source file: mills2.txt Date: Sat, 2 Mar 1996 09:35:28 -0800 Subject: Positive Systems From: non12@delta1.deltanet.com (John Chalmers) John Pusey: I apologize for my own errors and would like to stipulate that we use Bosanquet's definitions. Positive fifths are thus those that are sharper than 700 cents, negative, less than 700 and positive systems are those with positive fifths, etc. I think that J. M. Barbour was so committed to 12-tet that it is not surprising that he sometimes thought of the perfect fifth as having 700 cents. Thus, I view his definition of positive as sharper than 3:2 rather than 7 degrees of 12-tet as a lapse, not a fundamental misunderstanding. BTW, he did grudgingly admit that a piece by Bach sounded magnificent in the 1/6th comma "meantone" tuning (notes to the Musurgia series of didactic recordings he and Kuttner put put in the 60's). As for "doubly positive," see pages 60-63 of Bosanquet. _"A regular cyclic system is of the rth order, negative or positive, if 12 fifths fall short or exceed 7 octaves by r degrees of the system"_. Hence for 22-tet, 12 x 13 = 156 (12 x the 22-tet fifth) is two degrees of 22-tet sharper than 7 x 22 =154. Therefore, 22-tet is a doubly positive system. R is thus the difference between the diatonic (5 Fifths) and chromatic (7 Fifths) semitones in ET degrees or alternatively, the difference between B# and C if the cycle is started on C. In other words, r is the number of degrees in the Pythagorean ditonic comma in each ET (see Paul Rapoport's article "The Notation of Equal Temperaments," in XH16, 1995). Another parameter useful for characterizing ET's is the number of degrees in the apotome or Pythagorean chromatic semitone (the #) (Rapoport, XH16, 1995), i.e., 7 Fifths minus 4 Octaves. This quantity is the number of degrees between C# and C, 1 in the case of 12 and 19-tets, 2 in 17, 24 and 31-tets, and 3 in 22, 29, 36 and 50-tets. To my knowledge there is no accepted name for this number as a parameter (Erv Wilson simply calls the systems singulary, binary, ternary, etc. as the number is 1, 2, or 3, etc., but it is essential to fitting various ET's to the Bosanquet or multiple Bosanquet keyboards or rationally assigning accidentals (Wilson, XH2, 1974; Rapoport, XH16, 1995). Of course, this discussion presumes 12-tet is the standard system. If one were to base one's classification on 19-tet, for example, then "r" would be calculated relative to the 19-tet fifth (19 Fifths -11 Octaves) and the systems would be arranged according to their ditonic commas (12 Fifths - 7 Octaves). Similarly, one might chose 17, 22, 31 or even 5 and 7 (Wilson, XH3, 1975; XH1, 1974) as bases for notation and keyboards. With less historical and musical justification, one might choose cycles of other intervals than the 4th or 5th such as 5/4, 7/4 and a low number ET which approximates them well. One could even use some tempered interval like 8 or 10 degrees of 13-tet (and higher homologs) to define r and the (pseudo) chromatic semitone. --John Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 2 Mar 1996 22:17 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA29885; Sat, 2 Mar 1996 13:17:04 -0800 Date: Sat, 2 Mar 1996 13:17:04 -0800 Message-Id: <960302211308_71670.2576_HHB56-7@CompuServe.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu