Pythagorean comma in the context of Enharmonic

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Traditionally, there are two most common commata; the syntonic comma (80:81), "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma (524288:531441, approximately 73:74), "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Pitches separated by either comma are considered the same note because conventional notation does not distinguish Pythagorean intervals from 5-limit intervals. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53 TET, the harmonic seventh B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.

Meantone temperaments are musical temperaments; that is, a variety of tuning systems constructed, similarly to Pythagorean tuning, as a sequence of equal fifths, both rising and descending, scaled to remain within the same octave. But rather than using perfect fifths, consisting of frequency ratios of value ${\displaystyle 3:2}$, these are tempered by a suitable factor that narrows them to ratios that are slightly less than ${\displaystyle 3:2}$, in order to bring the major or minor thirds closer to the just intonation ratio of ${\displaystyle 5:4}$ or ${\displaystyle 6:5}$, respectively. Among temperaments constructed as a sequence of fifths, a regular temperament is one in which all the fifths are chosen to be of the same size.

Twelve-tone equal temperament (12 TET) is obtained by making all semitones the same size, with each equal to one-twelfth of an octave; i.e. with ratios √2  : 1. Relative to Pythagorean tuning, it narrows the perfect fifths by about 2 cents or ⁠1/ 12 ⁠ of a Pythagorean comma to give a frequency ratio of ${\displaystyle 2^{7/12}:1}$. This produces major thirds that are wide by about 13 cents, or ⁠1/ 8 ⁠ of a semitone. Twelve-tone equal temperament is almost exactly the same as ⁠1/ 11 ⁠ syntonic comma meantone tuning (1.955 cents vs. 1.95512).