source file: m1362.txt
Date: Sun, 22 Mar 1998 20:37:24 -0600 (CST)

Subject: Re: Well temperaments, just diminished

From: mr88cet@texas.net (Gary Morrison)

>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.