source file: mills2.txt Date: Sat, 14 Dec 1996 00:17:36 -0800 Subject: Brian McLaren's Group Theory From: Lindsay Shaw and Paul Turner >From Paul Turner Brian McLaren's December 10 post on Group Theory needs some tidying up. One doesn't like to be pedantic over idiosyncrasies of terminology, nor risk giving offence by talking about minor inaccuracies. On the other hand, to say nothing might be considered antisocial in another sense: both xenharmonic theory and mathematics could fall into disrepute and good people could get confused. There are, I think, two main issues that need clarification. Much of Brian's article depends on his interpretation of the mod function and on the relevance of the concepts of relative primeness and compositeness. ______ To begin, a group is not a group until a set is defined together with a binary operation on the elements of the set that satisfies the group axioms. We must take it as read in Brian's article that the group under consideration is the set of n-tet intervals measured in scale-steps together with the binary operation of addition modulo n. It is important to mention this because other ways of defining a group in this setting are possible. (It is true, however, that other formulations may have essentially the some group structure, even though the set elements and the binary operation are defined differently.) In speaking about the elements 5 and 7 in 12-tet, and also the elements 1 and 11, Brian claims that in modulo 12 arithmetic, 7 5 and 11 1. This is not true. It is true that 12 5(mod7) and 12 7(mod5) but neither of these facts is particularly helpful. Closer to the mark is the fact that 5 -7(mod 12). What is needed here is the idea of the 'inverse'. In this group 5 and 7 are each other's inverse, as are 1 and 11. The significance of this is that an element and its inverse each generate the same cyclic subgroup. In fact they generate the elements of the subgroup in opposite order to one another. It should be noted that the idea of 'order' in the sense of 'permutation', is not intrinsic to groups. Indeed, elements 1,5,7,11 generate exactly the same subgroup, namely the group itself. But the sequence in which the elements appear in the process of generating the subgroup is different in each case, as Brian points out. Now, every element of a group generates a cyclic subgroup of some size or other (always dividing n). In the case of the particular type of group that arises in the n-tet context, i.e. the cyclic groups of order n, the size of the subgroup generated by a particular interval is n/d where d is the greatest common divisor of n and the number of scale-steps in the interval. E.g. in 12-tet, the element 6 generates a subgroup of order 2, the element 10 generates a subgroup of order 6, and 5 generates a subgroup of order 12 - the whole group. Clearly, if the number of scale steps in the generating interval is 1 or a number relatively prime to n, the whole group will be generated. Contrary to Brian's assertion, the compositeness of the number of scale steps is neither here nor there. For example, in 16-tet, the 3-step interval generates the whole group but so does the 9-step interval, notwithstanding the fact that 9 is composite. Only the permutation of the elements is different: 0 3 6 9 12 15 2 5 8 11 14 1 4 7 10 13 0 9 2 11 4 13 6 15 8 1 10 3 12 5 14 7 Similarly, the number of different 'cycles of fifths' in an n-tet tuning doesn't depend on the compositeness or otherwise of the numerical size of the generating intervals. It is their relative primeness to n that counts. __________ The full significance of Brian's post probably becomes apparent when read in conjunction with the paper by Gerald J. Balzano: The Group-theoretic Description of 12-Fold and Microtonal Pitch Systems; Computer Music Journal, Vol. 4, No. 4, Winter 1980; MIT which, in turn, should be read in conjunction with an introductory text on abstract algebra, if necessary. (The one by Frahleigh would be a good choice.) Brian's main thesis, if I have inferred it correctly, is that a group-theoretic view can be helpful in making sense of the structural possibilities in n-tet tunings. In this I thoroughly concur. I have to admit defeat in trying to unravel Brian's statements about 'tiling'. I suspect there is a connection here with the Balzano article. It may be that the relevant essence is the fact that some cyclic groups are isomorphic to direct products of smaller groups and some are not: the group corresponding to 12-tet _is_ while the groups corresponding to 16-tet and 19-tet are not. (This seems to have a bearing on the kinds of chordal relationships that can be found.) PT Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 14 Dec 1996 20:37 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04192; Sat, 14 Dec 1996 20:39:41 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA04226 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id LAA28461; Sat, 14 Dec 1996 11:39:38 -0800 Date: Sat, 14 Dec 1996 11:39:38 -0800 Message-Id: <199612141935.LAA29999@sunatg1> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu