Under a comma is understood in music theory a small interval ( much smaller than a semitone), which is the difference of different combinations of pure intervals. The term is closely related to the mood systems. When trying to gain as large a number of useful musical tones and intervals, one or more commas are always balanced.
Particularly well known is the Pythagorean comma, which is seemingly the circle of fifths: the sequence of 12 pure fifths leads to a tone that is slightly different from (corresponding octave ) output sound.
Twelve pure fifths superimposed achieve a sound that has the seventh octave of the fundamental tone at a distance approximately one quarter semitone, the Pythagorean comma:
Four pure fifths (3/2 ) superimposed achieve a sound that is at a distance from the Pythagorean third of the second octave of the fundamental. This third is about to fifth semitone, the syntonic comma or didymische point is greater than the pure third.
The Pythagorean third in comparison to the pure third:
The syntonic comma:
The major whole tone ( 9/8) differs from the small whole tone (10 /9) to the syntonic comma:
The schism is the difference between the Pythagorean comma and the syntonic comma:
The exact frequency ratio is
Andreas Werckmeister ( " Musicalische temperature", Quedlinburg 1691) considered the schism in the construction of his well-tempered tunings: Assuming h is a series of perfect fifths down to ces, is the last note - octave - a Pythagorean comma is deeper than h Go the other hand one by one syntonic comma lower h, we obtain einenTon, h ( depth comma h), which occurs in the pure major chord g, hd and different from Ces only to the schism. This difference is at the " limit of perceptible tonal differences " (See The pure harmonium ). One can therefore, h identify with ces: , h = ces, as well as the =, cis; it = dis; tot =, fis, as =, g #, b =, ais u.s.w.
The schism should not be confused with the twelfth part of the Pythagorean comma ( relevant to the mood systems ), even if all these numerical values in cents:
In Euclid's " division of the canon ," in which the theoretical knowledge of music of that time is summarized (ca. 3rd century BC), can be read as a set 14: "The octave is less than 6 whole tones ". The octave is the interval with the proportion ( today's interpretation: frequency ratio ) 2:1, and whole tone of the interval with the proportion 9:8. The difference ( six whole tones - octave ) is called the Pythagorean comma. Its proportion is specified in Euclid to 531441:524288 (though the term is not κόμμα in Euclid before ).
Only with the advent of polyphonic music during the Renaissance and Baroque playing the commas, especially for the voices of Tastinstrumenten, where only 12 pitches were present in the octave, a decisive role. It has developed a variety of mood systems where the commas were distributed differently on the scale degrees.
Other designated as a comma small intervals
If three major thirds strung together, so this result in equally- tempered tuning, one octave, in just intonation, however, a slightly smaller interval. The difference to the octave is called enharmonic comma or small DIESIS. The enharmonic comma is in meantone tuning exactly is the difference between the enharmonic change tones, such as Dis - It.
The exact frequency ratio is
As the Leipzig comma approximately 27.26 cents large interval is indicated by the frequency ratio 64:63, which between
- The natural seventh ( 7:4 approx 968.82 cents) and
- The minor seventh (16:9 about 996.08 cents)
The pure tuning is.