source file: mills2.txt Date: Thu, 21 Nov 1996 17:15:00 -0800 Subject: Fractional comma manifesto From: Kami Rousseau First things first. Acoustique musicale is copyright 1959 by editions du centre national de la recherche scientifique (Paris, France). Yes, there are two 31TET tunes on my page, they are now dowloadable from the "music" section. It is in perpetual construction, as are all web pages. The URL is http://www.interlinx.qc.ca/~kami . All the readers of this forum know about the "spiral of fifths." .. Cb 4096/2187 Gb 1024/729 Db 256/243 Ab 128/81 Eb 32/27 Bb 16/9 F 4/3 C 1/1 G 3/2 D 9/8 A 27/16 E 81/64 B 243/128 F# 729/512 C# 2187/2048 .. This sequence can be made more visual using exponantial notation : .. Cb ( 3^(-7) ) / ( 2^(-12) ) Gb ( 3^(-6) ) / ( 2^(-10) ) Db ( 3^(-5) ) / ( 2^(-8) ) Ab ( 3^(-4) ) / ( 2^(-7) ) Eb ( 3^(-3) ) / ( 2^(-5) ) Bb ( 3^(-2) ) / ( 2^(-4) ) F ( 3^(-1) ) / ( 2^(-2) ) C ( 3^(0) ) / ( 2^(0) ) G ( 3^(1) ) / ( 2^(1) ) D ( 3^(2) ) / ( 2^(3) ) A ( 3^(3) ) / ( 2^(4) ) E ( 3^(4) ) / ( 2^(6) ) B ( 3^(5) ) / ( 2^(7) ) F# ( 3^(6) ) / ( 2^(9) ) C# ( 3^(7) ) / ( 2^(11) ) .. We can generalize by saying P={3^x / 2^y | a=log2(3), x E Z, y = [ax]}. (x is an integer and [] means integer part.) This formula is cute, but we can generalize it to include meantone and equal temperaments. A meantone tuning is based upon a spiral of fifths, but each of these fifth is flattened by a certain quantity, to make a certain interval (and its multiples) just. For example, the intervals of 1/4-comma meantone are defined by M={ (3^x / 2^y) / (81/80)^(x/4) | a=log2(3), x E Z, y=[ax]}. The expression can be simplifed to M={ 5^x / 2^(y-x) | a=log2(3), x E Z, y=[ax]}. The first expression focuses on the comma fraction and the second one on the just interval (5/4). The equal temperaments also come from the spiral of fifths. For example, 12TET is T={ (3^x / 2^y) / (3^12 / 2^19)^(x/12) | a=log2(3), x E Z, y = [ax]} and 19TET is T={ (3^x / 2^y) / (3^19 / 2^30)^(x/19) | a=log2(3), x E Z, y=[ax]}. You can see that with x=19, the interval is 1/1. This is where the circle of fifths closes. By studying this procedure, we discover that each ET is generated by a special enharmonic relation. 5TET B=C 7TET C#=C 12TET B#=C 19TET BX=C As the number of degrees increases, the comma gets smaller and the fifths get purer. We use 31TET, 53TET and 72TET to approximate just intervals, like 5/4 and 7/4. But what are we _really_ doing? Stacking up an infinity of 3/2's! In a certain sense, using a ET scale means playing in an extended 3-limit tuning. In short, the formulas for 3-limit ratio, 1/n-comma meantone and nTET temperaments are P={3^x / 2^y | m E Z, a=log2(3), n = [ax]} M={(3^x / 2^y) / (81/80)^(x/n)| a=log2(3), (n,x) E Z, y = [ax]} T={(3^x / 2^y) / (3^n / 2^l)^(x/19)| a=log2(3), (n,x) E Z, l=[ak], y=[ax]}. The "law" could be extended to non-octave ET's if we stretch the ratios (2^1/13)^log2(3) = 3^1/13, but I do not think this is meaningful for analysis. A triave fifth would be (3^2)^log2(3) = 112 cents. Any suggestions? -Kami Rousseau, AKA the Unatuner. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 22 Nov 1996 02:47 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA03146; Fri, 22 Nov 1996 02:47:12 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA03190 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id RAA09002; Thu, 21 Nov 1996 17:47:10 -0800 Date: Thu, 21 Nov 1996 17:47:10 -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