we mean well......

Bob asks:
<>>Meantone now, but I keep seeing references to Well. It's 12-tone, right?

Yes, Well temperament generally refers to the 12 note division of the
octave.
Meantone tuning, ( which got its name long after it was no longer used, I
think), display little key color, as most of the usable keys are tuned alike,
with the last 1/3 of the combinations absorbing all of the comma. Keys are
really harmonious or really bad.
Well temperament, on the other hand, spread the comma out far enough to
make even the worst keys acceptable to the 17th and 18th century ears. The
elegance of a well temperament was judged by how even a tonal palette
resulted, and the difference of tempering in the thirds provided what I have
labeled a "Harmonic Toolbox" for the composers of the era. They would select
a key for particular harmonic character to fit their composition.
There were quite a few meantone tunings that spread the comma beyond the
first four fifths: there was 1/5 comma, 1/6, etc., all of which gradually
tempered more and more of the intervals. If I had to make a hard edge
distinction, I would say that a well temperament provided a variety of
tempering in the root tonic thirds, in a progression demonstrated by
modulating by fourths or fifths, with no third tempered by more than a
syntonic comma. The more accidentals, the more dissonance in the thirds. The
Young,(1799) has the F#-C# third at 21.5 cents, but that is the limit. Some
of the earlier well temperaments had several thirds that were tempered this
much, but no more.
The well temperaments were, of course, hijacked by those that touted the
'doctrine of affections", or the "Character of the Keys", etc. These
philosophies had their origin in Plato and his buddies idea of "Ethos", and
through the ages, whatever was current was fair game for "proof".
I have written a long piece for all of this, which can be found at
http://www.airtime.co.uk/forte/history/edfoote.html
If you want to hear well temperament on a modern Steinway, email me off
list for an address, and I can send you ordering info.

Regards,
Ed Foote
Precision Piano Works
Nashville, Tn.

>Okay, I'm starting to get the gist of what well temperaments are all about,
>but I don't really see how they are constructed.

I don't claim to be the kind of expert that, for example Ed Foote is,
but from what I understand, you specify a well-temperament in terms of how
many (pythagorean) commas you temper each fifth.

The premise of a 12 toned scale is that a stack of twelve fifths, which
would bring you from (for example) a Ab all the way around to a G#, should
land you on what is conceptually the same note. If you try that with exact
3:2 perfect fifths, you have Pythagorean tuning, wherein the resulting,
would-be G# overshoots the Ab by a ratio of 531441:524288, or about 23.5
cents.

That error is called a Pythagorean comma, and is strikingly close -
purely coincidentally - to the "syntonic" comma (21.5 cents). The
difference between the two is called a skhisma, which (curiously) is
strikingly close to the amount by which a 12TET P5 is flat of a 3:2.

So anyway, Pythagorean tuning, when limited to 12 tones per octave, has
a wolf fifth from G# to Ab, which is 1 Pythagorean comma too small. It
sounds dreadful. So, the question then comes down to "how do we distribute
that (pythagorean) comma's-worth of error more evenly, rather than lumping
it all into the one, last fifth?"

In short, everybody had his own favorite way to distribute that comma
around, for example, Werckmeister apparently used these temperings of the
fifths around the circle (and probably others):

Werckmeister III: Werckmeister IV: Werckmeister VI:
Eb-Bb 0 +1/3 0
Bb-F 0 -1/3 -1/7
F-C 0 0 0
C-G -1/4 -1/3 -1/7
G-D -1/4 0 -4/7
D-A -1/4 -1/3 +1/7
A-E 0 0 0
E-B 0 -1/3 0
B-F# -1/4 0 -1/7
F#-C# 0 -1/3 -2/7
C#-G# 0 0 0
G#-Eb 0 +1/3 +1/7

The fractions here are what fraction of a (pythagorean) comma that fifth is
flat of a 3:2 ratio. You will notice that they all add up to -1, which is
required for the circle to close at 12 fifths.

As far as I'm aware, there's no "science" as such behind these choices,
other than that the fifths must be off from a 3:2 by very much, and
preferably not differ in size from each other by very much. Beyond that
and the fact that they have to add up to -1, it was more a matter of what
gave each key a certain, desired sound.





>On another subject, some of the music I play contains "diminished 7th"
>chords (stacked minor thirds). Is this chord a modern invention?

Reasonably modern; it's not diatonic, and was fairly heavily explored
in the Romatic era.

But it can make sense in other than 12TET, provided that you don't
stipulated that all of the intervals be the same. Of course some of the
games that Romantic composers played with it more or less assume that all
of the intervals are the same, or at least close to the same. Tritone
substitutions were one of those games.

5:6:7 is one just-intonation possibility for tuning a diminished triad.
You can make the top minor third be 6:5 or 7:6 - whichever you'd prefer.
Each will sound a little bit different.