Meantone temperament
From Academic Kids

Meantone temperament is a system of musical tuning. In it, the major third is tuned to a particular ratio (5:4), 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,
 <math> {(3/2)^2 \over 2} = {9/4 \over 2} = {9 \over 8}, <math>
four fifths up and two octaves down make a major third in meantone temperament,
 <math> {(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}, <math>
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.
 <math> 4:1 \times 5:4 = 5:1 <math>
 <math> \left( {3 \over 2} \right)^4 = {81 \over 16} \approx {80 \over 16} = 5. <math>
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,
 <math> x^4 = 5 \ <math>
therefore
 <math> x = \sqrt[4]{5}\, <math>
so that
 <math> {x^2 \over 2} = {\sqrt{5} \over 2} = \hbox{wholetone}.\, <math>
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 1/4comma meantone.
 <math> 5^{1/4} = 1.495348 = 696.578428 \ \hbox{cents}, <math>
 <math> 3/2 = 1.5 = 701.955001 \ \hbox{cents}, <math>
 <math> 696.578428  701.955001 = 5.376572 \ \hbox{cents}, <math>
 <math> 5.376572 \times 4 = 21.506290 = 1200 \lg (81/80), <math>
since
 <math> 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) <math>
 <math> = 1200 \lg \left( {81/16 \over 5} \right) = 1200 \lg {81 \over 80}. <math>
This system gives whole tones in the ratio <math>\sqrt{5}:2<math>, diatonic semitones in the ratio <math>8:5^{5 \over 4}<math>, and perfect fifths in the ratio of <math>5^{1 \over 4}:1<math>, 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, 2^{3}/x^{5} = 2^{3}:(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 1/4comma 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 number, so is (3R+1)/(5R+2), 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 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.
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 1/11comma of equal temperament, well temperaments and eventually equal temperament became more popular.
Contents 
Construction of the quartercomma meantone diatonic scale
As discussed above, in the 1/4comma meantone temperament, the ratio of a tone is <math> \sqrt{5}:2 <math>, the ratio of a semitone is <math> 8:5^{5/4} <math>, and the ratio of a fifth is <math> 5^{1/4} <math>. 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:
 <math> 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. <math>
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 

C  1  1  0 
D  T  1.118033  193.156856 
E  T^{2}  1.25  386.313713 
F  T^{2} S  1.337480  503.421571 
G  P  1.495348  696.578428 
A  P T  1.671850  889.735285 
B  P T^{2}  1.869185  1082.892142 
C'  P T^{2} S  2  1200 
Construction of the quartercomma meantone chromatic scale
Construction of a 1/4comma meantone chromatic scale can proceed by taking the 1/4comma meantone diatonic scale as its foundation. The tones in the diatonic scale can be divided into pairs of semitones. However, S^{2} is not equal to T. Instead, let
 <math> \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}. <math>
Thus, each tone can be divided into a pair of unequal semitones, the major one being S, and the minor one being <math>\bar{S}<math>. Notice that S is equal to 117.107 cents, and that <math>\bar{S}<math> 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 <math>\bar{S}<math> 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 quartermeantone chromatic scale was constructed by Pietro Aaron in 1523:
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 — <math> \bar{S} <math>. 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 <math>\bar{S}<math> to move up by a minor semitone.
Note  Formula  Ratio  Cents 

C  1  1  0 
C#  <math>\bar{S}<math>  1.04490672653  76.0489992634 
D  T  1.11803398875  193.156856932 
D#  T S  1.19627902498  310.264714601 
E  T^{2}  1.25  386.313713865 
F  T^{2} S  1.33748060995  503.421571534 
F#  T^{3}  1.39754248594  579.470570797 
G  P  1.49534878122  696.578428466 
G#  <math> P \ \bar{S} <math>  1.5625  772.62742773 
A  P T  1.67185076244  889.735285399 
A#  P T S  1.788854382  1006.84314307 
B  P T^{2}  1.86918597653  1082.89214233 
C'  P T^{2} 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.
Triads in the quartermeantone 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  <math> S^2\cdot \bar{S}^2 <math>  C—G  <math> S^4 \cdot \bar{S}^3 <math>  C—D#  <math> S^2 \cdot \bar{S} <math> 
C#—F  <math> S^3 \cdot \bar{S} <math>  C#—G#  <math> S^4 \cdot \bar{S}^3 <math>  C#—E  <math> S^2 \cdot \bar{S} <math> 
D—F#  <math> S^2\cdot \bar{S}^2 <math>  D—A  <math> S^4 \cdot \bar{S}^3 <math>  D—F  <math> S^2 \cdot \bar{S} <math> 
D#—G  <math> S^2\cdot \bar{S}^2 <math>  D#—A#  <math> S^4 \cdot \bar{S}^3 <math>  D#—F#  <math> S \cdot \bar{S}^2 <math> 
E—G#  <math> S^2\cdot \bar{S}^2 <math>  E—B  <math> S^4 \cdot \bar{S}^3 <math>  E—G  <math> S^2 \cdot \bar{S} <math> 
F—A  <math> S^2\cdot \bar{S}^2 <math>  F—C  <math> S^4 \cdot \bar{S}^3 <math>  F—G#  <math> S \cdot \bar{S}^2 <math> 
F#—A#  <math> S^3 \cdot \bar{S} <math>  F#—C#  <math> S^4 \cdot \bar{S}^3 <math>  F#—A  <math> S^2 \cdot \bar{S} <math> 
G—B  <math> S^2\cdot \bar{S}^2 <math>  G—D  <math> S^4 \cdot \bar{S}^3 <math>  G—A#  <math> S^2 \cdot \bar{S} <math> 
G#—C  <math> S^3 \cdot \bar{S} <math>  G#—D#  <math> S^5 \cdot \bar{S}^2 <math>  G#—B  <math> S^2 \cdot \bar{S} <math> 
A—C#  <math> S^2\cdot \bar{S}^2 <math>  A—E  <math> S^4 \cdot \bar{S}^3 <math>  A—C  <math> S^2 \cdot \bar{S} <math> 
A#—D  <math> S^2\cdot \bar{S}^2 <math>  A#—F  <math> S^4 \cdot \bar{S}^3 <math>  A#—C#  <math> S \cdot \bar{S}^2 <math> 
B—D#  <math> S^3 \cdot \bar{S} <math>  B—F#  <math> S^4 \cdot \bar{S}^3 <math>  B—D  <math> S^2 \cdot \bar{S} <math> 
First, look at the column of fifths in the middle. All the fifths except one have a ratio of
 <math> S^4 \cdot \bar{S}^3 = 1.495348 = 696.578 \ \hbox{cents} <math>
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
 <math> S^5 \cdot \bar{S}^2 = 1.531237 = 737.637 \ \hbox{cents} <math>
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
 <math> S^2 \cdot \bar{S}^2 = 1.25 = 386.313 \ \hbox{cents} <math>
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
 <math> S^3 \cdot \bar{S} = 1.28 = 427.372 \ \hbox{cents} <math>
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
 <math> S^2 \cdot \bar{S} = 1.196279 = 310.264 \ \hbox{cents} <math>
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
 <math> S \cdot \bar{S}^2 = 1.168241 = 269.205 \ \hbox{cents} <math>
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#.
See also
External links
 Tonalsoft Encyclopaedia of Tuning (http://tonalsoft.com/enc/index2.htm?meantone.htm)
 Kyle Gann's Introduction to Historical Tunings (http://home.earthlink.net/~kgann/histune.html) has an explanation of how the meantone temperament works.
 List of some of the meantone systems (http://www.harmonics.com/lucy/lsd/mean.html)
 Tuning system derived from π and the writings of John 'Longitude' Harrison (http://www.lucytune.com)fr:Tempérament mésotonique