source file: mills2.txt Date: Fri, 3 Jan 1997 10:42:38 -0800 Subject: RE: Food for thought - a puzzle From: PAULE Gary wrote, >But for whatever it's worth, I don't think that 9 would be considered a >limit on the grounds that it's not prime. I, for one, completely disagree. I would define the limit as the largest odd number that occurs in ratios that are considered consonant. I differ with many JI advocates in that I do not think dissonances should be represented as ratios -- in 5-limit diatonic music, a major second is a major second whether you call it 9/8 or 10/9. (Not to deny that one could create a 9-limit diatonic style in which these would be two distinct consonances -- this is hinted at by the use of chords such as C-D-E-G as consonances; i.e., 8:9:10:12.) Harry Partch falls into my camp in this debate; please read Genesis of a Music if you haven't already. The simplest equal temperament that clearly distinguishes all 9-limit intervals is 41-equal. In 41-equal, Glen Peterson's scale is 0 ? ? 6 7 8 9 11 13 15 17 20 21 24 26 28 30 32 33 34 35 ? ? 41 or steps of (6) 1 1 1 2 2 2 2 3 1 3 2 2 2 2 1 1 1 (6) A feature that sticks out here is 3-1-3, so I'd probably want to repeat that at the 3/2 above, which turns out to be the 2/1. Transposing down to the 1/1, or 0, we add 0, 3, 4, and 7, giving us 0 3 4 6 7 8 9 11 13 15 17 20 21 24 26 28 30 32 33 34 35 ? ? 41 But now the 3-1-3 at the bottom is a 3-1-2-1. Putting this pattern back in the original position of the 3-1-3 (by adding note 23 and taking liberties with the original question), this becomes 0 3 4 6 7 8 9 11 13 15 17 20 21 23 24 26 28 30 32 33 34 35 ? ? 41 or steps of 3 1 2 1 1 1 2 2 2 2 3 1 2 1 2 2 2 2 1 1 1 (6) I'd probably split up the (6) into (2+2+2), and then add the notes 10 and 31 for symmetry: 0 3 4 6 7 8 9 10 11 13 15 17 20 23 24 26 28 30 31 32 33 34 35 37 39 41 or steps of 3 1 2 1 1 1 1 1 2 2 2 3 1 2 1 2 2 2 1 1 1 1 1 2 2 2. So I have added 7 notes instead of 4. I would not attach ratios to these notes; although they form dissonances with the 1/1, they may form various consonances with the other notes -- one of the beauties of temperament! You can see this scale as based on a "tetrachord" of 1 1 1 1 2 2 2 3 1 2 1 with the extra "whole tone" filled in as 2 2 2 1. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 3 Jan 1997 22:30 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00660; Fri, 3 Jan 1997 22:33:28 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA00658 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA00444; Fri, 3 Jan 1997 13:33:25 -0800 Date: Fri, 3 Jan 1997 13:33:25 -0800 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu