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.