Meantone temperament

Meantone temperaments are musical temperaments;^[1] 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. 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 ¹²√2  : 1. Relative to Pythagorean tuning, it narrows the perfect fifths by about 2 cents or ⁠1/ 12 ⁠th 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 ⁠th 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).