Nice Chord

[Richard Moody]

>
> Greetings,
>     Most of the temperaments documented by Barbour,(and later by
Jorgensen)
> have at least one third that is tempered by the full syntonic comma (21.5
> cents).  This is quite a bit different than the 13.7 cents found in an ET
> third.
> Ed Foote RPT


I believe you will find that these temps are constructed to start out in
Meantone with greatly narrowed 5ths, (4 cents) and then go to pure 5ths to
"off set"
so  when you get to G#--Eb you no longer have a wolf.   Or others, (I think
the Young Valotti)  start with a series of pure 5ths and end up with narrow
5ths to close the circle.  Of course these temps were devised with one
object in mind, the elimination of the wolf.   If they kept one or two pure
3rds, then yes some other 3rds will be greater.   If they were designed for
ease of tuning as in series of pure 5ths then the thirds fall where they may
depending on how the tuner tunes up and down from the starting note.
    I am searching for the one temp that goes on the theory that if a
certain number of 1/4 meantone 5ths are tuned, and then the rest of the 5ths
are tuned pure, the circle will "come out" ie no wolf at Eb--G#.   If this
can be done with at least 4 MT 5ths then one pure third will exist.  Lets
see.
     This would give 386.3 cents for the pure 3rd leaving  1200 - 386.3  or
813.7 cents left for two 3rds to make up. If these were equal then 813.7 / 2
= 406.85, which is    20.55 cents sharp (from pure) for each. The Syntonic
comma (21.5) is a 3rd produced by tuning four pure 5ths of the circle.  So
if 4 Meantone 5ths make a pure 3rd or 386.3 cents and 4 Just (pure) 5ths
make a "pythagorean 3rd" or 407.8 (sharp from pure by 21.5) then 405.9 is
needed to complete the octave of 1200 cents.  A second series would put it 2
cents over the octave. So theoretically  you could tune the first four 5ths
of 1/4 comma meantone and the rest of the 5ths pure and end up with the
circle being 1.9 cents sharp of the octave, (a schisma? the same amount an
ET 5th is flat) or leaving the last 5th  2 cents narrow which would more
than satisfy the objective of  no wolf 5th in the tuning scheme.   Hmmm I
gotta give this a try.  "..up from C to G#....down from C to Eb"   HEY this
should satisfy everybody.   a pure 3rd, lots of pure 5ths, one ET 5th, and
at least one ET 3rd, but don't ask me where.

    Of course the people figuring out temperaments way back when, did not
use cents since cents did not come along until abt 1860.   With the aid of a
mono-chord a scheme like the above might have been figured,  or much simpler
with a tuning hammer and a harpsichord.  It doesn't take much rocket science
to tune 4 meantone 5ths and then six pure 5ths, especially when the pattern
of C up G# by 5ths and C down to Eb by 5ths was centuries old.  ---ric