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Notice that, in quarter-comma meantone, the seventeenth is 81 / 80 times narrower than in Pythagorean tuning. This difference in size, equal to about 21.506 cents, is called the syntonic comma. This implies that the fifth is a quarter of a syntonic comma narrower than the justly tuned Pythagorean fifth.
Syntonic comma on C Pythagorean comma on C . In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. [1] Traditionally, there are two most common comma; the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven ...
The Pythagorean comma shown as the gap (on the right side) which causes a 12-pointed star to fail to close, which star represents the Pythagorean scale; each line representing a just perfect fifth. That gap has a central angle of 7.038 degrees, which is 23.46% of 30 degrees. The size of a Pythagorean comma, measured in cents, is
The standard tempered fifth has a frequency ratio of 2 7/12:1 (or about 1.498307077:1), approximately two cents narrower than a justly tuned fifth. Ascending by twelve justly tuned fifths fails to close the circle by an excess of approximately 23.46 cents, roughly a quarter of a semitone, an interval known as the Pythagorean comma.
In neither the twelve tone equitemperament nor the quarter-comma meantone is the fifth a rational fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of the equal temperaments ( "N TET" ), in which the octave is divided into some number (N) of equally wide intervals.
In quarter-comma meantone, the frequency ratio for the fifth is 4 √ 5 , which is about 3.42157 cents flatter than an equal tempered 700 cents, (or exactly one twelfth of a diesis) and so the wolf is about 737.637 cents, or 35.682 cents sharper than a perfect fifth of ratio exactly 3:2, and this is the original "howling" wolf fifth.