Fred writes:
<< But in no instance were there
instructions for grading the sizes of the M3s (in surviving "practical
instructions"). So that, from a practical historical perspective,
there would not be an even gradient of sizes of M3, but rather a
number of "equally good" and a few "equally bad" ones.<<
There are many instructions that, if followed, created an orderly,
(not necessarily perfect), increase in the size of the thirds. The Prinz,
Jean Jousse, Rousseau, several Kirnbergers, as well as Stanhope's improvement on
the Kirnberger, (Pg. 285 in "Tuning"). This version of Stanhope's temperament
has two pure M3, 6 M3's at 20 cents, and steps of 6 and 13 cents for the D
and A thirds, and 7 and 14 cents for the F-A and Bb-D respectively. This is one
of the rougher temperaments, but still follows the general order that almost
all others have written.
Even Handel's temperament (pg 167 in "Tuning"), graduates the M3's from 9
cents at the C-E to 19 cents at the C#-F, with each additional accidental in
the key signature corresponding to a larger degree of tempering in the tonic
M3. I don't know where one would decide the M3 changes from "good" to "bad,"
but there is an undeniable balance shared with the overwhelming majority of
Jorgensen's documented temperaments.
To say that, " from a practical historical perspective, there would not be
an even gradient of sizes of M3, but rather a number of 'equally good" and a
few "equally bad" ones" doesn't address our question here. A good example was
Rousseau, who according to Jorgensen, produced a temperament of 1779 which
progresses around the circle with every key's M3 from G all the way around to C
progressing in increasing then decreasing fashion. Perfection of progression is
not required to obey the virtually universal form of WT.
<<While the neat symmetry of Valotti and Young is lovely and
intellectually satisfying, they do not provide a good model of
historical practice, as they are exceptions appearing on the fringes. >>
I submit that they are the most highly refined versions of what the
rest of the theorists and tuners were attempting to do in historical practice.
(DeMorgan excepted).
Regards,
Ed Foote RPT
http://www.uk-piano.org/edfoote/index.html
www.uk-piano.org/edfoote/well_tempered_piano.html
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