Re: Reply to William S. Annis

Hello, there, and greetings to William S. Annis, who writes the
following:

> Under the influence of your web page on 13th century
> polyphony, I have started to catalog the harmonic resources of any
> new tuning I use into four categories: consonant, moderately
> consonant, dissonant and moderately dissonant.

Here I might comment that while 13th-century Western European music is
one fine example of these grades of concord/discord, there are various
styles where it can nicely apply. Ludmila Ulehla, _Contemporary
Harmony_, has a very interesting discussion on "The Control of
Dissonance" where the section on 20th-century music might be of
special interest. Of course, as you note, you might have a few things
to add about how different interval ratios might rate in your own
scheme.

It's a pleasure to see history serving as a catalyst for people to
explore various possible approaches to concord/discord.

> I am quite new to JI and I'm surprised to find how I'm categorizing
> intervals.. For example, in dissonant and moderately dissonant. I
> am quite new to JI and I'm surprised to find how I'm categorizing
> intervals. For example, in many timbres I consider the septimal or
> subminor third (7/6) *more* consonant than the minor third (6/5).
> Even more strangely, the ditone (81/64) doesn't bug me too much.

As to 7:6, or its close approximations in systems such as extended
Pythagorean or meantone, opinions can vary, but I've heard your view.
In 1555, Vicentino says that the ~7:6 on his archicembalo (a diesis
narrower than a usual minor third) seems to "resemble a second," and
therefore lean toward dissonance, in comparison with his ~11:9, which
he considers an acceptable concord. However, I find that at least in
some timbres, the 7:6 sounds considerably more concordant.

Interestingly, even for conventional composed music of the 18th
century which would generally be considered 5-limit, tuning theorists
such as Owen Jorgensen suggest that 7:6 is an acceptable concord, so
that he would accept a very narrow minor third, although he would
consider an equally wide major sixth approaching 12:7, say, as a
"Wolf" (not a "playable" substitute for a normal major sixth) in the
same music.

As to the 81:64, I would say that it is a regular major third in a
medieval setting, a _relatively_ concordant interval. It can "bug"
people if they expect something close to a 5:4, as would be true in
16th-century music, assuming that the fifth partial is prominent in
the timbre of the lower note. I've found that in music where the 81:64
is the norm, a 5:4 (or the Pythagorean diminished fourth, 8192:6561,
very close) can actually sound "strange," maybe a bit like a 7:4 or
7:6 in music where it's not expected.

> I'm no longer too concerned about quantitative solutions to
> the question of consonance. I find W. Sethares's dissonance curves
> cover my own perception sufficiently.

Just to clarify my original meaning: I was thinking of ways to
evaluate the "fit" of a tuning such as an equal temperament
(e.g. 22-tet or 53-tet) to an ideal intonation system such as
Pythagorean. I was wondering if one might invite some mathematical
measure of "fit," for example, which would recognize that 53-tet has
excellent equivalents not only for the stable 3:2 and 4:3 of medieval
polyphony, for example, but for the mildly unstable 81:64.

I warmly agree that William Sethares has made a very great
contribution to this area, including the immediate qualification I'd
attach to the last paragraph: any "rating" of the numbers in the
abstract might be modified by the actual timbres involved in a
specific realization of the scale.

> I have to say, making those initial steps in JI, especially
> harmonicly, can be quite a large step. I managed to join the JI
> Network, buy a few books and join this list before putting down the
> first note outside 12tet. My own predisposition to contrapuntal
> textures has not made this process any easier, either.

Interestingly, much medieval music based on Pythagorean JI, and also
Renaissance music based on 5-limit JI (or meantone, a useful
approximation for keyboards), would be described as strongly
"contrapuntal" in the sense of having individualized and often
rhythmically quite independent voices. In fact, with forms such as
13th-14th century motets, many of the polyphonic songs of Guillaume de
Machaut (c. 1300-1377), and the technique of the _Ars subtilior_ at
the end of the 14th century, this horizontal diversity has sometimes
received so much (quite deserved) attention that the beauties of the
vertical dimension may be underemphasized.

You could also consider the polyphony of Ockeghem in the 15th century,
although here the treatment of concord/discord is a bit more
restrained (the Renaissance is probably the most "smooth" or
homogenous era of Western European composition when it comes to the
approach to concord/discord).

In the 20th century, both a vertical interest in new approaches to
concord/discord and a taste for melodic independence in counterpoint
have contributed to the kind of developments which Ludmila Ulehla
discusses.

Anyway, I would say that by taking a contrapuntal approach, you are
following a very nice historical tradition.

> I've been keeping a lot of notes of my explorations, in
> additions to pages of manuscript exploring the functional
> implications of various sorts of interval progressions. Perhaps
> I'll turn some of that into a web page and see if the processes I've
> employed are at all useful to other beginners.

This could be a very interesting resource, and I'd much encourage you
to experiment with various intervals and tunings and reach your own
conclusions.

Most appreciatively,

Margo Schulter
mschulter@value.net