source file: mills2.txt Date: Mon, 25 Nov 1996 19:11:29 -0800 Subject: Fractional Comma Manifesto, Part 2 From: Kami Rousseau As Jonathan Walker said, my equations were not simplified. (He also pointed that I forgot the frational part of an exponent.) The reason I did not simplify the expressions was that I wanted the isolate the 3^m/2^n factor. But the equation of a chain of meantone or ET fifths can be expressed in a simplified way. I find that the long expressions helps us understand where the simplifications come from. The mth meantone fifth is given by 2 ^ (4m/k - n) * 3 ^ (m - 4m/k) * 5 ^ (-m/k) where m is an integer, k is the "inverse of the comma fration" (for 1/4 comma meantone, k=4). n is the integer part of log2(3)m. The 4's are the exponents of 2 and 3 in the syntonic comma, 81/80. For example, 1/4 comma meantone is 2 ^ (4m/4 - n) * 3 ^ (m - 4m/4) * 5 ^ (m/4) or 2^(m-n) * 5^(m/4). Why is a power of 3 included in the general equation? Because the 3's do not always simplify. Take a look at 1/3 comma meantone : 2 ^ (4m/3 - n) * 3 ^ (m - 4m/3) * 5 ^ (m/3) When m= 3, we get 2^(4*3/3 -4) * 3^(3- 4*3/3) * 5^(3/3) = 5/3. ******* Now let's look at ET's. Let a=log2(3) and [x] mean integer part of x. The equation for the mth ET fifth (from my first post) simplifies to : 2 ^ ( [ak]m/k - [am] ) For the 12TET fifth, we have 2 ^ ( [a*12]*1/12 - [a*1] ) = 2^ ( 19/12 - 1 ) = 2^ 7/12. Surprised? The ET fifth is not only a random division of the octave, it is derived from a circle of fifths. For nTET, are there n fifths in the circle? Not always. When I began my little investigation, I believed that you could get all the notes from the circle of fifths, but sometimes this circle closes before you get all the scale degrees. Here is a list of fifths, and the ET's they are used in, for 0 id IAA27547; Tue, 26 Nov 1996 08:09:20 -0800 Date: Tue, 26 Nov 1996 08:09:20 -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