31 Tone Equal Temperament

Just Intonation

Just intonation, also called just tuning, is the musical tuning system in which intervals correspond exactly to whole number ratios in the harmonic series. Intervals in just tuning are called just intervals. Because just intervals are the intervals that naturally occur in the harmonic series, they provide a reference point to compare intervals to, to measure the degree to which intervals in a scale match the natural harmonics. One could accurately say that just intervals are perfectly in-tune.

In just intonation, it is not possible to return to the same note via a cycle or circle of the same interval.

Naming and Ratio Notation

Intervals in just intonation are notated by ratios, and are also given names. For example, 3:2 is a perfect fifth, meaning that this interval is the difference between the third and second harmonics. It is a convention that the larger number is usually listed first. These names sometimes correspond to the intervals in various equally tempered tuning systems, but in many cases, the just intervals are given more specific names. For example, 12-ET has only one tritone, whereas in just intonation, there are countless tritones, including the septimal tritones (7:5 and 10:7) and undecimal tritones (11:8 and 16:11).

Just Interval Counterparts of 31-ET Intervals

All of the intervals in 31-ET have their counterparts in just intonation. None of the intervals in 31-ET correspond exactly to the just intervals, but some are very close, whereas others are farther. With the exception of the whole tone and the semi-diminished fourth, the intervals in 31-ET correspond uniquely to just intervals. Because 31-ET is a meantone tuning, the whole tone is intermediate in size (close to halfway) between the major tone (9:8) and the minor tone (10:9).