The two commas

[Richard Moody]

>Why are all of the Meantones named by
> the fraction of the Syntonic Comma that all of the 5ths are tempered by?
I> know this to be a fact and that is why tuning a chain of 5ths by that
> fraction works.  But the aural tuning
> instructions always concentrate on 3rds and only involve 4ths and 5ths
when> you run out of 3rds to tune.
> In other words, the Meantones are all about how the 3rds sound but the
name of the temperament is always the fraction of the Syntonic Comma that
the
5ths are tempered by.  Why is that so? And why was the Syntonic Comma chosen
as
a value rather than anything else?
> Bill Bremmer RPT

There are 2 commas, the Pythagorean and the syntonic.  Ignoring the span of
octaves,  the Pythagorean comma is that of twelve 5ths over the octave or
23.460 cents sharp.   The syntonic comma is four pure 5ths over the just 3rd
or 21.5 cents sharp.    (From C  go up four  5ths on the keyboard and you
come to E.) Both are based on cycles of 5ths.
   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.
    Now this series of narrow fifths usually starting from C goes up eight
5ths to G#.   Then down from C, three 5ths to Eb. The next 5th would be Ab
but it has already been tuned as G#. The interval Eb--G# then
is the "odd man out" and is called the "wolf"  because when played it
literally howls
because of the dissonance and beats.     Why they go up 8 and down 3 is
tradition, but I suppose there is a musical reason that musicologists would
be happy to explain---ric