More about commas

[Richard Moody]

Once upon a time twas written about Pythagorean tuning.......
> There are no just 3rds.

OK I know it was a long time ago but I happened to catch this after I
learned that there ARE intervals that sound like pure thirds in Pythagorean
tunings.  In a Tuner's Supply publication (op?) John Link claims that in
Pythagorean tuning, diminished fourths sound like pure major 3rds.   He says
they are off by a schisma, for example the difference a 5th is flat in ET.
(2 cents). .   Thus G#--C should sound like a pure 3rd.   And it is true,
when I experiemented with Pythagorean tunings I did hear a pure third here
and there but could not put a system to it.    I suppose one way to prove
this is to do the math.  As Robert mentioned, the Pythagoream 3rd is 81/64
as opposed to the pure 3rd of 80/64 or simply 5/4.   Thus 81/64 times 81/64
should leave close to 5/4 away from 2/1.   (81/64)^2 * 5/4 = 2.002258300781.
A series of pure fifths would  tune out resulting in C--E
(81/64);  E--G#(81/64) and leaving G#--c slightly smaller than 5/4.  If the
series were continued, (G#--D#--A#--E#);   D#--G, A#--C, and E#--A should
all sound as pure 3rds.   Crazy eh?  Of course the "wolf" fifth caused by
the pythagorean comma would then be between "F" (actually E#)  and
.  ---ric

----- Original Message -----
From: Robert A. Anderson <fndango@azstarnet.com>
To: <pianotech@ptg.org>
Sent: Tuesday, April 11, 2000 11:57 AM
Subject: More about commas




>, 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  41 cents. ... The reason is that
> the Pythagorean scale is constructed using only just 5ths(or 4ths).
> There are no just 3rds.

> Bob Anderson
> Tucson, AZ