source file: m1400.txt Date: Thu, 30 Apr 1998 10:10:20 -0500 (CDT) Subject: Re: Synthesizer Resolution From: Paul Hahn On Tue, 28 Apr 1998, Fred Kohler wrote: > Because of the digital circuitry involved, synthesizer manufacturers will be > inclined to select a number of steps per octave in which powers of 2 are a > majority. Is there some mathematical reason that such numbers as 768, 864, > 1024, 1152 and 1280 have poor consistency? [snip] I don't have any deep or rigorous explanation for it, but whenever one finds a ET number with high consistency for its size (like 12) its multiples tend to drop off rapidly in consistency level. > Is there a number that is mostly powers of 2 that would qualify as having > high consistency that would make the synth manufacturers happy? Here's how Fred's suggested numbers look: (limit) 3 5 7 9 --------------------- 768| . . . 864| . . 1024| 312 . . 1152| 4 3 2 . 1280| 2 . (1152 looks the best of those, although 1024 is a power of 2 and has great 3/2s--level 312 is a bit of overkill, though. 8-)> ) Of the multiple-of-12 numbers in my first list, their prime factorizations are: 612 = 17 * 3^2 * 2^2 624 = 13 * 3 * 2^4 684 = 19 * 3^2 * 2^2 Compare these to the numbers in Fred's list: 768 = 3 * 2^8 864 = 3^3 * 2^5 1024 = 2^10 1152 = 3^2 * 2^7 1280 = 5 * 2^8 1200, BTW, is 5^2 * 3 * 2^4. I don't have time just at the moment (maybe this weekend) to do a more thorough search, but others are welcome to look through my consist.txt table and suggest better compromises. --pH http://library.wustl.edu/~manynote O /\ "Churchill? Can he run a hundred balls?" -\-\-- o NOTE: dehyphenate node to remove spamblock. <*>