JPropp>I originally misread this as "comparing 7*sqrt(8) vs. 8*sqrt(7)", Analogously, suppose you want 2^(1/12) and have 2^(1/3) and 2^(1/4), but subtract instead of dividing. (E.g., maybe you can only get the beat frequency.) "No problem." If a:=2^(1/3)-2^(1/4), then 1/12 11 10 9 8 7 2 = (- 126360 a + 76707 a - 28614 a + 1050384 a + 45198 a 6 5 4 3 2 - 277596 a - 3218520 a + 26497700 a - 15686172 a + 12188460 a + 18908336 a + 7114424)/8030072 "I ask for a half tone and I get a bleeping cadenza!" (Btw, a~~sqrt(2)/20.) On a related note, in the late 70s I found a suite (>= six) of identities like 5 a 2 1 1 | a, --- + -, a - -, a + - | 3 3 5 5 | 125 F ( | ---) = 4 3 5 a 1 5 a 1 5 a 3 | 128 --- - -, --- + -, --- + - | 3 3 4 4 4 4 | a 7 a 3 a pi (- - --)! (- - --)! 8 csc(--) 2 10 2 10 5 ------------------------------ . a 3 a 2 2 (- - -)! (- - -)! 2 5 2 5 My Merriam-Webster gives 125/128 as a definition of "enharmonic diesis", but refers to it as a difference! (That's OK, I have identities at 3/128 as well.) Are there any other named fractions besides "one half", "one quarter", and "enharmonic diesis"? Maybe "cent" and "percent"? --rwg cart horse - horsecart - orchestra sacher torte - orchestrate panettones - nonseptate - ten past one - pentatones supersonic - percussion higher test - High Street - eighth rest ciguateras - guitar case guitermanites - time signature ostentous - sostenuto necrotic - concerti pangolins - plainsong snoutiest - sostenuti canotier - anoretic - conarite - cortinae - creation - reaction - cantorie actinon - contain - cantion - cantino typhonian - antiphony Dantean - andante poetaster - operettas hyponyms - symphony rhetorics - chorister costumiers - music store normalcies - sermonical - minor scale impairers - primaries - primieras - impresari abrotine - beta iron - obtainer - baritone - taborine monometer - monotreme - metronome separator - sea parrot - opera star paternoites - proteinates - street piano Aaronic - Nicarao - conaria - ocarina underarms - snare drum tragions - roasting - organist assorting - Gastornis - organists partimento - armipotent - portamenti band shell - handbells aerophores - horse opera (and of course the maraschino Chorasmian harmonicas) .ucf.edu andrews@math.psu.edu elh@osots.com jford343@sbcglobal.net rudy@rudyrucker.com BuckonBass@aol.com adr@flex.com wrn@verizon.net ef@cmu.edu