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equitone tuning

[Margo Schulter, Yahoo tuning group message]

(Possible draft as starting point for dictionary item)

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.

(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.


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

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