Quarter-comma meantone

Quarter-comma meantone was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this tuning the perfect fifth is tempered by one quarter of a syntonic comma in order to obtain just major thirds (5:4). It was described by Pietro Aron (also spelled Aaron), in his "Toscanello de la Musica" of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

Construction

The just major third is divided in half to make two whole tones of equal size. Since two fifths up and an octave down make up a whole tone,

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

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

:$\{(3/2)^4\; over\; 4\}\; =\; \{81/16\; over\; 4\}\; =\; \{81\; over\; 64\}\; approx\; \{5\; over\; 4\}\; =\; \{5\; imes\; 16\; over\; 4\; imes\; 16\}\; =\; \{80\; over\; 64\},$

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.

:$4:1\; imes\; 5:4\; =\; 5:1$

:$left(\; \{3\; over\; 2\}\; ight)^4\; =\; \{81\; over\; 16\}\; approx\; \{80\; over\; 16\}\; =\; 5.$

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

Letting "x" be the ratio of the flattened fifth, it is desired that four fifth have a ratio of 5:1, :$x^4\; =\; 5$

therefore

:$x\; =\; sqrt\; [4]\; \{5\},$

so that

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

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, Cmusic|sharp' 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 3:2. It is this that gives the system its name of "quarter comma meantone" or "1/4-comma meantone".

:$5^\{1/4\}\; =\; 1.495348\; =\; 696.578428\; hbox\{cents\},$:$3/2\; =\; 1.5\; =\; 701.955001\; hbox\{cents\},$:$696.578428\; -\; 701.955001\; =\; -5.376572\; hbox\{cents\},$:$5.376572\; imes\; 4\; =\; 21.506290\; =\; 1200\; lg\; (81/80),$since:$4\; left(\; 1200\; lg\; \{3\; over\; 2\}\; -\; 1200\; lg\; 5^\{1/4\}\; ight)\; =\; 1200\; left(\; lg\; left(\{3over\; 2\}\; ight)^4\; -\; lg\; 5\; ight)$::$=\; 1200\; lg\; left(\; \{81/16\; over\; 5\}\; ight)\; =\; 1200\; lg\; \{81\; over\; 80\}.$

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/"x"⁵ = 2³:(5^1/4)⁵ = 8 : 5^5/4.)

Construction of the diatonic scale

As discussed above, in the quarter-comma meantone temperament, the ratio of a tone is $sqrt\{5\}:2$, the ratio of a semitone is $8:5^\{5/4\}$, and the ratio of a fifth is $5^\{1/4\}$. Let these ratios be represented by letters: "T" for the tone, "S" for the semitone and "P" for the fifth.

It can be verified through calculation that three tones and one semitone equals a fifth:

:$T^3\; cdot\; S\; =\; \{5^\{3/2\}\; over\; 2^3\}\; cdot\; \{8\; over\; 5^\{5/4\; =\; 5^\{6/4\; -\; 5/4\}\; =\; 5^\{1/4\}\; =\; P.$

A diatonic scale can be constructed by starting from the fundamental note and multiplying it either by "T" to move up by a tone or by "S" to move up by a semitone. The result is shown in the following table:

First, look at the column of fifths in the middle. All the fifths except one have a ratio of:which deviates by -5.377 cents from the just 3:2 = 701.955 cents. Five cents is small and acceptable. On the other hand, the fifth from Gmusic|sharp to Dmusic|sharp has a ratio of:which deviates by +35.683 cents from the just fifth. Thirty five cents is beyond the acceptable range.

Now look at the column of major thirds on the left. Eight of the twelve major thirds have a ratio of:which is exactly a just 5:4. On the other hand, the four major thirds with roots at Cmusic|sharp, Fmusic|sharp, Gmusic|sharp and B have a ratio of:which deviates by +41.059 cents from the just M3. Thirty five cents is unacceptable even for the more forgiving fifth, forty one cents sounds even less in tune.

Major triads are formed out of both major thirds and fifths. If either of the two intervals is a wolf interval in a triad, then the triad is not acceptable. Therefore major triads with root notes of Cmusic|sharp, Fmusic|sharp, Gmusic|sharp and B are not used in meantone scales whose fundamental note is C.

Now look at the column of minor thirds on the right. Nine of the twelve minor thirds have a ratio of:which deviates by -5.377 cents from the just 6:5 = 315.641 cents. Five cents is acceptable. On the other hand, the three minor thirds whose roots are Emusic|flat, F and Bmusic|flat have a ratio of :which deviates by −46.436 cents from the just minor third. These minor thirds will not sound good.

Minor triads are formed out of both minor thirds and fifths. If either of the two intervals go out of whack in a triad, then the triad will not sound good. Therefore minor triads with root notes of Emusic|flat, F, Gmusic|sharp and Bmusic|flat are not used in the meantone scale defined above.

The following major triads are usable: C, D, Emusic|flat, E, F, G, A, Bmusic|flat.
The following minor triads are usable: C, Cmusic|sharp, D, E, Fmusic|sharp, G, A, B.
The following root notes are useful for both major and minor triads: C, D, E, G and A. Notice that these five pitches form a major pentatonic scale.
The following root notes are useful only for major triads: Emusic|flat, F, Bmusic|flat.
The following root notes are useful only for minor triads: Cmusic|sharp, Fmusic|sharp, B.
The following root note is useful for neither major nor minor triad: Gmusic|sharp.

Chain of fifths

The fifth of quarter-comma meantone, expressed as a fraction of an octave, is log₂(5)/4. This number is irrational and in fact transcendental; hence a chain of meantone fifths, like a chain of pure 3/2 fifths, never closes. However, the continued fraction approximations to this irrational fraction number allow us to find equal divisions of the octave which do close; the denominators of these are 1, 2, 5, 7, 12, 19, 31, 174, 205, 789 ... From this we find that 31 quarter-comma meantone fifths come close to closing, and conversely 31 equal temperament represents a good approximation to quarter-comma meantone.