Re: A just (and well-tempered) reconciliation?

Hello, there, and having followed a rather busy day on the Tuning
List, I'd like to offer a few comments in the hope of promoting
harmony and reconciliation between the precious people contributing to
this list, if not necessarily between their sometimes conflicting
views.

Maybe I should begin with two quick observations which may typify my
response to some of the recent colloquies.

First, I find it very easy to tune a pentatonic scale in JI without
any comma problems that I can identify, for example (placing a minor
third as the lowest interval of the octave, and arbitrarily measuring
intervals from this note):

D F G A C D
1:1 32:27 4:3 3:2 16:9 2:1
0 ~294 ~498 ~702 ~996 1200
|------|------|------|-----|-----|
32:27 9:8 9:8 32:27 9:8

To borrow the title of a novel by Gertrude Stein, "Q.E.D."

Secondly, granted that it's easy to generate a JI pentatonic scale
simply by making a chain of five pure 3:2 fifths, does this mean that
we can't/wouldn't/shouldn't do it any other way? Hardly, I would say.

Recently I found myself improvising a kind of two-voice counterpoint
with a pentatonic scale in 27-tone equal temperament (27-tet), with
lots of fifths and octaves, imaging that I was hearing the _gender_ in
a Javanese or Balinese gamelan playing in _slendro_ (something like
5-tet, maybe not so far from 27). The "vibraphone" timbre maybe fit
this effect -- I originally chose it for a Setharean smoothening
of this scale as a neo-Gothic hypermeantone.

The JI (3-limit) version is beautiful, as is the 27-tet version, as is
the actual _slendro_ (varying from gamelan to gamelan, with 5-tet only
one possible generalized approximation).

The last I checked, the universe is based not only on the simple
arithmetic of the harmonic series, but also on inharmonicity, and also
more generally on the "language" of differential equations, a language
we express using many irrational as well as rational qualities
(including, for example, Euler's e).

More generally, it has been asked whether there is a kind of natural
or possibly cultural "trend" toward JI. From experience, I would say
(and hope) that there may be a trend toward diversity _including_ not
only JI systems in the standard sense but what I'd call JR (standing
for Just Ratio) systems -- systems with _some_ just intervals, maybe
judiciously mixed with tempered ones.

Playing Gothic music in Pythagorean JI -- using only integer ratios --
is a great joy for me. Here there are generally no daunting comma
problems even using a simple 12-tone tuning, and not too much larger
tunings can overcome the problems that might occur in a few
experimental pieces which go beyond a 12-note range. Since the
syntonic comma isn't an issue for this kind of music, the familiar
arguments for temperament in 5-limit or higher music don't apply.

However, both as someone who plays and as someone who likes to
theorize, I have found that a familiar theme can invite variations.
Thus my exploration of what I call "neo-Gothic" tunings, especially
hypermeantones with fifths wider than pure.

In other words, playing in a very beautiful JI system has led me to
experiment with temperament. It's not necessarily of matter of "I'd
like to do this particular musical trick but can't," but rather, "What
might it sound like if I somewhat accentuate _this_ aspect of the
tuning by altering the size of this or that interval a bit...?"

To me, different just and tempered intervals might be compared to
primary and mixed colors. Why not a musical palette including both
kinds of colors -- not to mention that Pythagorean and other JI
systems themselves in effect achieve "virtual temperament" by
generating complex ratios which might be heard as variants of simpler
ones.

For example, a Pythagorean major third is 81:64, and in Gothic or
similar musics is at once unstable and relatively concordant. However,
might this be heard in effect, for example, as a 19:15 (about 1.42
cents larger?) -- and a major seventh at 243:128 likewise as a 19:10
(larger by the same amount)? The 81:64-19:15 resemblance occurred to
me when I first ran into a mention of the latter interval and its
size, while someone posted a while ago on the 243:128/19:10
resemblance.

Please let me express my special admiration and awe for those of you who
indeed practice what I call "polyhedral" JI, building intricate
multidimensional lattices from various prime factors, and using these
artful geometries to make beautiful music.

At the same time, I would like to suggest that one can value just
intervals -- simple or complex -- and use them _in combination_ with
tempered ones to develop often somewhat simpler and more "regular" or
symmetrical systems with many shades of concord and tension.

We can weave the intonational fabric around some just ratios while
filling out the design with a "patchwork" of tempered intervals. The
two categories need not be antagonistic -- they _can_ get along quite
nicely.

One technical aside: as a player of Renaissance music, I can confirm
that the syntonic comma problem with 5-limit JI isn't limited to
major/minor tonality or its 18th-20th century manifestations. It also
hits you where you live if you happen to be playing a 16th-century
piece in Dorian, Mixolydian, or any other mode which happens to call
for concordant fifths at both G-D and D-A. Yes, I've played a keyboard
with two versions of D which can solve this problem -- and sometimes
the comma shifts aren't at all unpleasing, but it's a demanding art
even for many quite simple pieces. Thus I don't disagree with Zarlino
when, after describing such an experimental keyboard, he found
temperaments for fixed-pitch instruments much more practical for this
music.

Temperaments need be neither equal nor regular, of course, and I would
tend to associate Bach with the well-temperaments described by theorists
such as Werckmeister (who also recognized 12-tet as a possibility). As
Paul Erlich has noted, the documented history of 12-tet goes back (in
Europe) to the lutes of the 16th century, and musicians recognized that
different timbres might call for different temperaments. Thus Vicenzo
Galilei championed 12-tet on the lute, and found it in theory the most
perfect tuning, but preferred 2/7-comma meantone on harpsichord despite
the resulting "defect" (in his view) that Ab could not be equivalent to
G#, etc., as on the lute.

In my view, the "problem" with 12-tet is not as a tuning but as a
universal standard. Any other kind of tuning placed in this
unfortunate role -- including polyhedral JI lattice systems -- would
be subject to different but similar objections.

People on the list, you are all precious to me. May we have
stimulating arguments, but in friendship and mutual appreciation.

In friendship,

Margo Schulter
mschulter@value.net