Definitions

# Meantone temperament

Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a chain of perfect fifths, but in a meantone, each fifth is narrowed by the same amount (or equivalently, each fourth widened) in order to make the other intervals like the major third closer to their ideal just ratios.

Quarter-comma meantone is the best known type of meantone temperament, and the term meantone temperament is often used to refer to it specifically.

## Meantone temperaments

Though quarter-comma meantone is the most common, other systems which flatten the fifth by differing amounts but which still equate the major whole tone, which in just intonation is 9/8, with the minor whole tone, tuned justly to 10/9, are also called meantone systems. Since (9/8) / (10/9) = (81/80), the syntonic comma, the fundamental character of a meantone tuning is that all intervals are generated from fifths, and the syntonic comma is tempered to a unison. While the term meantone temperament refers primarily to the tempering of 5-limit musical intervals, optimum values for the 5-limit also work well for the 7-limit, defining septimal meantone temperament.

Meantones can be specified in various ways. We can, as above, specify by what fraction (logarithmically) of a syntonic comma the fifth is being flattened, what equal temperament has the meantone fifth in question, or what the ratio of the whole tone to the diatonic semitone is. This ratio was termed "R" by American composer, pianist and theoretician Easley Blackwood, but in effect has been in use for much longer than that. It is useful because it gives us an idea of the melodic qualities of the tuning, and because if R is a rational number, so is (3R+1)/(5R+2), which is the size of fifth in terms of logarithms base 2, and which immediately tells us what division of the octave we will have. If we multiply by 1200, we have the size of fifth in cents.

In these terms, some historically important meantone tunings are listed below. The relationship between the first two columns is exact, while that between them and the third is closely approximate.

Meantone tunings
R Size of the fifth in octaves Fraction of a (syntonic) comma
9/4 31/53 Zero (Pythagorian Tuning)
2 7/12 1/11 (12-tone Equal Temperament - 1/12 Pythagorian comma)
9/5 32/55 1/6
7/4 25/43 1/5
5/3 18/31 7/29
33/20 119/205 1/4
8/5 29/50 2/7
3/2 11/19 1/3

### Equal Temperaments

Some equal temperaments can be considered meantone temperaments. Of course, standard equal temperament has only one size of whole tone (although people may use the term meantone to exclude this).

19-ET is the smallest division of the octave (more than 12) that produces a useful meantone temperament. 31-ET is another tuning which can be considered as a meantone temperament that offers thirds close to Just and also fits more intervals with higher overtones. The equal temperaments of 26, 38, 43, and 50 divisions of the octave can also be considered meantone temperaments, but these tunings are rarely used.

7-tone equal temperament (to which some non-Western tunings are an approximation) may be interpreted as meantone as well: "letter" notes are equally spaced, sharps and flats are ignored (the diatonic semitone is the same size as the whole tone, while the chromatic semitone has zero size).

## Wolf intervals

A whole number of just perfect fifths will never add up to a whole number of octaves, because they are incommensurable (see Fundamental theorem of arithmetic). Therefore, a chromatic scale in Pythagorean tuning must have one fifth that is out of tune by the Pythagorean comma, called a wolf fifth. Most meantone temperaments share this problem, except for the case where the fifth is exactly 700 cents (tempered by approximately 1/11 of a syntonic comma) and the meantone becomes the familiar 12-tone equal temperament. This appears in the table above when R=2.

Because of this wolf fifth which arises when twelve notes to the octave are tuned to a meantone with fifths significantly flatter than the 1/11-comma of equal temperament, well temperaments and eventually equal temperament (a special case of the former) became more popular.

In fact, using standard music names, twelve fifths equal six octaves plus one augmented seventh; seven octaves are equal to eleven fifths plus one diminished sixth. Given this, three "minor thirds" are actually augmented seconds (for example, B to C), and four "major thirds" are actually diminished fourths (for example, B to E). Several triads (like B–E–F and B–C–F) contain both these intervals and have normal fifths.

Despite the logical consistency of this, these intervals may be better considered unusually narrow minor thirds and unusually wide major thirds. With many meantone temperaments, they are close to the septimal minor third and septimal major third respectively. Therefore they are more consonant than expected from "standard" music theory, which does not consider intervals using the seventh harmonic.

## Extended meantones

Meantone, in general, has an indefinite number of notes in each octave, that is, seven natural notes, seven sharp notes (F to B), seven flat notes (B to F), double sharp notes, double flat notes, and so on. (The exception is when the meantone temperament is also equal temperament. There are only so many different notes, for example, 19 or 31; further ones are just duplicates.)

Almost all problems are caused by a restriction to twelve notes per octave. For example, if we want a piano to play in C minor, we need three flat notes (B, E, A); if we want a piano to play in A major, we need three sharp notes (F, C, G). But this is already a problem since the keys for A and G should occupy the same place.

Another way to solve the problem of the wolf fifth is to forsake enharmonic equivalence (so, for example, G♯ and A♭ are actually different pitches) and use a temperament with more than 12 pitches to the octave. This is known as extended meantone. Its advantage is the ability to modulate into arbitrarily distant keys without wolf fifths, but an obvious disadvantage is the necessity of using instruments capable of playing more than twelve pitches in an octave, such as fretless string instruments, lutes with tied frets, or modified keyboard instruments with extra keys, like the archicembalo.

The existence of the "wolf fifth" is one of the reasons why, before the introduction of well temperament, instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between G♯/A♭[G sharp/A flat] and D♯/E♭[D sharp/E flat]). Some period harpsichords and organs have split D♯/E♭[D sharp/E flat] keys, such that both Emajor/C♯minor [E major/C sharp minor](4 sharps) and E♭major/Cminor[E flat major/C minor] (3 flats) can be played without wolf fifths.

## Use of Meantone temperament

The first Meantone tunings are described in late 16th century treatises by Francisco Salinas and Gioseffo Zarlino. Salinas (in De musica libra septum) describes three different mean tone temperaments: the 1/3 comma system, the 2/7 comma system, and the 1/4 comma system. He is the likely inventor of the 1/3 system, while he and Zarlino both wrote on the 2/7 system, apparently independently. Lodovico Fogliano mentions the 1/4 comma system, but offers no discussion of it. These formulations were often more theoretical than practical, as scientific methods to precisely determine the pitch of a string from its physical attributes were inadequate, and tuning had to be done by ear. For instance, to achieve the 1/4 comma system Salinas recommends tuning the 5ths as low as the ear will allow.

Although Meantone is best known as a tuning environment associated with earlier music of the Renaissance and Baroque, there is evidence of continuous usage of meantone as a keyboard temperament well into the middle of the 19th century. Meantone temperament has had considerable revival for early music performance in the late 20th century and in newly-composed works specifically demanding meantone by composers including György Ligeti and Douglas Leedy.

## New uses of Meantone tunings

Meantone tunings are particularly well-suited for use with an isomorphic keyboard, because such keyboards offer transpositional invariance and tuning invariance across the syntonic temperament's tuning continuum, which includes the entire range of extended meantone tunings.