>. If however > we want a beatless 3rd from C by tuning a series of four 5ths, we must > diminish (temper) each of the FOUR 5ths to arrive at a > pure 3rd.. This diminished amount is 1/4 of the syntonic comma and thus the > name > Quarter Comma Meantone. The old theorists proposed temperaments from cycles > of fifths and since the syntonic comma was used to get the thirds, most of > meantones were expressed as some fraction of that comma. > Thank you, this is the best explanation I have ever heard. It also tells me > how, if instead of pure 3rds, you allow 3rds which are open, you get > differing fractions of the comma from 1/5, 1/6, 1/7, all the way to 1/11, the > equivilent oe ET where the last 5th ends up beling tempered 2 cents narrow > just like all the others. There would also be other possibilities, other > fractions of the comma. > > Bill Bremmer RPT > Madison, Wisconsin Beyond 1/4 comma to get pure thirds I have yet to fathom the thinking of 1/5 comma, 1/6, 2/7, 2/9 etc, and how they figured out from that how to make the thirds. It goes so deep as to get out of the 12 note scale as Lindley , New Groves, remarks, "According to Mersenne, Titelouze had a harpsichord with 19 notes per octaved tuned in equal microtones -- the equilivent, as it happens, of 1/3 comma mean-tone." How did they know this? "But mean-tones with major 3rds larger than pure also flourished. Cerone attributed it in 1613 to master organ builders. Lemme Rossi (1666) described 1/5 comma and mentioned 2/9 comma. In 1707 and 1711 the acoustician Joseph Souveur, using an elaborate monochord to compare 1/4, 1/5, 1/6 comma mean-tone, reported that organ and harpsichord makers adhered more closely to 1/5 comma evenhouugh 'les musiciens ordinaires' used 1/6 comma. I don't understand how commas were set up on the monochords, and how from that they could determine what flavor meantone was being used. I could put them in my spread sheet quick enough, but that doesn't tell me how the early theorists could realize temperaments from how much the syntonic comma was altered. (execpt for 1/4 tone) ---ric
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