Microtemperament: Difference between revisions

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~~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.

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 [[~~Schismatic|~~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|~~ennealimmal ~~temperament~~]], which tempers out the [[~~2401/2400|~~breedsma]] 2401/2400 and [[~~4375/4374|~~ragisma]] 4375/4374, is also 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 {{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]].

[[Category:Microtemperaments| ]]

== See also ==

* [[Unnoticeable comma]]

[[Category:Microtemperaments| ]]

[[Category:Regular temperament theory]]

[[Category:Method]]

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-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.
+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 [[Schismatic|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|ennealimmal temperament]], which tempers out the [[2401/2400|breedsma]] 2401/2400 and [[4375/4374|ragisma]] 4375/4374, is also 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 {{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]].
-[[Category:Microtemperaments| ]]
+== See also ==
-<!-- main article -->
+* [[Unnoticeable comma]]
+[[Category:Microtemperaments| ]] <!-- main article -->
 [[Category:Regular temperament theory]]
 [[Category:Method]]