Re: In search of unison vectors -- equistep tunings?

Hello, there, Dan Stearns, Dave Keenan, Paul Erlich and everyone.

Often I'm hesitant to respond to a post addressing important conceptual
questions without taking some time for reflection, but maybe here a quick
and very tentative response might be helpful.

As I suspect that both Paul and Dave may especially appreciative, my
purpose in proposing the term "equitone" was to suggest the "regular"
features of a meantone tuning while avoiding the historical and musical
implications of "meantone" of major and minor thirds near 5:4 or 6:5.

The advantage of equitone is that it can be used in analogous manner for
regular tunings favoring other sizes of major and minor thirds, neo-Gothic
equitones providing an obvious example (most typically Pythagorean through
17-tone equal temperament or 17-tET, and sometimes heavier temperaments of
the fifth in the wide direction such as 22-tET with its almost pure 9:7
major thirds).

From this perspective, a meantone is an equitone where major and minor
thirds happen to be at or near 5:4 and 6:5, in contrast, say, to a
neo-Gothic equitone where they might be at or near 81:64 and 32:27
(Pythagorean) or 14:11 and 33:28 (around 46-tET).

In my own concept of an equitone, as you suggest, Paul, there is an
implication that the fifth is at or near 3:2, and that two such fifths
form a "tone" somewhere roughly between 8:7 and 10:9, say.

With meantones, the step size, as Paul has remarked, is generally not far
from the mean of 9:8 and 10:9 (the precise size in 1/4-comma meantone,
which some would say is _the_ "meantone" in the most proper sense). With
typical neo-Gothic equitones from Pythagorean to 17-tET, the size ranges
from 9:8 to somewhat larger; in 22-tET, the whole-tone of 4/22 octave is
just a bit larger than the mean of 9:8 and 8:7, making the regular ~9:7
major third (~436.36 cents, 8/22 octave) just a tad larger than pure.

What equitones as a superset share with the narrower set of meantones are
the following main features:

(1) The fifth is not too far from 3:2, say between 7-tET and 5-tET
(Blackwood's "perfect fifth" generator), with the implication
of a "recognizable diatonic scale";

(2) Four fifths up less two octaves form some kind of "major third,"
typically in the general region between ~5:4 and ~9:7; and

(3) Two fifths form a "whole-tone" between ~10:9 and ~8:7, actually
most often in the somewhat narrower range suggested by (2).

For regular tunings using other kinds of generators, maybe the term
"equistep" would suggest a superset of equitones. The idea might be that
we have a chain of generators, but not necessarily "traditional fifths or
fourths."

The term "equistep" might emphasize the idea that a usual kind of diatonic
structure implied by "tone" does not necessarily apply, but that a chain
of identical generators does.

This is just a first impression, and I'd invite a lot of further
discussion.

In discussing "meantone" and "equitone," I should also duly note one
important distinction between the terms. "Meantone," although apparently
coined sometime after the 16th century if I'm not mistaken (terms like
_participatio_ for "temperament" were common, but I'm not aware of
"meantone" or "mesotonic" so early), carries lots of historical
baggage. This is a very strong argument for not speaking of "neo-Gothic
meantones," for example, but rather of "neo-Gothic equitones."

In contrast, "equitone" is new, and if the term attracts a widespread
usage, the "accepted" meaning may be shaped by that.

My own inclination might be to suggest "equistep" for the yet larger set
of tunings we're discussing, but that is again a tentative first response,
maybe more timely than carefully considered.

Most appreciatively,

Margo Schulter
mschulter@value.net