source file: mills3.txt Date: Tue, 30 Dec 1997 20:11:31 +0100 Subject: MOS Theory From: John Chalmers MOS: Finding the generators of M-toned MOS is not a simple task, but I'll try to explain my procedure. MOS are generated by intervals which do not divide the octave or other Interval of Equivalence evenly. They are also characterized by a quasi-periodic pattern of two intervals (A and B) whose sizes may have any proportion, but the most familiar ones have 1/2 <= A/B <= 2 as these are Rothenberg-proper. The two intervals A and B are arranged "maximally-evenly" (see papers by John Clough and colleagues, Clampitt, Carey, Douthett et al.) and any given M, there are several possible patterns such that the numbers P of A and Q of B sum to M and have no common factors. Thus for 7 tone MOS (M=7), the possible patterns are 6A+1B (and 1A +6B), 5A+2B (2A +5B) and 4A+3B (3A+4B). In an N-tone ET, PA +QB = N, and P +Q =M. Both G and the complement of G with respect to the IE generate the same MOS, though modally rotated. Hence one can obtain the diatonic scale in 12-tet by a cycle of either 4ths or 5ths. To find the Generator G of each class of MOS of M tones, I do the following. For the simplest case where P= M-1 and Q =1, the limits are 0 to N/M in cents or logs and from 0 to [N/M] where [] is the integer <= N/M in the case of equal temperaments. The logic is that if A/B were infinite and A>B, then B-> 0 and there would be only P=M-1 tones per octave (or IE). In the case of M=7, the largest possible generator would be 200 cents as that would yield 6 whole tones and a zero-width interval for B. In the A+6B case, if G were infinitesimal, then there would be 6 very small intervals of G and a large remainder. Hence for the M-1, 1 type of MOS, G lies between 0 and Octave/(M-1). For the 5A+2B=7 MOS case. the logic is similar. IF A/B is infinite, then there are only 5 tones per octave (IE), if infinitesimal, then only 2. Hence intervals between 600 and 720 cents generate 7-tone MOS of the 5A+2B and 2A+5B form. In 9-tet, G= 5 degrees or 666.667 cents and a cycle of 5 degrees produces a MOS 1 1 1 2 1 1 2. The corresponding scale in 12-tet is 2 2 2 1 2 2 (as ascending successive degrees). The 4+3 scales are produced by cycles of neutral thirds (or 6ths) such as 7 degrees of 24-tet. To find the generator(s) of a particular class of MOS in a given N-tone ET, one simply chooses the interval(s) which lie in the ranges calculated above. Needless to say, not every N-tone ET contains every M-tone MOS even when M < N. Remember that PA + QB = N and P+Q = M. A few examples: If one wishes a 9-tone MOS of the 5+4 class and whose intervals are in the ratio of 2/1, then 5*2 + 4*1= 14 and the MOS exists in 14-tet. One may find the MOS by distributing the intervals as evenly as possible by inspection: 2 1 2 1 2 1 2 1 2 and immediately see that G must be 3 degrees as the repeating block in the MOS spans 3 degrees of 14 (2+1). Alternatively, one can use the fact that G must lie between 1200/5 and 1200/4 cents (from 5+4=9) or 240-300 cents. The only interval in 14-tet in this range is 3 degrees (257 cents) (one could also use 11 degrees). --John SMTPOriginator: tuning@eartha.mills.edu From: John Chalmers Subject: Stellate Hexanies (14-anies) PostedDate: 30-12-97 20:12:10 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 30-12-97 20:09:49-30-12-97 20:09:50,30-12-97 20:09:13-30-12-97 20:09:13 DeliveredDate: 30-12-97 20:09:13 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C125657D.00694350; Tue, 30 Dec 1997 20:11:41 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA28256; Tue, 30 Dec 1997 20:12:10 +0100 Date: Tue, 30 Dec 1997 20:12:10 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA28119 Received: (qmail 17507 invoked from network); 30 Dec 1997 11:11:48 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 30 Dec 1997 11:11:48 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu