Bell pitches / cycle of fifths

Bill Hibbert bill at ...
Tue Jun 14 20:17:41 BST 2005

Nigel Taylor:

> Bill Hibbert is wrong about Gb being flatter than F#; in
> 19 note mean-tone (that is literally 19 notes to an octave),
> Gb is 41 cents sharper than F#.

I almost don't want to take this one on, because I fear I'm going to 

As an example of what I had in mind, take A = 440 and do a cycle of 
fifths in both directions using 3/2 as the frequency ratio. Going up 
via A -> E -> B -> F# -> C# -> G# -> D# and folding back into the 
core octave gives D# = 313.2 Hz. Going down via A -> D -> G -> C -> 
F -> Bb -> Eb and folding up gives Eb = 309.0 Hz.

Thus in these circumstances (and I know no practical tuning would use 
perfect fifths like this) D# is sharper than Eb by 23 cents, the 
Pythagorean comma, and the order of pitches really is D, Eb, D#, E. I 
KNEW I had seen this worked example but had to reconstruct it to be 

All a bit of fun really, of no practical importance!


Bill H


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