source file: mills2.txt Date: Sun, 16 Mar 1997 10:50:34 -0800 Subject: melodic coordinate systems From: gbreed@cix.compulink.co.uk (Graham Breed) >From reading the list, it looks like I'm not the only person using harmonic axes. However, I haven't seen any mention of the full algebraic treatment I apply to them. I will outline some of my ideas here, and someone can tell me whether they are original and, if not, how they are notated. First a column matrix needs to be defined to correspond to harmonic axes: oct (log(2)) H ----- (log(3)) log(2) (log(5)) H can be generalised to have as many dimensions as the matrix you are multiplying it by. Using this, a fifth is (-1 1)H, a major third (-2 0 1)H and a subminor seventh (-2 0 0 1)H. H is useful for defining harmonies. For melodies, though, it is more convenient to use the smallest intervals that occur in a scale. For a diatonic scale with comma shifts, these will be: (t) ( 1 -2 1) (s) ( 4 -1 -1) H (p) (-4 4 -1) Where t is a minor tone, s a diatonic semitone and p a syntonic comma. These intervals completely define a 3-D JI, and so the matrix can be inverted: ( 5 2 3) (t) H ( 8 3 5) (s) (12 4 7) (p) This will be true of any three intervals whose defining matrix has a determinant of +/- 1. A fifth can now be written: ( 5 2 3)(t) (t) (-1 1 0)H (-1 1 0)( 8 3 5)(s) (3 1 2)(s) (12 4 7)(p) (p) Other coordinate systems can be defined from t, s and p or directly from H. One I find useful is: (r5) (-3 -1 2 0) (12 7 3 0)(r5) (q5) ( 7 0 -3 0)H H (19 11 5 0)(q5) (p5) (-4 4 -1 0) (28 16 7 0)(p5) (p7) ( 1 2 -3 1) (34 19 8 1)(p7) You can add commas for as many harmonic dimensions as you like. Equally tempered scales can then be defined according to the number of steps in these intervals: nTET r5 q5 p5 p7 12 1 0 0 0 15 1 0 1 0 19 1 1 0 0 22 1 1 1 1 31 2 1 0 0 34 2 1 1 0 41 2 2 1 1 46 3 1 1 0 53 3 2 1 1 This is the clearest representation I know of. From the defining matrix, nTET 12*r5 + 7*q5 + 3*p5. That'll do for the time being. Graham Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 16 Mar 1997 19:52 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA30779; Sun, 16 Mar 1997 19:52:24 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA30778 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id KAA27175; Sun, 16 Mar 1997 10:50:07 -0800 Date: Sun, 16 Mar 1997 10:50:07 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu