Microtemperament: Difference between revisions
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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. | |||
Putting 118 and 171 together, we see that the 5-limit [ | 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 {{EDOs|118, 171, 270 …}} are [[5-limit]] microtemperaments; that {{EDOs|171, 270, 441, 612, 643 …}} are [[7-limit]] microtemperaments; {{EDOs|764, 836, 1084, 1106 …}} are [[11-limit]] microtemperaments, and that {{EDOs|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 == | |||
* [[Unnoticeable comma]] | |||
* [[Very high accuracy temperaments]] | |||
[[Category:Regular temperament theory]] | |||
[[Category:Method]] | |||
Latest revision as of 07:12, 23 April 2026
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.