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
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