The two commas

[Billbrpt@AOL.COM]

In a message dated 4/17/00 8:20:00 AM Central Daylight Time, 
remoody@midstatesd.net (Richard Moody) writes:

<<  If we want a pure 3rd from C we simply tune E beatless to C.  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.   The practice of
 tuning them was something else.  It is rather impossible to tune four narrow
 fifths and come to a good pure 3rd.  So you cheat and tune the pure third
 first. >>

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