5th-octave temperaments: Difference between revisions

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* [[Trisedodge family|Trisedodge temperaments]]

== ~~Slendrismic~~ ==

== Slendroschismic ==

~~: ''~~See also~~: [[~~No-fives subgroup temperaments #~~Slendrismic]] and~~ [[~~Slendrisma~~]]~~''~~

In ~~slendrismic~~, the period (1\5) is given a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])2 = [[8/7]] * [[1029/1024|S7/S8]], which is a significant interval as it is the "harmonic 5edostep" in that it's a [[rooted]] (/2^n) interval that approximates 1\5 very well. The generator is [[1029/1024]], the difference between [[8/7]] and [[147/128]] and therefore between 3/2 and (8/7)3. The temperament is named for the very "slender" generator as well as as a ~~pun~~ on "[[slendric]]~~" (which it shouldn't be confused with)~~. One can consider this as a microtemperament counterpart to [[cloudy]], which equates them.

Slendroschismic tempers out the [[slendroschisma]]. In this temperament, the period (1\5) is given a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])2 = ([[8/7]])⋅([[1029/1024|S7/S8]]), which is a significant interval as it is the "harmonic 5edostep" in that it is a [[rooted]] (/2''n'') interval that approximates 1\5 very well. The generator is [[1029/1024]], the difference between [[8/7]] and [[147/128]] and therefore between 3/2 and (8/7)3. The temperament is named for the very "slender" generator as well as as a reference on [[slendric]]. One can consider this as a microtemperament counterpart to [[cloudy]], which equates them.

A possible extension to the full 7-limit is given by the [[hemipental]] temperament.

[[Subgroup]]: 2.3.7

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[[Comma list]]: 68719476736/68641485507

: Mapping generators: ~147/128 ~~= 1＼5~~, ~262144/151263

: Mapping generators: ~147/128, ~262144/151263

[[Optimal tuning]] ([[CTE]]): ~8/7 = 230.9930 (or ~1029/1024 = 9.0080)

[[Optimal tuning]]s:

* [[CTE]]: ~147/128 = 240.0000, ~8/7 = 230.9930 (~1029/1024 = 9.0080)

* [[POTE]]: ~147/128 = 240.0000, ~8/7 = 231.0094 (~1029/1024 = 8.9906)

{{Optimal ET sequence|legend=1| 130, 135, 265, 400, ~~1065~~, ~~1465~~, ~~1865~~ }}

{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 935, 1335, 1735, 3070, 4805d }}

[[Badness]]: 0.~~013309~~

[[Badness]] (Sintel): 0.456

== Thunderclysmic ==

@@ Line 8: / Line 8: @@
 * [[Trisedodge family|Trisedodge temperaments]]
-== Slendrismic ==
+== Slendroschismic ==
-: <small>''See also: [[No-fives subgroup temperaments #Slendrismic]] and [[Slendrisma]]''</small>
+{{See also| No-fives subgroup temperaments #Slendroschismic }}
-In slendrismic, the period (1\5) is given a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])<sup>2</sup> = [[8/7]] * [[1029/1024|S7/S8]], which is a significant interval as it is the "harmonic 5edostep" in that it's a [[rooted]] (/2^n) interval that approximates 1\5 very well. The generator is [[1029/1024]], the difference between [[8/7]] and [[147/128]] and therefore between 3/2 and (8/7)<sup>3</sup>. The temperament is named for the very "slender" generator as well as as a pun on "[[slendric]]" (which it shouldn't be confused with). One can consider this as a microtemperament counterpart to [[cloudy]], which equates them.
+Slendroschismic tempers out the [[slendroschisma]]. In this temperament, the period (1\5) is given a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])<sup>2</sup> = ([[8/7]])⋅([[1029/1024|S7/S8]]), which is a significant interval as it is the "harmonic 5edostep" in that it is a [[rooted]] (/2<sup>''n''</sup>) interval that approximates 1\5 very well. The generator is [[1029/1024]], the difference between [[8/7]] and [[147/128]] and therefore between 3/2 and (8/7)<sup>3</sup>. The temperament is named for the very "slender" generator as well as as a reference on [[slendric]]. One can consider this as a microtemperament counterpart to [[cloudy]], which equates them.
+A possible extension to the full 7-limit is given by the [[hemipental]] temperament.
 [[Subgroup]]: 2.3.7
@@ Line 16: / Line 19: @@
 [[Comma list]]: 68719476736/68641485507
-{{Mapping|legend=1|5 0 18|0 2 -1}}
+{{Mapping|legend=1| 5 0 18 | 0 2 -1 }}
-: Mapping generators: ~147/128 = 1＼5, ~262144/151263
+: Mapping generators: ~147/128, ~262144/151263
-[[Optimal tuning]] ([[CTE]]): ~8/7 = 230.9930 (or ~1029/1024 = 9.0080)
+[[Optimal tuning]]s:
+* [[CTE]]: ~147/128 = 240.0000, ~8/7 = 230.9930 (~1029/1024 = 9.0080)
+* [[POTE]]: ~147/128 = 240.0000, ~8/7 = 231.0094 (~1029/1024 = 8.9906)
-{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 1065, 1465, 1865 }}
+{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 935, 1335, 1735, 3070, 4805d }}
-[[Badness]]: 0.013309
+[[Badness]] (Sintel): 0.456
 == Thunderclysmic ==