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.

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 {{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.

Higher-rank microtemperaments are also useful as simplifications of just intonation. 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. A list of microtemperaments can be found at [[:Category: Microtemperaments]].

The opposite of a microtemperament is an [[exotemperament]].

The opposite of a microtemperament is an [[exotemperament]], which is very inaccurate and tempers out large and/or high-damage commas, such as [[16/15]], [[25/24]], or [[28/27]].

== Practical notes ==

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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.
+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 {{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.
+Higher-rank microtemperaments are also useful as simplifications of just intonation. 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. A list of microtemperaments can be found at [[:Category: Microtemperaments]].
-The opposite of a microtemperament is an [[exotemperament]].
+The opposite of a microtemperament is an [[exotemperament]], which is very inaccurate and tempers out large and/or high-damage commas, such as [[16/15]], [[25/24]], or [[28/27]].
 == Practical notes ==