Төрттен Бір Коммалық Орта Тондық Темперамент

Quarter comma meantone, or comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma ( 81 : 80 ), with respect to its just intonation used in Pythagorean tuning (frequency ratio 3 : 2 ); the result is or a fifth of 696.578 cents. (The 12th power of that value is 125, whereas 7 octaves is 128, and so falls 41.059 cents short.) This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds (with a frequency ratio equal to sesquiquartum 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." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

The whole chromatic scale (a subset of which is the diatonic scale), can be constructed by starting from a given base note, and increasing or decreasing its frequency by one or more 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 ) 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). When tuned to a 12 note keyboard many notes must be left out, and unless the tuning is "tempered" to gloss over the missing notes, keyboard players who substitute the available nearest pitch note (which is always the wrong pitch) for the actual appropriate quarter comma note (which would sound consonant, if it were available) create dissonant notes in place of the consonant quarter comma note. The construction table below illustrates how the pitches of the notes are obtained with respect to D (the base note), in a D based scale (see Pythagorean tuning for a more detailed explanation). For each note in the basic octave, the table provides the conventional name of the interval from D (the base note), the formula to compute its frequency ratio, and the approximate values for its frequency ratio and size in cents. Note Interval from D Formula Freq. ratio Size(cents) Size (31 ET) A diminished fifth 1.4311 620.5 16.03 E minor second 1.0700 117.1 3.03 B minor sixth 1.6000 813.7 21.02 F minor third 1.1963 310.3 8.02 C minor seventh 1.7889 1006.8 26.01 G perfect fourth 1.3375 503.4 13.00 D unison 1.0000 0.0 0.00 A perfect fifth 1.4953 696.6 18.00 E major second 1.1180 193.2 4.99 B major sixth 1.6719 889.7 22.98 F major third 1.2500 386.3 9.98 C major seventh 1.8692 1082.9 27.97 G augmented fourth 1.3975 579.5 14.97

In the formulas, x = = 5 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. As in Pythagorean tuning, this method generates 13 pitches, with A and G nearly a quarter tone apart. To build a 12 tone scale A is typically discarded.

A just intonation version of the quarter comma meantone temperament may be constructed in the same way as Johann Kirnberger's rational version of 12 TET. The value of 5 · 35 is very close to 4, which is why a 7 limit interval 6144 : 6125 (which is the difference between the 5 limit diesis 128 : 125 and the septimal diesis 49 : 48), equal to 5.362 cents, appears very close to the quarter comma of 5.377 cents. So the perfect fifth has the ratio of 6125 : 4096, which is the difference between three just major thirds and two septimal major seconds; four such fifths exceed the ratio of 5 : 1 by the tiny interval of 0.058 cents. The wolf fifth there appears to be 49 : 32, the difference between the septimal minor seventh and the septimal major second.

The perfect fifth of quarter comma meantone, expressed as a fraction of an octave, is  log2 5. Since log2 5 is an irrational number, a chain of meantone fifths never closes (i. e. never equals a chain of octaves). 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.