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
lose!

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
sure.

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

Cheers,

Bill H

```