source file: mills2.txt Date: Fri, 18 Oct 1996 20:01:15 -0700 Subject: More beating Intervals From: John Chalmers While trying to determine the source of the discrepancies between my numbers and Manuel's, I recomputed my earlier table and got the same results. However, it occurred to me that my approach does not distinguish between intervals that are sharp from those that are flat, or in other words, between intervals beating because they are to narrow or too large. Hence, there are two solutions to each of the relations; one where the beat rates are due to intervals both sharp or flat, and one where they are dissimilar. I have thus added a column of "Equal and Opposite" tunings for these new results. Beating Intervals Equal Equal & Opposite 1. 4th = min 6th 695.8096 697.1769 2. 4th = Maj 6th 696.2958 692.1612, too flat? 3. min 6th= Maj 6th 696.0063 698.1741 4. 5th = Maj 3rd 695.6304 697.2785 5. 5th = min 3rd 695.8096 693.3588, too flat? 6. Maj 3rd = min 3rd 695.7294 (629.9748, unacceptable!) 7. min 3rd = min 6th 698.8781 695.9340 8. 4th = Maj 3rd 697.4747 695.2284 9. 5th = min 6th 697.0390 696.0237 My approach is based on Rasch's beat rate formulae where x is the temperer and S=81/80. The tempered fifths thus equal 3x/2. The "register-free" beat rates of the various intervals are as follows: Fifth 3x - 3 Fourth 4/x - 4 Major 3rd 5Sx^4 - 5 Minor 3rd 6/Sx^3 - 6 Major 6th 5Sx^3 - 5 Minor 6th 8/Sx^4 - 8 Tunings are defined by taking pairs of these relations and setting one equal to the other or to the negative of the other. Thus #4 where the Fifth beats at the same rate as the Major third is defined by the formulae 3x - 3 = 5Sx^4 - 5 and 3x - 3 = 5 - 5Sx^4. Solve these equations for x, then multiply by 3/2 to get the decimal fraction for the tempered fifth and then multiply this value by some convenient pitch base such as C261.625 hz. From this frequency, one may compute the difference between the 3rd partial of the tonic and 2nd of the tempered fifth. This value is 0 for x=1. Beat rates of the other intervals are calculated by generating them from tempered fifths by the traditional Pythagorean cycles. Beat rates are then determined from the lowest (nearly) coincident partials as above. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 19 Oct 1996 18:41 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA03481; Sat, 19 Oct 1996 18:43:12 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA03475 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA25310; Sat, 19 Oct 1996 09:43:09 -0700 Date: Sat, 19 Oct 1996 09:43:09 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu