5th-octave temperaments: Difference between revisions

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[[5edo]] is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of [[12edo]].

{{Infobox fractional-octave|5}}[[5edo]] is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of [[12edo]].

The most notable 5th-octave family is [[limmic temperaments]] – [[tempering out]] [[256/243]] and associates 3\5 to [[3/2]] as well as 1\5 to [[9/8]], producing temperaments like [[blackwood]]. Equally notable among small equal divisions are the [[Cloudy clan|cloudy temperaments]] – identifying [[8/7]] with one step of 5edo.

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* [[Trisedodge family|Trisedodge temperaments]]

== Slendroschismic ==

~~== Slendrismic ==~~

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.

~~: ''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.

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, ~262144/151263

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

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

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

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

[[Badness]] (Sintel): 0.456

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

[[Tp tuning #T2 tuning|RMS error]]: 0.0212 cents

== Thunderclysmic ==

Thunderclysmic is a weak extension of ~~slendrismic~~ (above), [[rainy]] and [[cata]], with a generator of a slightly sharp ~6/5 (befitting of any [[kleismic]] temperament), three of which making [[26/15]]~[[19/11]]. More interesting though is that the period is [[5edo|1\5]]; ~~slendrismic~~ gives this 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 in the 2.3.7 subgroup that approximates 1\5 very well). Thunderclysmic gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of Thunderclysmic. In fact, Thunderclysmic combines many convergents and semiconvergents to intervals of [[5edo]] into a single, high-limit temperament:

Thunderclysmic is a weak extension of slendroschismic (above), [[rainy]] and [[cata]], with a generator of a slightly sharp ~6/5 (befitting of any [[kleismic]] temperament), three of which making [[26/15]]~[[19/11]]. More interesting though is that the period is [[5edo|1\5]]; slendroschismic gives this 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 in the 2.3.7 subgroup that approximates 1\5 very well). Thunderclysmic gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of Thunderclysmic. In fact, Thunderclysmic combines many convergents and semiconvergents to intervals of [[5edo]] into a single, high-limit temperament:

1\5 = [[23/20]] = [[31/27]] = [[85/74]] = [[54/47]] (which Thunderclysmic also equates with [[63/50]]), and 2\5 = [[33/25]] = [[95/72]] = [[29/22]] = [[62/47]] = [[128/97]] (which Thunderclysmic also equates with [[37/28]] and [[120/91]]).

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{{Optimal ET sequence|legend=1| 15, 95bc, 110, 125, 140, 265, 405 }}

[[Badness]] (~~Dirichlet~~): 3.009

[[Badness]] (Sintel): 3.009

=== 11-limit ===

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{{Optimal ET sequence|legend=1| 15, 95bce, 110e, 125, 140, 265e, 405ee }}

[[Badness]] (~~Dirichlet~~): 1.856

[[Badness]] (Sintel): 1.856

=== 13-limit ===

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[[Badness]] (~~Dirichlet~~): 1.458

[[Badness]] (Sintel): 1.458

=== 17-limit ===

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[[Badness]] (~~Dirichlet~~): 1.493

[[Badness]] (Sintel): 1.493

=== 19-limit ===

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[[Badness]] (~~Dirichlet~~): 1.507

[[Badness]] (Sintel): 1.507

=== 23-limit ===

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[[Badness]] (~~Dirichlet~~): 1.424

[[Badness]] (Sintel): 1.424

=== 29-limit ===

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[[Badness]] (~~Dirichlet~~): 1.318

[[Badness]] (Sintel): 1.318

=== 31-limit ===

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[[Badness]] (~~Dirichlet~~): 1.501

[[Badness]] (Sintel): 1.501

=== 37-limit ===

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[[Badness]] (~~Dirichlet~~): 1.537

[[Badness]] (Sintel): 1.537

=== 37-limit add-47 add-97 ===

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[[Badness]] (~~Dirichlet~~): 1.715

[[Badness]] (Sintel): 1.715

== Pentonismic (rank-5) ==

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: Mapping generators: ~9/8, ~7

~~{{Multival|legend=1| 0 0 5 0 8 12 }}~~

[[Optimal tuning]]s:

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[[Badness]]: 0.048312

== Obscenity ==

Obscenity maps [[32/21]] to 3\5 and was named in subtle reference to [[Syntonic–chromatic equivalence continuum #Absurdity|absurdity]] as a kind of septal (2.3.7) analog to it.

