31 Tone Equal Temperament

Meantone Tuning

A meantone tuning or meantone temperament is a musical tuning that, if viewed as being constructed from stacked perfect fifths, makes the fifths narrow relative to the true harmonic intervals, so as to achieve a closer match to major and minor thirds. Meantone tunings can be seen as a compromise tuning, moving the tuning of fiths slightly farther from perfect, in order to move other musical intervals closer to their perfect intervals.

31-ET is a meantone tuning, and there are other equally tempered meantone tunings as well, the best-known of which is 19 tone equal temperament.

Properties of Meantone Tunings

List of Equally Tempered Meantone Tunings

This table illustrates the degree of closeness of match of certain key intervals to their true intervals in just intonation, for all the equally tempered meantone tunings between 19 and 50 divisions of the octave.

The numbers here represent the difference in cents between the intervals in these tunings, and the natural intervals. A negative value means the intervals are narrower than in just intonation, and a positive value means the intervals are wider.

Steps in OctaveWidth of Whole Tone (in Cents)FifthMajor ThirdMinor Third
12 (also 24,36)200.00-1.95+13.69-15.64
19 (also 38)189.47-7.22-7.37+0.15

Note that, as in all meantone tunings, the fifths are always narrower than the true interval.

This table gives a clear demonstration of the superiority of 31-ET to other equally tempered meantone tunings. 19-ET does provide a better match to the minor third, at the expense of a flatter fifth and much flatter major third. 31-ET's match to the major third is extraordinarily close, close enough for the distance to just intonation to be completely imperceptible to most people.

43-ET and 50-ET improve slightly on the matches to fifths and minor thirds, at the expense of the major third, but these tunings have far more divisions of the octave and are sub-optimal for other reasons. In the neighborhood of these tunings, 41-ET and 53-ET offer compelling advantages in terms of better matches for numerous natural intervals.

Note on 12, 24, 36, and 48-ET

I have included 12, 24, and 36 in this table because these tunings are technically meantone tunings, as the whole tones in these scales are narrower (although only by a tiny bit) than the true 9:8 ratio. I did not include 48 because, although 48-ET can technically be used as a subdivision of 12-ET, it offers a closer match to both the major and minor thirds that is not a result of the standard meantone construction (i.e. stacking flat fifths to achieve an in-tune major or minor third), but rather, a match made by using intervals peculiar to this tuning, and not able to be made by stacking fifths in this tuning.

Further Reading