source file: mills2.txt Date: Tue, 25 Mar 1997 13:43:11 -0800 Subject: RE: NPD 36 (Paul E) From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE I e-mailed to find out more about this system. I got the following response: >The NPD 36 system is reforming to the proportion number: 4, 5, 6, 7, 8. >Intervals h(n) are generators change amplitude these proportion >frequency. I'm not sure what that means, but it sounds like a 7-limit system is the intent. 36-equal is in fact consistent through the 7-limit, but not the 9-limit. The accuracy of 36-equal within the 7-limit is actually slightly greater than that of 22-equal (though 22-equal is consistent through the 11-limit). >The intervals round has 36 sounds and based on the interval 19 >microtons. >Therefor C major-0 key have intervals:0, 2, 4, 6, 8, 10, 12, 14, 15, 17, >19, 21, 23, 25, > 27, 29, 31, 33, 34, (36). which, we may recall, Tomasz has named >C, Q, Z, D, K, R, E, L, F, M, T, G, N, U, A, O, V, B, W, c. So the scale is generated by an interval of 633 cents, which represents 10/7 in this system, starting on F or 15. Since three consecutive 633-cent intervals mod the octave give a 700-cent interval, 19 notes from the cycle of 633-cent intervals are needed to obtain a traditional diatonic scale (7 notes plus 6*2 intermediate notes). There are seven 4:5:6:7 chords in this scale, and seven 1/7:1/6:1/5:1/4 chords as well. All six consonant triads of the diatonic scale have been "septimalized," in other words, consonant tetrads that are built on these triads exist in the scale. Here is a little progression using Tomasz's letter names, M indicates a 4:5:6:7 chord (4 is the root), and m indicates a 1/7:1/6:1/5:1/4 chord (1/6 is the root) CM Om TM Em QM Vm GM CM 00-02-04-04-02-04-06-00 12-14-12-12-14-16-14-12 21-21-19-21-23-23-21-21 29-29-31-33-31-31-33-29 This is a clever system and definitely seems worth exploring by microtonal composers. It seems unlikely, though, that one could maintain a sense of "key" unless the diatonic notes are treated as the main notes and the other notes used only for "septimalization" or for passing tones or chords. Thanks to Tomasz Liese for his original contribution, and please reply if he disagrees or has additional comments. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 26 Mar 1997 03:19 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA17546; Wed, 26 Mar 1997 03:19:09 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA17387 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id SAA25579; Tue, 25 Mar 1997 18:17:37 -0800 Date: Tue, 25 Mar 1997 18:17:37 -0800 Message-Id: <199703252110_MC2-1347-9C85@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu