Microtemperament

Any system of temperament in which all of the intervals considered to be consonances of the system are so closely approximated as to be effectively just is a microtemperament. There is no agreed on value for the threshold to microtempering, and here we will use less than a cent in error as our line of demarcation, which some people would call a nanotemperament. Using this definition and the tonality diamond as our consonance set, we find that the equal temperaments 118, 171, 270 … are 5-limit microtmeperaments; that 171, 270, 441, 612, 643 … are 7-limit microtemperaments; 764, 836, 1084, 1106 … are 11-limit microtemperaments, and that 1171, 1178, 1186 … are 13-limit microtemperaments.

Putting 118 and 171 together, we see that the 5-limit schismatic temperament, which tempers out the schisma of 32805/32768, is a microtemperament when appropriately tuned. Putting 171 and 270 together, we see that ennealimmal temperament, which tempers out the breedsma 2401/2400 and ragisma 4375/4374, is also a microtemperament.

The opposite of a microtemperament is an exotemperament.

Tuning microtemperaments can be challenging due to the sheer number of pitch materials, but one may take advantage of MIDI channels. See Tuning per channel.