Meantone Temperament - Meantone Temperaments

Meantone Temperaments

Though quarter-comma meantone is the most common type, other systems that flatten the fifth by some amount, but that still equate the major whole tone (9/8 in just intonation) with the minor whole tone (10/9 in just intonation), are also called meantone systems. Since (9/8) / (10/9) = (81/80)—the syntonic comma—the fundamental characteristics of meantone systems are that all intervals are generated from fifths, and the syntonic comma is tempered to a unison.

All meantone temperaments fall on the syntonic temperament's tuning continuum, and as such are "syntonic tunings." The distinguishing feature of each unique syntonic tuning is the width of its generator in cents, as shown in the central column of Figure 1. Historically notable meantone temperaments, discussed below, occupy a narrow portion of the syntonic temperament's tuning continuum, ranging from approximately 695 to 699 cents. The criteria which define the limits (if any) of the meantone range of tunings within the syntonic temperament's tuning continuum are not yet well-defined.

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. In Figure 1, the valid tuning ranges of 5-limit, 7-limit, and 11-limit syntonic tunings are shown, and can be seen to include many notable meantone tunings.

Meantone temperaments can be specified in various ways: by what fraction (logarithmically) of a syntonic comma the fifth is being flattened (as above), what equal temperament has the meantone fifth in question, the width of the tempered perfect fifth in cents, or the ratio of the whole tone to the diatonic semitone. This last 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 N/D, so is (3R+1)/(5R+2) or (3N+D)/(5N+2D), 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 notable 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 1/315 (nearly Pythagorean Tuning)
2/1 7/12 1/11 (1/12 Pythagorean comma)
9/5 32/55 1/6
7/4 25/43 1/5
5/3 18/31 1/4
8/5 29/50 2/7
3/2 11/19 1/3

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Other articles related to "meantone temperaments, meantone, temperaments, meantone temperament, temperament":

Meantone Temperaments - Equal Temperaments
... Neither the just fifth nor the quarter-comma meantone fifth is a rational fraction of the octave, but several tunings exist which approximate the fifth by ... Equal temperaments useful as meantone tunings include (in order of increasing generator width) 19-ET, 50-ET, 31-ET, 43-ET, and 55-ET ... The farther the tuning gets away from quarter-comma meantone, however, the less related the tuning is to harmonic timbres, which can be overcome by tempering the timbre to ...
Musical Tuning - Tuning Systems - Systems For The Twelve-note Chromatic Scale
... Meantone temperament A system of tuning which averages out pairs of ratios used for the same interval (such as 98 and 109) ... The best known form of this temperament is quarter-comma meantone, which tunes major thirds justly in the ratio of 54 and divides them into two ... or lesser degree than this and the tuning system will retain the essential qualities of meantone temperament historical examples include 1/3- and 2/7th ...

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