Fearless fifths

[Paul Erlich TD 532:3]
> Roughly speaking, in the West until ~1420, due to Pythagorean tuning,
sharps
> were higher than flats: e.g., G# was higher than Ab, D# higher than Eb,
etc.
> From ~1480 to ~1790, due to meantone tuning, the situation was reversed,
and
> Ab was higher than G#, Eb was higher than D#, etc. Many keyboards were
built
> which allowed these distinctions, particularly in the 16th-17th
centuries.
>As we know, though, the 18th-19th century shift to closed 12-tone systems
> was kind of messy, like all revolutions in tuning.

Thanks to Paul Erlich, again, for being the only person who took the time
here to really explain, rather than obfuscate, something. Paul you would
be a great teacher... sign me up for Xenharmonics 401 anyday... (on-line or
desk-line) Pay real money, too...

One other *fearless* dumb question... if fifths in Pythagorean are all
*pure*, why does this 'multiple naming" happen?? Is it a naming convention
and the "second time" the fifths go around in the spiral (due to the
Pythagorean comma) they are called something else??... am I getting the
drift or just drifting.

(Oui or wee-wee? -- just wanted to have Jerry laugh again...)

Joseph Pehrson

Joseph Pehrson wrote,

>One other *fearless* dumb question... if fifths in Pythagorean are all
>*pure*, why does this 'multiple naming" happen?? Is it a naming convention
>and the "second time" the fifths go around in the spiral (due to the
>Pythagorean comma) they are called something else??...

Precisely. The naming convention for both Pythagorean and meantone tunings
takes the chain of fifths as:

...Fbb-Cbb-Gbb-Dbb-Abb-Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D
#-A#-E#-B#-F##-C##-G##-D##-A##-E##-B##...

In Pythagorean tuning, moving twelve fifths to the right along this chain
leads to a pitch rise of a Pythagorean comma; in meantone tuning, moving
twelve fifths to the right along this chain leads to a pitch drop of a
diesis.