Re: Equitones and meantones -- definitional questions

Hello, there, Paul and Monz and everyone, and please let me suggest a
few fine points regarding the question of defining a meantone.

Actually, I might suggest a set of narrowing categories like this:

(1) REGULAR TUNING. A regular tuning is built from a chain of
identically sized fifths -- or sometimes from two or more such chains
(e.g. 24-tET, 51-tET). This is Dave Keenan's "chain-of-fifths-tunings"
category, with some implication that the generator is indeed a "fifth"
with a size somewhere between that of 7-tET and 5-tET (not to rule out
greater musical latitude, a la Jacky Ligon or otherwise -- just to take
note of a usual assumption).

(2) EQUITONE. An equitone is a regular tuning used in a given musical
context, or in established practice, so that the regular major third
is derived from four fifths up (e.g. F-A from F-C-G-D-A), and thus
from two equal whole-tones, or "equi-tones" (e.g. F-A from F-G-A).

(3) MEANTONE. A meantone is a negative equitone (diesis negative, or
12 pure fifths falling short of 7 pure octaves) where four fifths up
provide the best approximation of a pure 5:4. The second condition,
proposed as a criterion by Dave Keenan, sets a lower limit at around a
fifth size of 691.51 cents, while either condition sets an upper limit
at 700 cents (12n-tET).

----------------------------------------------------
1. All meantones are equitones -- but not vice versa
----------------------------------------------------

One very important implication of this scheme is that not all
equitones are meantones: an equitone may aim at some simple or complex
ratio for regular thirds other than the 5:4 and 6:5 of meantone.

In typical Gothic/neo-Gothic equitones (Pythagorean to 17-tET), for
example, major thirds have complex ratios ranging from 81:64 (~407.82
cents) to ~423.53 cents; this complexity fits the unstable role of
these intervals, contrasting with the pure or near-pure fifths and
fourths (the primary concords in this music).

In what might be termed "septimal" equitones or paultones around the
22-tET region, regular thirds have ratios at or near 9:7 and 7:6. In
addition to providing a stimulating variation for neo-Gothic music,
the equitonal thirds and minor sevenths of this region could ideally
fit a style treating 12:14:18:21 or 14:18:21:24 as a stable tetrad. A
chain of only 4 fifths or fourths would provide all the intervals of
such a sonority!

To sum up, a meantone is a special case of an equitone where regular
major and minor thirds are near 5:4 and 6:5, or as you have said,
Paul, a tuning which disperses the syntonic comma so as to achieve a
regular structure with reasonable 5-limit approximations.

---------------------------------------------------------------
2. All equitones are regular tunings, but not always vice-versa
---------------------------------------------------------------

While all equitones are by definition regular tunings, or
"chain-of-fifths" tunings, a given regular tuning may or may not be
used in an equitonal fashion in a given musical context or setting.
A few tunings may readily illustrate this point.

Consider, for example, 46-tET, a very characteristic neo-Gothic
equitone with major thirds at almost precisely 14:11 (~417.51 cents).
In a neo-Gothic setting, where these complex thirds are a major
attraction, this temperament provides a superb diatonic scale as a
kind of accentuated flavor of Pythagorean tuning. Usual note and
interval spellings fit theory and practice.

If 46-tET is used as a 7-limit tuning for a guitar or "srutar"
approximating the 22 srutis, for example, however, then it is no
longer an equitone: the complex thirds of neo-Gothic style no longer
define the "usual" intervals in use, and a more intricate approach to
scale formation and mapping intervals to steps is required.[1]

The same situation holds for 22-tET, a superb equitonal realization of
the diatonic scale in a neo-Gothic setting, or a posited setting where
tetrads of 12:14:18:21 serve as stable concords. Regular note and
interval spellings present no complications: this equitonal situation
seems synonymous to me with your "Pythagorean" 22-tET modes, Paul.

However, in a decatonic/tetradic style based on the stable 4:5:6:7, or
for that matter a Classic/Romantic 5-limit style based on 4:5:6, we
have again entered a nonequitonal world, since the equitonal major
third very accurately approximates 9:7 rather than 5:4.

To sum up, to describe a tuning as an "equitone" is to say both that
it has an intrinsic regular structure ("chain-of-fifths tuning"), and
that it is being _used_ in a "regular" fashion ("a major third equals
four fifths up"). The second assumption may be very decidedly
style-dependent.

Specifically, when various authors remarks that a tuning such as
46-tET or 22-tET has the disadvantage of unconventional spelling, they
are (often implicitly) assuming styles where major thirds should be
tuned at or near 5:4 rather than 14:11 or 9:7.

While such assumptions may often be obvious enough, articulating them
may make our definitions more responsive to the variety of practices.

------------------------------
3. Usage: a retrospective note
------------------------------

Having shared with others on this List in the process of seeking some
mutually agreeable definition of "meantone" -- or at least one not
needlessly disagreeable -- I have concluded that a focus on the
approximation of 5:4 and 6:5 thirds indeed fits both history and
typical current usage.

In contrast, an "equitone" is free to use four fifths up to attain or
approximate a major third of any desired size.

The terms "meantone" and "equitone" are in one way synonymous: both
point to the formation of a major third from two "mean" or "equal"
whole-tones, a feature of interest from a melodic as well as vertical
standpoint.

The usage and connotations of these terms, however, differs on one
critical point: "meantone" suggests both in its origins and in its
usual associations a specific concern with dispersing the syntonic
comma to achieve pure or near-pure 5:4 and 6:5 thirds.

The advantage of "equitone" is that it does not carry this kind of
baggage, but describes the "two equi-tones make a major third" feature
of a tuning.

Also, from an historical point of view, we may find it very convenient
to say, "Conventional Western European notation has evolved in a
setting of equitonal tunings, namely Pythagorean and meantone."[2]

We could also try out typologies like this:

----------------------------------------------------------------------
Type of regular tuning Equitonal use Nonequitonal use
----------------------------------------------------------------------
Negative (meantone) The norm Likely rare
(~5-based thirds) (~9:7 remapping?)
----------------------------------------------------------------------
Pythagorean region Gothic/neo-Gothic Skhismic tunings
(complex thirds) (5-limit or higher)
----------------------------------------------------------------------
Septimal region neo-Gothic decatonic, etc.
(22-tET/paultone) (7-flavor thirds) (7-limit or higher)
----------------------------------------------------------------------

Anyway, maybe the "equitone" concept can lead to further discussion.

-----
Notes
-----

1. Following the example of others here, I would prudently add that
one might wisely make a distinction between the traditional system of
22 srutis in the classical music of India, and various recent systems
and offshoots taking the 22-sruti tradition as an inspiration, but not
necessarily based on the same intonational and stylistic framework.

2. Here Ed Foote might not be disappointed if I add that historical
well-temperaments, although not equitonal, are cleverly designed so as
to be compatible with basic equitonal notations even while providing a
variety of transpositional colors.

Most appreciatively,

Margo Schulter
mschulter@value.net