"Well" Temperaments from Meantone

[Richard Moody]

>That is to say, in a Well temperament, the
> C-E, G-B, D-F#, and F-A thirds are always favored over the other thirds,
> right? 
> John Formsma
> Blue Mountain, MS
> 

The "Wells" seem to have evolved from Meantone, combining "highly" tempered fifths
with pure fifths in a scheme to eliminate the wolf.  Recently I visited  the Shrine
to Music Museum in Vermillion South Dakota. http://www.usd.edu/smm/index.html 
This museum welcomes researchers. I was very grateful to be given a copy of an
article from the journal, Early Music; "Instructions for the Clavier Diversely
Tempered", by Mark Lindley.  (Jan. 1977). 
	Mark Lindley is the author of articles in the New Groves, including
"Temperament",  the best overall reference to the subject that I have
seen. The brief 1977  article in itself is very informative. It goes through the
"evolutions"
of temperament so to speak, complete with musical examples for the very earliest. 
	Lindley mentions Werckmeister's "preferred tuning" in which the first three fifths
are
tempered as in Meantone. C--G--D--A.  1/4 comma flat.  Continuing in the cycle of
fifths the next, A--E--B are tuned pure.  Returning to Middle C and going down
through the cycle, F--Bb--Eb--------etc all the way to to Gb are tuned pure. Now Gb
as F# is compared to B and "should beat nearly the same as C--G a semitone higher". 
 Thus the wolf is only heard as meantone fifth. In actuality only three
notes are tempered, as the B--F# "tempered" fifth results from the remaning fifth
that "closes the circle". 
	If E had been tuned flat 1/4 comma flat from its fifth, A, C--E would be pure.  But
E is tuned as a pure fifth, therefore C--E beats slightly. So tuning pure fifths in
a meantone scheme where 1/4 comma flat fifths should have been, gives wider thirds
than the pure thirds of traditional meantone. The first two thirds above C (by
fifths) beat slower than ET as do the first two below C. (C--E, G-B; F--A, Bb--D).  
With a spread sheet is it is easy to get beat calculations for all the intervals if
the tempering of the fifths is known. For those interested I would be glad to
furnish the set up formulas. 
	Quoting from Lindley, (tuning the Werckmiester "perferred") 
	"....make middle C beat three or four times per second with A (the ambiguity
	stems from Werckmeister's own mathematics)....
	...the 3rds of the most remote triads, such as Ab, Db and F# among the 
	major tirads, were much more heavily tempered; and E-G#-B and perhaps
	Eb-G-Bb sounded about as in equal temperament.
	Dozens of 18th century musicians praised this kind of tuning.  It gave 
	variety and nusance to 'transpositions', tha is, nodulations and sequences
	and the different keys gained a distinctivenerss of acoustic character.  
	One such tuning, praised by Tartini in 1754 for its qualities of *Chairoscuro*,
	was used at Padua by Vallotti from the 1720's until his death in 1780. 

Next LIndley explains the "Valotti" starting as a series of pure fifths from F to
Gb/F#.
Now "dividing in half the remaining series by placing D as a major 3rd between Bb
and F#" 
This series "among the diatonic notes." is "tempered equally among themselves."

>From the diagram he gives it appears two notes forming fifths from D to F must be
tempered, ie G and C.   Now from D to B there are two more notes that make fifths
between,  A and E, which must be tempered.  

This may seem ambiguious, but if you have tried to aurally tune Meantone by setting
C--E pure and then "juggling" G, D and A to  make the C--G--D--A--E  fifths "sound
more or less the same", (tempered equally among themselves") you should "get the
picture" of how Lindley is explaining the Valotti scheme.  It should be interesting
to try tuning this from only these instructions. The first question would be how
should the Bb--D--F# thirds sound?  Could it possibly be possibly be that these
would be pure??    hmmmm.  This I could figure out faster tuning than with a spread
sheet. Oh well, for another time... ---ric