Quarter-comma meantone

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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.

1 Construction
- 1.1 Construction of the diatonic scale
- 1.2 Construction of the chromatic scale
2 Triads in the chromatic scale
3 Chain of fifths

[edit] 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,

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

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.

$4:1 \times 5:4 = 5:1$

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

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

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

$3/2 = 1.5 = 701.955001 \ \hbox{cents},$

$696.578428 - 701.955001 = -5.376572 \ \hbox{cents},$

$5.376572 \times 4 = 21.506290 = 1200 \lg (81/80),$

since

$4 \left( 1200 \lg {3 \over 2} - 1200 \lg 5^{1/4} \right) = 1200 \left( \lg \left({3\over 2}\right)^4 - \lg 5 \right)$

$= 1200 \lg \left( {81/16 \over 5} \right) = 1200 \lg {81 \over 80}.$

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/x⁵ = 2³:(5^1/4)⁵ = 8 : 5^5/4.)

[edit] 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:

Note	Formula	Ratio	Cents	EQT Cents
C	1	1.0000	0.0	0
D	T	1.1180	193.2	200
E	T²	1.2500	386.3	400
F	T² S	1.3375	503.4	500
G	P	1.4953	696.6	700
A	P T	1.6719	889.7	900
B	P T²	1.8692	1082.9	1100
C'	P T² S	2.0000	1200.0	1200

[edit] Construction of the chromatic scale

Construction of a 1/4-comma meantone chromatic scale can proceed by taking the 1/4-comma meantone diatonic scale as its foundation. The tones in the diatonic scale can be divided into pairs of semitones. However, S² is not equal to T. Instead, let

$\bar{S} = {T \over S} = {\sqrt{5} / 2 \over 8 / 5^{5/4}} = {5^{1/2} \cdot 5^{5/4} \over 8 \cdot 2} = {5^{7/4} \over 16}.$

Thus, each tone can be divided into a pair of unequal semitones, the major one being S, and the minor one being $\bar{S}$ . Notice that S is equal to 117.107 cents, and that $\bar{S}$ is equal to 76.048 cents. These can be compared to the just intonated ratio 18/17 which equals 98.954 cents. S deviates from it by +18.153 cents, and $\bar{S}$ deviates from it by -22.906 cents. These two deviations, in cents, may be considered acceptable: they are comparable to the syntonic comma.

The following quarter-meantone chromatic scale was constructed by Pietro Aaron in 1523:

C    C#   D    Eb   E    F    F#   G    G#   A    Bb   B    C'
|----|----|----|----|----|----|----|----|----|----|----|----|
  _              _         _         _              _
  S    S    S    S    S    S    S    S    S    S    S    S

The scale is a series of 12 semitones, each of which may be either major ( $S$ ) or minor ( $\bar{S}$ ).

The chromatic scale is also presented in the following table, which has been constructed by starting from the fundamental note and multiplying it either by S to move up by a major semitone or by $\bar{S}$ to move up by a minor semitone.

Note	Formula	Ratio	Cents	EQT Cents	Delta
C	1	1.0000	0.0	0	0.0
C#	$\bar{S}$	1.0449	76.0	100	-24.0
D	T	1.1180	193.2	200	-6.8
Eb	T S	1.1963	310.3	300	+10.3
E	T²	1.2500	386.3	400	-13.7
F	T² S	1.3375	503.4	500	+3.4
F#	T³	1.3975	579.5	600	-20.5
G	P	1.4953	696.6	700	-3.4
G#	$P \ \bar{S}$	1.5625	772.6	800	-27.4
A	P T	1.6719	889.7	900	-10.3
Bb	P T S	1.7889	1006.8	1000	+6.8
B	P T²	1.8692	1082.9	1100	-17.1
C'	P T² S	2.0000	1200.0	1200	0.0

Notice that this scale is an extension of the diatonic scale shown in the previous table. Only five notes have been added: C#, Eb, F#, G# and Bb. These five pitches form a pentatonic scale: the difference between a chromatic scale and a diatonic scale is a pentatonic scale.

[edit] Triads in the chromatic scale

The major triad can be defined by a pair of intervals from the root note: a major third and a fifth. The minor triad can likewise be defined by a pair of intervals: a minor third and a fifth.

A chromatic scale has twelve different major thirds, twelve minor thirds, and twelve fifths. In an equally tempered chromatic scale, all major thirds have the same size, all minor thirds have the same size, and all fifths have the same size. In the meantone temperament, intervals of the same type may have different sizes (e.g. not all major thirds are equal). Thus it is necessary to examine each of the possible intervals, to examine their sizes, and to see how much each of these intervals deviates from their justly intoned ideal ratios. If the deviation is too large, then the given interval is not usable, either by itself or in a chord.

The examination will be done in the following table, in which each row has three intervals of different types but which have the same root note. Each interval is specified by a pair of notes. To the right of each interval is listed the formula for the ratio of the interval. (Wolf intervals have been marked in red.)

M3	Ratio	P5	Ratio	m3	Ratio
C—E	$S^2\cdot \bar{S}^2$	C—G	$S^4 \cdot \bar{S}^3$	C—Eb	$S^2 \cdot \bar{S}$
C#—F	$S^3 \cdot \bar{S}$	C#—G#	$S^4 \cdot \bar{S}^3$	C#—E	$S^2 \cdot \bar{S}$
D—F#	$S^2\cdot \bar{S}^2$	D—A	$S^4 \cdot \bar{S}^3$	D—F	$S^2 \cdot \bar{S}$
Eb—G	$S^2\cdot \bar{S}^2$	Eb—Bb	$S^4 \cdot \bar{S}^3$	Eb—F#	$S \cdot \bar{S}^2$
E—G#	$S^2\cdot \bar{S}^2$	E—B	$S^4 \cdot \bar{S}^3$	E—G	$S^2 \cdot \bar{S}$
F—A	$S^2\cdot \bar{S}^2$	F—C	$S^4 \cdot \bar{S}^3$	F—G#	$S \cdot \bar{S}^2$
F#—Bb	$S^3 \cdot \bar{S}$	F#—C#	$S^4 \cdot \bar{S}^3$	F#—A	$S^2 \cdot \bar{S}$
G—B	$S^2\cdot \bar{S}^2$	G—D	$S^4 \cdot \bar{S}^3$	G—Bb	$S^2 \cdot \bar{S}$
G#—C	$S^3 \cdot \bar{S}$	G#—Eb	$S^5 \cdot \bar{S}^2$	G#—B	$S^2 \cdot \bar{S}$
A—C#	$S^2\cdot \bar{S}^2$	A—E	$S^4 \cdot \bar{S}^3$	A—C	$S^2 \cdot \bar{S}$
Bb—D	$S^2\cdot \bar{S}^2$	Bb—F	$S^4 \cdot \bar{S}^3$	Bb—C#	$S \cdot \bar{S}^2$
B—Eb	$S^3 \cdot \bar{S}$	B—F#	$S^4 \cdot \bar{S}^3$	B—D	$S^2 \cdot \bar{S}$

First, look at the column of fifths in the middle. All the fifths except one have a ratio of

$S^4 \cdot \bar{S}^3 = 1.495348 = 696.578 \ \hbox{cents}$

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 G# to D# has a ratio of

$S^5 \cdot \bar{S}^2 = 1.531237 = 737.637 \ \hbox{cents}$

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

$S^2 \cdot \bar{S}^2 = 1.25 = 386.313 \ \hbox{cents}$

which is exactly a just 5:4. On the other hand, the four major thirds with roots at C#, F#, G# and B have a ratio of

$S^3 \cdot \bar{S} = 1.28 = 427.372 \ \hbox{cents}$

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 C#, F#, G# 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

$S^2 \cdot \bar{S} = 1.196279 = 310.264 \ \hbox{cents}$

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 Eb, F and Bb have a ratio of

$S \cdot \bar{S}^2 = 1.168241 = 269.205 \ \hbox{cents}$

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 Eb, F, G# and Bb are not used in the meantone scale defined above.

The following major triads are usable: C, D, Eb, E, F, G, A, Bb.
The following minor triads are usable: C, C#, D, E, F#, 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: Eb, F, Bb.
The following root notes are useful only for minor triads: C#, F#, B.
The following root note is useful for neither major nor minor triad: G#.

[edit] 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.

v • d • e Tunings

Pythagorean · Just intonation · Harry Partch's 43-tone scale

Equal temperaments	12-tone · 19-tone · 22-tone · 24-tone · 31-tone · 34-tone · 41-tone · 53-tone · 72-tone · 88-tone

Non-equal temperaments	Meantone (quarter-comma, Lucy tuning, septimal) · Schismatic · Miracle

Irregular temperaments	Well temperament