source file: m1441.txt Date: Tue, 9 Jun 1998 08:53:35 -0500 (CDT) Subject: Re: Disappearing Commas From: Paul Hahn On Tue, 9 Jun 1998, Patrick Ozzard-Low wrote: > Could anyone tell me - is there a unique theoretical limit at which a > comma (of different kinds?) may be said to disappear? Are there > strict and not so strict definitions of which systems (or musical > sequences?) of which this can be said to pertain/occur? Is there a > single accepted definitive reference which defines the 'disappearance > of a comma'? (Helmholtz?) These are things I missed in my education. An interval vanishes in a particular ET (or quasi-ET like a well-temperament) if it is represented by 0 scale degrees. In face, to specify which intervals vanish is a fairly practical way to specify a particular system. See Fokker's writings for this. As an example: working in the 3-5 harmonic lattice, deciding that one wants the syntonic comma 81/80 and the lesser diesis 128/125 to vanish commits one to working in a 12-ish system. Conveniently, one can represent this (as Graham Breed has posted before, following Fokker) with matrices: the 81/80 is represented by the vector (4 -1) (3 ^ 4 * 5 ^ -1, omitting 2s) and the lesser diesis by (0 3). Put the vectors together to form a matrix, and the determinant of the matrix is (4 * 3 - -1 * 0), or 12. This also works in higher dimensions. Geometrically, one can imagine the unison vectors (Fokker's terminology) specifying a way to collapse the infinite harmonic plane into a 2-manifold. The number of pitches in the ET is the area of the parallelogram bounded by the unison vectors, or the "real" size of the manifold--I forget what the formal mathematical term for that is. (This also extends to higher dimensions, with parallelepipeds etc.) --pH http://library.wustl.edu/~manynote O /\ "Churchill? Can he run a hundred balls?" -\-\-- o NOTE: dehyphenate node to remove spamblock. <*>