Fractional-octave temperaments: Difference between revisions

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~~All~~ temperaments ~~on this page~~ have a ~~fractional-~~octave ~~period~~.

'''Fractional-octave temperaments''', when viewed from a regular temperament theory perspective, are temperaments which have a period which corresponds to a just interval mapped to a fraction of the octave, that is one step of an [[EDO]].

Temperaments discussed ~~elsewhere~~ include:

== Theory ==

Fractional-octave temperaments are valuable with regards to [[Polysystemic|polysystemicism]] and polychromatics. They are acoustically significant with regards to containing modes of limited transposition, as well as their ability to expand on the harmony of the equal division they are a superset of. Such temperaments are also a way of introducing less common and harmonically less performing equal divisions into music that prefers consonance and is based on regular temperament theory.

=== Terminology ===

The terminology was developed by [[Eliora]]. The equal division containing the mos scale of such a temperament, starting from the tonic, is referred to as a ''wireframe'', and individual notes of that equal division are called ''hinges''. Thus in this context, the wireframe is the tuning consisting of only stacks of the period and no stacks of the generator. Temperament-agnostically, this can be used to refer to any structure embedded in an (x,y)-ET which repeats y times within that period, its "wireframe" is y-ET.

The most common way to produce a fractional-octave temperament is through an excellent approximation of an interval relative to the size of the wireframe edo. For example, [[compton family]] tempers out the Pythagorean comma and maps 7 steps of 12edo to [[3/2]]. Likewise, a lot of 10th-octave temperaments have a [[13/8]] as 7\10, and 26th-octave temperaments often have a [[7/4]] for 21\26.

== Temperament collections ==

Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:

* 1\2 period temperaments

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* [[Tritrizo clan #Schisennealimmal|Schisennealimmal]] (1\171 period)

* [[Tritrizo clan #Lunennealimmal|Lunennealimmal]] (1\441 period)

== 44th-octave temperaments ==

One step of 44edo is very close to the septimal comma, [[64/63]]. The relationship is preserved even up thousands of edos.

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-All temperaments on this page have a fractional-octave period.
+'''Fractional-octave temperaments''', when viewed from a regular temperament theory perspective, are temperaments which have a period which corresponds to a just interval mapped to a fraction of the octave, that is one step of an [[EDO]].
-Temperaments discussed elsewhere include:
+== Theory ==
+Fractional-octave temperaments are valuable with regards to [[Polysystemic|polysystemicism]] and polychromatics. They are acoustically significant with regards to containing modes of limited transposition, as well as their ability to expand on the harmony of the equal division they are a superset of. Such temperaments are also a way of introducing less common and harmonically less performing equal divisions into music that prefers consonance and is based on regular temperament theory.
+=== Terminology ===
+The terminology was developed by [[Eliora]]. The equal division containing the mos scale of such a temperament, starting from the tonic, is referred to as a ''wireframe'', and individual notes of that equal division are called ''hinges''. Thus in this context, the wireframe is the tuning consisting of only stacks of the period and no stacks of the generator. Temperament-agnostically, this can be used to refer to any structure embedded in an (x,y)-ET which repeats y times within that period, its "wireframe" is y-ET.
+The most common way to produce a fractional-octave temperament is through an excellent approximation of an interval relative to the size of the wireframe edo. For example, [[compton family]] tempers out the Pythagorean comma and maps 7 steps of 12edo to [[3/2]]. Likewise, a lot of 10th-octave temperaments have a [[13/8]] as 7\10, and 26th-octave temperaments often have a [[7/4]] for 21\26.
+== Temperament collections ==
+Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:
 * 1\2 period temperaments
@@ Line 83: / Line 92: @@
 * [[Tritrizo clan #Schisennealimmal|Schisennealimmal]] (1\171 period)
 * [[Tritrizo clan #Lunennealimmal|Lunennealimmal]] (1\441 period)
 == 44th-octave temperaments ==
 One step of 44edo is very close to the septimal comma, [[64/63]]. The relationship is preserved even up thousands of edos.