More about commas

[Robert A. Anderson]

Just to clean up some inadvertent errors, Bill Bremmer wrote:

"The 
Pythagorean Comma is the amount in excess of an octave that you get when
you 
tune 12 pure 5ths, 24 cents."

Make that "in excess of 7 octaves."

Ed Foote wrote:

" A syntonic comma is the difference between four just fifths and two
Just 
octaves. See Jorgensen, "Tuning", pg 777"

Make that "two just octaves plus one just major third." Now that makes
the ratio 81:80.
OR 
"Ditone: a Pythagorean major third created by the excess of four just
fifths over two just octaves."(TUNING, p.770).
 There's also more than one way to find the syntonic comma. Jorgensen's
definition:
"It is the difference between a ditone (Pythagorean major third) and a
just major third. Its ration is 81:80, and it is 21.50629 cents in
size."(TUNING, p. 777).

Ed also wrote:

"The difference between three Just 
contiguous thirds and an octave is a diesis, and it is approx. 41
cents."

That is accurate, but it might be helpful to some to add that not every
diesis is the same. Jorgensen's definition is broader:

"the difference between a sharp and a flat tuned on the same
key."(TUNING, p. 770.)

In Ed's example, if you started on C4 you would get to C5 by a direct
octave, ratio 2:1. If you used major 3rds you would get to B#4 (C-E,
E-G#, G#-B#), ratio 5:4 x 5:4 x 5:4 = 125:64. The difference between 2:1
and 125:64 is the diesis, in this case about 41 cents. However, as
Jorgensen points out(TUNING, p.770): "The diesis in Pythagorean tuning
is the ditonic comma which is 23.46 cents in size." The reason is that
the Pythagorean scale is constructed using only just 5ths(or 4ths).
There are no just 3rds. Anyway, if you span 7 octaves by way of 12 just
5ths, you get to B#, but a different B# from Ed's example. 

Bob Anderson
Tucson, AZ