Meantone temperament

Meantone temperament is a system of musical tuning. In it, the major third is tuned to a particular ratio (), and then divided in half to make two whole tones of equal size. Since two fifths up and an octave down make up a whole tone,

Related Topics:
Musical tuning - Major third - Whole tone

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: {(3/2)^2 over 2} = {9/4 over 2} = {9 over 8},

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four fifths up and two octaves down make a major third in meantone temperament,

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: {(3/2)^4 over 4} = {81/16 over 4} = {81 over 64} pprox {5 over 4} = {5 imes 16 over 4 imes 16} = {80 over 64},

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and hence four fifths in meantone temperament make an interval of a seventeenth (5+(5−1)+(5−1)+(5−1) = 20−3 = 17), which is two octaves (4:1) above the major third (5:4), and so has a ratio at or about 5:1, i.e.

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: 4:1 imes 5:4 = 5:1

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: left( {3 over 2} ight)^4 = {81 over 16} pprox {80 over 16} = 5.

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Meantone tuning involves flattening the fifth so as to bring the seventeenth more nearly, or exactly, equal to this ratio.

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Letting x be the ratio of the flattened fifth, it is desired that four fifth have a ratio of 5:1,

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: x^4 = 5

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therefore

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: x = sqrt{5},

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so that

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: {x^2 over 2} = {sqrt{5} over 2} = hbox{whole-tone}.,

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The most common form of meantone temperament tunes all the major thirds to the just ratio of 5:4 (so, for instance, if A is tuned to 440 Hz, C#' is tuned to 550 Hz). This is achieved by tuning the perfect fifth a quarter of a syntonic comma flatter than the just ratio of . It is this that gives the system its name of quarter comma meantone or 1/4-comma meantone.

Related Topics:
Just - Hz - Syntonic comma

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: 5^{1/4} = 1.495348 = 696.578428 hbox{cents},

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: 3/2 = 1.5 = 701.955001 hbox{cents},

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: 696.578428 - 701.955001 = -5.376572 hbox{cents},

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: 5.376572 imes 4 = 21.506290 = 1200 lg (81/80),

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since

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: 4 left( 1200 lg {3 over 2} - 1200 lg 5^{1/4} ight) = 1200 left( lg left({3over 2} ight)^4 - lg 5 ight)

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:: = 1200 lg left( {81/16 over 5} ight) = 1200 lg {81 over 80}.

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This system gives whole tones in the ratio sqrt{5}:2, diatonic semitones in the ratio 8:5^{5 over 4}, and perfect fifths in the ratio of 5^{1 over 4}:1, which is 1.495349.., compared with a justly tuned fifth of 3:2, which is 1.5. (A semitone is equal to three octaves up and five fifths down, since the octave equals 12 semitones and the fifth equals 7 semitones, so that 3×12 − 5×7 = 36 − 35 = 1 semitone (see limma). Then, in terms of ratios, 23/x5 = 23:(51/4)5 = 8 : 55/4.)

Related Topics:
Semitone - Limma

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: {2^3 over x^5} = 2^3 : (5^{1/4})^5 = 8 : 5^{5/4} .)

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