jul 1996 01:52 +0100
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id QAA01147; Sat, 20 Jul 1996 16:52:53 -0700
Date: Sat, 20 Jul 1996 16:52:53 -0700
Message-Id:
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu