Meantone Temperament Construction Of The Quarter comma Meantone

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.

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,

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.

This system gives whole tones in the ratio , diatonic semitones in the ratio , and perfect fifths in the ratio of , 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 3x12 − 5x7 = 36 − 35 = 1 semitone (see lemma). Then, in terms of ratios, 2³/x⁵ = 2³:(5^1/4)⁵ = 8 : 5^5/4.)

One of the fifths will be a wolf interval, which means it will be so sharp it will not sound at all the same as a perfect fifth, and will not normally be used in music of the common practice period. This is because twelve perfect fifths, each flattened by a quarter of a syntonic comma, do not add up to an exact number of octaves.

The term meantone temperament is sometimes used to refer specifically to 1/4-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.

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 numberIn mathematics, a rational number (or informally fraction is a ratio of two integers, usually written as the vulgar fraction a ''b where b is not zero. The set of all rational numbers is denoted by Q or in blackboard bold. Using the set-builder notation i, so is (3R+1)/(5R+2), which is the size of fifth in terms of logarithmIn mathematics, the logarithm functions are the inverses of the exponential functions. Logarithms are numbers that are substituted in computation for other numbers, to which they bear such a relation that the operations to be performed on the latter are rs 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	Equal temperament	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

Because of the wolf interval 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 temperamentWell temperaments are a form of musical tuning, also known as circular temperaments or irregular temperaments . Any tuning system which is not equal temperament but which tunes the chromatic scale in a way suitable for a keyboard instrument and in such as and eventually equal temperament became more popular.

1 Construction of the quarter-comma meantone diatonic scale

As discussed above, in the 1/4-comma meantone temperament, the ratio of a tone is , the ratio of a semitone is , and the ratio of a fifth is . 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:

A diatonic scaleIn Music theory, the diatonic major scale is a fundamental building block of the Western musical tradition. It contains seven notes to the octave, corresponding to the white keys on a piano, obtained from a chain of six successive fifths in some version o 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
C	1	1	0
D	T	1.118033	193.156856
E	T²	1.25	386.313713
F	T² S	1.337480	503.421571
G	P	1.495348	696.578428
A	P T	1.671850	889.735285
B	P T²	1.869185	1082.892142
C'	P T² S	2	1200

2 Construction of the quarter-comma meantone chromatic scale

Construction of a 1/4-comma meantone chromatic scaleThe chromatic scale is a musical scale that contains all twelve pitches of the Western tempered scale. All of the other scales in traditional Western music are currently subsets of this scale. Each pitch is separated from its upper and lower neighbors by 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

Thus, each tone can be divided into a pair of unequal semitones, the major one being S, and the minor one being . Notice that S is equal to 117.107 cents, and that 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 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 1523Events April Battle of Villalors Forces loyal to Emperor Charles V defeat the Comuneros, a league of urban bourgeois rebelling against Charles in Spain. June 6 Gustav Vasa becomes King of Sweden, establishing finally its full independence from Denmark.:

C C# D D# E F F# G G# A A# B C' |----|----|----|----|----|----|----|----|----|----|----|----| _ _ _ _ _ S S S S S S S S S S S S nov dec jan feb mar apr may jun jul aug sep oct

The scale is a series of 12 semitones, each of which may either by major — S — or minor — . The diagram compares the semitones to months of a year: months with 31 days correspond to major semitones, and months with 30 days or less correspond to minor semitones.

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 to move up by a minor semitone.

Note	Formula	Ratio	Cents
C	1	1	0
C#		1.04490672653	76.0489992634
D	T	1.11803398875	193.156856932
D#	T S	1.19627902498	310.264714601
E	T²	1.25	386.313713865
F	T² S	1.33748060995	503.421571534
F#	T³	1.39754248594	579.470570797
G	P	1.49534878122	696.578428466
G#		1.5625	772.62742773
A	P T	1.67185076244	889.735285399
A#	P T S	1.788854382	1006.84314307
B	P T²	1.86918597653	1082.89214233
C'	P T² S	2	1200

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

3 Triads in the quarter-meantone 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	P5	m3
C—E	C—G	C—D#
C#—F	C#—G#	C#—E
D—F#	D—A	D—F
D#—G	D#—A#	D#—F#
E—G#	E—B	E—G
F—A	F—C	F—G#
F#—A#	F#—C#	F#—A
G—B	G—D	G—A#
G#—C	G#—D#	G#—B
A—C#	A—E	A—C
A#—D	A#—F	A#—C#
B—D#	B—F#	B—D

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 G# to D# 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 C#, F#, G# and B have a ratio of

which deviates by +41.059 cents from the just M3. If thirty five cents is not acceptable, then neither is forty one cents.

Major triads are formed out of both major thirds and fifths. If either of the two intervals go out of whack 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

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 D#, F and A# 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 D#, F, G# and A# are not used in the meantone scale defined above.

The following major triads are usable: C, D, D#, E, F, G, A, A#.
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: D#, F, A#.
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#.

4 See also

Pythagorean tuning | just intonation | well temperament | equal temperament

5 External links

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

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