Definitions, pure fifths tuning

Ed Remler wrote...
> 3. Mean Temperament has two general meanings. First it means a
system of
> tuning by fifths such that four ascending fifths and two descending
> octaves make up a perfect third. It also means any one of many
slight
> variations about this basic system.
> 4. Bach's WTC refers to a mean tempered clavichord; in particular,
> well-tempered means mean tempered. Since the WTC starts in the key
of C,
> it seems that Bach's thought of mean tempered in terms of ascending
> fifths started on C.

Regarding the scheme (if I understand it right) of up four fifths and
down two ocataves will give a pure third.

Using the notation that middle C c' and tuning from the octave
below, or c Tuning this on a piano gives a c - e third that beats
sharp by more than 5 beats per second, more like 7 or 8. After
trying two or three times and erring on the flat side for fifth going
up and sharp octaves comming down, the c - e third was still sounded
faster than the rate prescribed for ET or 5.1-- bps.

To check against paper, using the ratio for the 5th or 3/2, or 1.5,
and using A220, (a) for conveince, four fifths gives a frequency of
220*1.5^4 (as entered on the W95 calculator, OK so ^ is the x^y key)
or 1113.75. So two octaves down or /4 gives 278.4375 for c# ' .
Hmm already that is sharper than 277.183 called for by ET.

Where one begns the temperament is all important and I believe as Ed
suggests that it should start from C, especially for the reason he
gives of WTC starting at C.
Continuing the temperament started above, the b fifth to e is tuned.
Here is is intersting to note that in the octaves e - e', d - d',
and c - c', all of the fourths and fifths sound pure, execpt f which
has not been tuned, so tuning f from c' and Bb from f gives the
diatonic scales in both C and F with pure fourths and fifths. If one
keeps close to the circle of fifths as Ed mentions, it would be
logical to tune the "flat keys" next, so from Bb tune up two octaves
to bb' (I knew this notation system would be rough in ASCII), down a
fifth (eb') down a fifth (ab) down a fifth (db). octave up from db
(db' ) fifth down and here we are at gb and f # and the wolf is from
f # to B (fifth) and f # to b (fourth) The rests of the fourths and
fifths are as pure as the skill of the tuner. It is interesting to
hear the thirds some pure and some well.. nasty.

This also can be quickly tuned from (for the sake of our piano tuner
members)
C3-G3-D3-A3-E3-B3-(F#4), Then C3-F3-Bb2-Eb3-Ab3-Db4-Gb4(/F#4) and the
octaves Gb4-G3, Eb4-E3, Db4-D3. and the E and B octaves. This I
believe might be the temperament that the variations are based. To
get rid of the wolf, it would have to be spread out, which means
temper the fifths, or put in another place by tuning further into the
circle of fifths, or starting the temperament from another key.

Richard Moody

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>And yet, a later generation, that of Mozart
>and Haydn, would be again satisfied with the key variety offered by
>meantone.

They never actually used meantone though, right?

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