Microtemperament

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A microtemperament is a regular temperament in which all of the intervals considered to be consonances of the system are so closely approximated as to be effectively just.

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 microtemperaments; 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 (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 the ragisma (4375/4374), is also a microtemperament.

The opposite of a microtemperament is an exotemperament.

Practical notes

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.

See also