# Tempering Out

In tuning, the concept of **tempering out** refers to a tuning removing the distinction between two intervals close in size, or the identification with unison of a narrow interval. When this happen, it is said that the tuning **tempers out** the interval or the difference between the two intervals. The concept of tempering out intervals is usually used in reference to the intervals of just intonation.

The tempering out of a small interval to unison, and the tempering out the difference between two intervals are equivalent concepts. For example:

## Differences Tempered Out in 12-ET but not 31-ET:

This list is not exhaustive but serves just to give some examples:

- The minor third(6:5) and septimal minor third(7:6).
- The major third(5:4) and septimal major third(9:7).
- The major whole tone (9:8) and septimal whole tone (8:7).
- The diesis

Although technically true that 12-ET "tempers out" the difference between neutral seconds and neutral thirds, and the other intervals more closely matched by 12-ET, it doesn't necessarily make much sense to say this because these intervals fall almost exactly halfway in between the steps of the 12-ET scale.

## Differences that 31-ET Tempers Out:

31-ET tempers out a large number of intervals, but the main relevant ones are those involving harmonics up through the 11th.

- The lesser(12:11) and greater (11:10) neutral seconds, corresponding to the small interval 121:120. This is also equivalent to the difference between the (15:11) tritone and the lesser undecimal tritone (11:8).
- The minor (10:9) and major (9:8) whole tones, corresponding to the small interval 81:80.
- The undecimal major third (14:11) and septimal major third (9:7), corresponding to the small interval 99:98.
- The diatonic semitone(16:15) and septimal diatonic semitone (15:14), corresponding to the small interval 225:224.