The two commas

[Richard Moody]

>.  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