source file: mills2.txt Date: Tue, 30 Jul 1996 06:17:40 -0700 Subject: Response to McLaren's kind remarks From: Lindsay Shaw and Paul Turner Several kilobytes ago - TUNING 783 - Brian McLaren posted some agreable remarks on an idea I had communicated to him about applying group theory to understanding some properties of et_s. Brian mentioned some references to work that had already been done along these lines, for which I'm grateful. (I'd be even more grateful if I knew exactly _which_ number of the Computer Music Journal contained Gerald G. Bolzano's paper "The Group Theoretic Structure of 12-fold and Microtonal Tunings".) It might be worth saying here what the idea was that attracted Brian's comments. Quite likely I'm covering more or less familiar material with this and, in any case, the mathematics is not deep. Maybe others will have useful comments. ***** We say that two frequencies u and 2u differ by an octave and that a scale of n steps from u to 2u is equal-tempered if the frequencies 2^(1/n)u, 2^(2/n)u, ... , 2^(n-1/n)u are interpolated between u and 2u. The set of numbers {1 = 2^(0/n), 2^(1/n), ... , 2^((n-1)/n)} representing the multipliers of the frequency u, forms a group under the operation of adding the numerators of the respective indices, modulo n. I.e. 2^(p/n) * 2^(q/n) = 2^((p+q)/n), where p+q is reduced mod n. Identifying the numerators 0,1,...,n-1 with pitch-class numbers, we have the familiar group of pitch-classes under addition modulo n. The group formed in this way is the cyclic group of order n. It has (cyclic) subgroups of every order dividing n. This means, for example, that the groups corresponding to, say, 14-et, 21-et and 35-et all have subgroups of order 7 and hence these et_s possess _all_ the structure of 7-et. In fact they include 7-et both as a subgroup and as cosets of the subgroup of order 7. [E.g. in 21-et, the set of pitch classes {0, 3, 6, 9, 12, 15, 18} constitutes the subgroup of order 7, while the sets {1, 4, 7,...,19}, {2, 5, 8,...,20}, etc. are the cosets.] Similarly, 8-et contains transpositions of the 'diminished seventh' chord found in 12-et, and 9-et contains transpositions of the 'augmented' triad from 12-et, due to shared subgroups of orders 4 and 3 respectively. Practical consequences of this way of looking at et_s might include the recognition of: (1) the availability of smooth transitions between tunings by using a common subgroup as a kind of 'pivot', and (2) the need to avoid certain combinations or progressions nominally available in a given tuning, in order to express the _uniqueness_ of that tuning. In n-et, where n is not prime, consider the factorisation pqr...-et, where p,q,r,... are prime powers and n = pqr... . One might want to avoid the 'trap' of expressing p-et or q-et or r-et etc. rather than whatever is special about n-et itself. Of course, one might _not_ want to avoid these commonalities: indeed, one might want to exploit them. But to be aware of their existence, at least, seems important. The question arises as to where the specialness of any n-et, for a composite n, resides. Consider a sequence, of sufficient length, of pitches a,b,c,... . If the sequence does not lie entirely within any particular coset of any proper subgroup, then n is the smallest order of et_s that contains the sequence. The sequence is unique to n-et if we discount the fact that there are higher order et_s that also contain it. But what might 'of sufficient length' mean? And how difficult does it become to meet this criterion for uniqueness when n is highly composite (so that there are subgroups of many orders)? For example, in 12-et, a passage in the key of C-major consisting of the tones F-G-A-B, expresses 6-et until the tones -C-D-E are appended. Note that if C,D,E belong to one coset of the subgroup of order 6, then F,G,A,B belong to the other. Likewise, if F,B,D belong to one coset of the the subgroup of order 4 then C,A and E,G belong to the other two cosets respectively; and the sets {C,E}, {G,B}, {F,A}, {D} contain representatives of the cosets of the subgroup of order 3. It looks as if structures like the major scale express the unique features of 12-et rather well, while unadorned permutations of the sets {C,D,E,F#,G#,A#} or {C,D#,F#,A} or {C,E,G#} or {C,F#} or their transpositions, might not. ***** All of this is knowable through other approaches. It might even be trivial. But it seems to me that group theory provides a relatively tidy way of dealing with at least some tuning issues. Paul Turner Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 30 Jul 1996 15:28 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id GAA25139; Tue, 30 Jul 1996 06:28:49 -0700 Date: Tue, 30 Jul 1996 06:28:49 -0700 Message-Id: <9607301328.AA15941@ ccrma.Stanford.EDU > Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu