source file: mills2.txt Date: Sat, 20 Jul 1996 15:16:06 -0700 Subject: notation, 20-tET, etc. From: Paul Rapoport 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