Meantone temperament
Meantone temperament 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.
The term
meantone temperament is sometimes used to refer specifically to
quarter-comma meantone. However, 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 what fraction (logarithmically) of a syntonic comma the fifth is being flattened by, 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 comma |
|---|
| 2 | 7/12 | 1/11 |
| 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 |
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.
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 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♭ and D♯/E♭). Some period harpsichords and organs have split D♯/E♭ keys, such that both Emajor/C♯minor (4 sharps) and E♭major/Cminor (3 flats) can be played without wolf fifths.
Although Meantone is best known as a tuning environment associated with earlier music of the common practice era, 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.
*
Equal temperament*
Just intonation*
Interval*
Mathematics of musical scales*
Pythagorean tuning*
Semitone*
Well temperament*
Listen to music fragments played in different temperaments*
Tonalsoft Encyclopaedia of Tuning*
Kyle Gann's Introduction to Historical Tunings has an explanation of how the meantone temperament works.
*
List of some of the meantone systems*
Tuning system derived from π and the writings of John 'Longitude' Harrison
