source file: mills2.txt Date: Fri, 22 Nov 1996 15:56:31 -0800 Subject: 13th harmonic just intonation From: PAULE When I visited Johnny last weekend, he played me some very beautiful music his wife had composed and the couple had performed together. He couldn't tell me offhand what the tuning was, except that it used the number 13 predominantly. Let me assume that the scales Mayumi used were entirely built on cycles of 13:8 plus all octave equivalents. There would be no 13:6's or any prime numbers besides 2 and 13, and thus no possible perfect fifths. A cycle of seven 13:8s would produce the scale 0 121.6 359.5 481.1 718.9 840.5 962.1 (1200) which resembles a neutral diatonic scale and has quite a few usable harmonies, unlike a cycle of seven 11:8s 0 97.4 454.0 551.3 648.7 1005.3 1102.6 (1200), (sounds like 13-tET) seven 9:8s 0 203.9 407.8 611.7 792.2 815.6 996.1 (1200), (sounds like 6-tET) seven 7:4s 0 231.2 275.3 462.3 506.5 737.7 968.8 (1200), or seven 5:4s 0 345.3 386.3 427.4 772.6 813.7 1158.9 (1200). Johnny, I wonder if this has anything to do with what you were saying about the 13th harmonic at the Japanese restaurant. If so, I would decribe this as a "number-theoretic fluke" (to quote Manfred Schroeder) rather than anything intrinsic to the sound of 13:8 itself. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 23 Nov 1996 01:59 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04432; Sat, 23 Nov 1996 02:00:48 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA04748 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id RAA06887; Fri, 22 Nov 1996 17:00:45 -0800 Date: Fri, 22 Nov 1996 17:00:45 -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