[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 (12`n`-tET).

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.

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.

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

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

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augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

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blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

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doamu / 2mu (MIDI-unit)

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dorian (mode)

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enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

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Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

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Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

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jot (unit of interval measurement)

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locrian (mode)

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magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

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mina (unit of interval measurement)

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Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

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paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

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sk (unit of interval measurement)

skhismic / schismic (temperament family)

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tina (unit of interval measurement)

tone (interval, and other definitions)

tredek (unit of interval measurement)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)