source file: mills2.txt Date: Tue, 16 Jan 1996 10:00:57 -0800 Subject: Sine Mod ET's From: "John H. Chalmers" The Tuning List has been rather inactive recently so I thought I'd post some recent studies of mine. Just before last Xmas, Brian Thomson posted some examples of sine-modulated equal temperaments. SMET's are ETs in which the interval width is periodically varied to improve the consonance of certain intervals, at least in the mode starting on the note 0. Brian used the formula A*sin(pi*F*N/ET) where F is the frequency (restricted to even numbers in this case), A is the amplitude, N is the note index and ET is the number of notes per octave). The phase is 0 at tone #0. I wrote a simple 1-D search program BASIC program and minimized the sum of the abolute values of the errors between the closest SMET notes and the JI intervals 3/2, 4/3, 5/4, 6/5, 5/3 and 8/5, a somewhat different set than BT used. The summed absolute error, squared error, mean deviation, variance and standard deviations were computed in cents for each SMET. Other evaluation and statistical functions could have beed used, but these seemed appropriate to me. Results were obtained for the 12,15,17,19, 22, 24, and 31-tets. The errors of some unmodulated ET's are small enough that "improvement" is scarcely necessary and others are so inharmonic there is little point as even drastic amounts of modulation do not really make much difference. ET F A STD Un-MOD STD 12 10 0.132579 5.29859 12.05267 15 8 -0.240234 4.69927 13.31584 17 10 0.502893 5.53674 27.30851 19 8 0.1136078 0.48389 5.95502 22 20 0.1699097 2.94625 8.92950 24 10 0.265158 5.29859 12.05267 31 18 0.133739 1.467844 4.58336 A few caveats: the search program was somewhat sensitive to initial conditions and although I set the acceptable summed error to be less than 10 exp -7, even in double precision mode, I would not trust the numbers past the 6th figure. In a few cases, the best solutions had A's greater than 0.5, which would imply overlapping notes, except that in some cases two or 3 adjacent intervals varied sufficiently that no inconsistencies arose. These cases have been omitted from the table above except for 17. My solution for 19 is very similar to Brian's, the differences in the amplitude are probably due to my use of a different interval set and round-off in my computer. --John Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 16 Jan 1996 19:02 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA03635; Tue, 16 Jan 1996 10:02:48 -0800 Date: Tue, 16 Jan 1996 10:02:48 -0800 Message-Id: <199601132256.OAA11434@netcom13.netcom.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu