Quarter-comma meantone
The purpose is to obtain justly intoned major thirds (with a frequency ratio equal to 5:4).
It was described by Pietro Aron 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".
[1] Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.
In Pythagorean tuning, the size of a seventeenth is defined using a stack of four justly tuned fifths (frequency ratio 3 : 2 ): In quarter-comma meantone temperament, where a just major third (5:4) is required, a slightly narrower seventeenth is obtained by stacking two octaves and a major third: By definition, however, a seventeenth of the same size ( 5 : 1 ) must be obtained, even in quarter-comma meantone, by stacking four fifths.
This method is identical to Pythagorean tuning, except for the size of the fifth, which is tempered as explained above.
However, meantone temperaments (except for 12 TET) cannot fit into a 12-note keyboard; and like quarter-comma meantone, most require an infinite number of notes (although there is a very close approximation to quarter-comma that can fit into a keyboard with 31 keys per octave).
In the formulas, x = 4√5 = 51⁄4 is the size of the tempered perfect fifth, and the ratios x : 1 or 1 : x represent an ascending or descending tempered perfect fifth (i.e. an increase or decrease in frequency by x), while 2 : 1 or 1 : 2 represent an ascending or descending octave.
If the last step (here, G♯) is replaced by a copy of A♭ but in the same octave as G♯, that will increase the interval C♯–G♯ to a discord called a wolf fifth.
However, the construction table shows only 12 of them, in this case those starting from C. This is at the same time the main advantage and main disadvantage of the C-based asymmetric stack, as the intervals from C are commonly used, but since C is not at the center of this stack, they unfortunately include an augmented fifth (i.e. the interval from C to G♯), instead of a minor sixth (from C to A♭).
Notice that in the above-mentioned set of 144 intervals pure minor sixths are more frequently observed than impure augmented fifths (see table below), and this is one of the reasons why it is not desirable to show an impure augmented fifth in the construction table.
The C-based symmetric stack is rarely used, possibly because it produces the wolf fifth in the unusual position of F♯–D♭ instead of G♯–E♭, where musicians accustomed to the previously used Pythagorean tuning might expect it).
As discussed above, in the quarter-comma meantone temperament, The tones in the diatonic scale can be divided into pairs of semitones.
These two deviations are comparable to the syntonic comma (21.5 cents), which this system is designed to tune out from the Pythagorean major third.
However, since even the just intonated ratio 18:17 sounds markedly dissonant, these deviations are considered acceptable in a semitone.
In quarter-comma meantone, the minor second is considered acceptable while the augmented unison sounds dissonant and should be avoided.
The price paid, in this case, is that none of them is justly tuned and perfectly consonant, except, of course, for the unison and the octave.
Since they span the same number of semitones, perfect fifths and diminished sixths are considered to be enharmonically equivalent.
The following table focuses only on the above-mentioned three interval types, used to form major and minor triads.
S and X denote the ratio of the two abovementioned kinds of semitones (minor second and augmented unison).
On the other hand, the diminished sixth from G♯ to E♭ has a ratio of which deviates by +35.7 cents from the just perfect fifth.
On the other hand, the four diminished fourths with roots at C♯, F♯, G♯ and B have a ratio of which deviates by +41.1 cents from the just major third.
On the other hand, the three augmented seconds whose roots are E♭, F and B♭ have a ratio of which deviates by −46.4 cents from the just minor third.
These augmented seconds, though sufficiently consonant by themselves, will sound "exotic" or atypical when played together with a perfect fifth.
As discussed above, G♯ is a different pitch that A♭, as are all other "enharmonic" pairs of sharps and flats in quarter comma meantone: Each requires a separate key on the keyboard and neither can substitute for the other.
The limited chordal options is not a fault in meantone tunings; it is the consequence of needing more notes in the octave than is available on some modern equal tempered instruments.
This more conventional approach, similar to the D-based Pythagorean tuning system, explains the reason why the Χ and S semitones are arranged in the particular and apparently arbitrary sequence shown above.
