Yahoo post from 2001 by Graham Breed on a catalog of rank two temperaments
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More temperaments (was: white key scales (Dan, Miracle folk)) Posted By: [email protected] Fri May 25, 2001 10:25 am In-Reply-To: <[email protected]> Dave Keenan wrote: > --- In tuning@y..., graham@m... wrote: > > This must be similar to what Dave Keenan's generator-finding program > does. > > No. It works by brute force. It's an Excel spreadsheet. It simply does > the same calculation for every generator between 0 to 600c in 0.1c > increments. It calculates how many generators in a chain (up to some > maximum) give the best (octave equivalent) approximation for each of > 1:3, 1:5, 1:7, 1:11 and what the errors are. From those it calculates > the hexad-width and the MA and RMS errors over all 11-limit intervals > (consistency is guaranteed). Then it flags those that meet the set > criteria for hexad-width and errors. Oh, is that the thing I downloaded? I might have another look if I ever get Excel installed on this machine. I thought you mentioned it could be getting only local minima once, which would suggest a larger granularity and optimising from there. My program's working now. I rank temperaments by the product of the number of steps for a complete chord and the minimax error, then show the 10 best ones. Miracle actually comes out as third best for the 11-limit. I make it a 3.3 cent minimax error. The two better ones require more notes to get that all-important hexad. You can get to 1.3 cents accuracy with 49 notes or 2.4 cents with 31 notes. The simplest 15-limit temperament in the top 10 is defined like this, by octaves and generators: ( 1 0) ( 2 -1) (-1 8) (-3 14) (13 -23) (12 -20) That means a 3:1 is two octaves minus a generator, so the generator is a fourth. A 5:1 is 8 generators minus an octave. That looks like schismic to me. It's consistent with 41 and 94-equal. You need 37 notes from a chain of fourths to get a 15-limit chord, and it's accurate to 4.9 cents. The number one 15-limit temperament is consistent with 87 and 72, defined like this: ( 3 0) ( 6 -6) ( 8 -5) ( 8 2) (11 -3) (14 -14) It needs 16 generators for a complete chord, but because the octave divides into three I think that means 49 notes to the octave. So it's fairly complex, but accurate to 2.8 cents. I don't plan on tuning it up any time soon, but if anybody has a huge keyboard and wants to play with 15-limit harmony, well, now you know an accurate temperament exists. Graham