[[Subgroup]]: 2.3.7

[[Comma list]]: 4194304/4084101

: Mapping generators: ~512/441, ~3

[[Support]]ing [[ET]]s: {{EDOs|5, 65d, 70, 75, 80, 85, 90, 95}}

~~[[Badness]]: 0.048312~~

@@ Line 1: / Line 1: @@
-[[5edo]] is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of [[12edo]].
+{{Infobox fractional-octave|5}}[[5edo]] is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of [[12edo]].
 The most notable 5th-octave family is [[limmic temperaments]] – [[tempering out]] [[256/243]] and associates 3\5 to [[3/2]] as well as 1\5 to [[9/8]], producing temperaments like [[blackwood]]. Equally notable among small equal divisions are the [[Cloudy clan|cloudy temperaments]] – identifying [[8/7]] with one step of 5edo.
@@ Line 8: / Line 8: @@
 * [[Trisedodge family|Trisedodge temperaments]]
+== Slendroschismic ==
+{{See also| No-fives subgroup temperaments #Slendroschismic }}
-== Slendrismic ==
+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.
-: <small>''See also: [[No-fives subgroup temperaments #Slendrismic]] and [[Slendrisma]]''</small>
-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.
+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|legend=1|5 0 18|0 2 -1}}
+{{Mapping|legend=1| 5 0 18 | 0 2 -1 }}
+: Mapping generators: ~147/128, ~262144/151263
-: Mapping generators: ~147/128 = 1＼5, ~262144/151263
+[[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 tuning]] ([[CTE]]): ~8/7 = 230.9930 (or ~1029/1024 = 9.0080)
+{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 935, 1335, 1735, 3070, 4805d }}
-{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 1065, 1465, 1865 }}
+[[Badness]] (Sintel): 0.456
-[[Badness]]: 0.013309
+[[Tp tuning #T2 tuning|RMS error]]: 0.0212 cents
 == Thunderclysmic ==
-Thunderclysmic is a weak extension of slendrismic (above), [[rainy]] and [[cata]], with a generator of a slightly sharp ~6/5 (befitting of any [[kleismic]] temperament), three of which making [[26/15]]~[[19/11]]. More interesting though is that the period is [[5edo|1\5]]; slendrismic gives this 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 in the 2.3.7 subgroup that approximates 1\5 very well). Thunderclysmic gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of Thunderclysmic. In fact, Thunderclysmic combines many convergents and semiconvergents to intervals of [[5edo]] into a single, high-limit temperament:
+Thunderclysmic is a weak extension of slendroschismic (above), [[rainy]] and [[cata]], with a generator of a slightly sharp ~6/5 (befitting of any [[kleismic]] temperament), three of which making [[26/15]]~[[19/11]]. More interesting though is that the period is [[5edo|1\5]]; slendroschismic gives this 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 in the 2.3.7 subgroup that approximates 1\5 very well). Thunderclysmic gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of Thunderclysmic. In fact, Thunderclysmic combines many convergents and semiconvergents to intervals of [[5edo]] into a single, high-limit temperament:
 \5 = [[23/20]] = [[31/27]] = [[85/74]] = [[54/47]] (which Thunderclysmic also equates with [[63/50]]), and 2\5 = [[33/25]] = [[95/72]] = [[29/22]] = [[62/47]] = [[128/97]] (which Thunderclysmic also equates with [[37/28]] and [[120/91]]).
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 {{Optimal ET sequence|legend=1| 15, 95bc, 110, 125, 140, 265, 405 }}
-[[Badness]] (Dirichlet): 3.009
+[[Badness]] (Sintel): 3.009
 === 11-limit ===
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 {{Optimal ET sequence|legend=1| 15, 95bce, 110e, 125, 140, 265e, 405ee }}
-[[Badness]] (Dirichlet): 1.856
+[[Badness]] (Sintel): 1.856
 === 13-limit ===
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 {{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}
-[[Badness]] (Dirichlet): 1.458
+[[Badness]] (Sintel): 1.458
 === 17-limit ===
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 {{Optimal ET sequence|legend=1| 15, 125f, 140, 265ef, 405eef }}
-[[Badness]] (Dirichlet): 1.493
+[[Badness]] (Sintel): 1.493
 === 19-limit ===
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 {{Optimal ET sequence|legend=1| 15, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.507
+[[Badness]] (Sintel): 1.507
 === 23-limit ===
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 {{Optimal ET sequence|legend=1| 15, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.424
+[[Badness]] (Sintel): 1.424
 === 29-limit ===
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 {{Optimal ET sequence|legend=1| 15, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.318
+[[Badness]] (Sintel): 1.318
 === 31-limit ===
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 {{Optimal ET sequence|legend=1| 15k, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.501
+[[Badness]] (Sintel): 1.501
 === 37-limit ===
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 {{Optimal ET sequence|legend=1| 15k, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.537
+[[Badness]] (Sintel): 1.537
 === 37-limit add-47 add-97 ===
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 {{Optimal ET sequence|legend=1| 15ko, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.715
+[[Badness]] (Sintel): 1.715
 == Pentonismic (rank-5) ==
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 : Mapping generators: ~9/8, ~7
-{{Multival|legend=1| 0 0 5 0 8 12 }}
 [[Optimal tuning]]s:
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 {{Optimal ET sequence|legend=1| 5, 15ccd }}
+[[Badness]]: 0.048312
+== Obscenity ==
+Obscenity maps [[32/21]] to 3\5 and was named in subtle reference to [[Syntonic–chromatic equivalence continuum #Absurdity|absurdity]] as a kind of septal (2.3.7) analog to it.
+[[Subgroup]]: 2.3.7
+[[Comma list]]: 4194304/4084101
+{{Mapping|legend=1| 5 0 22 | 0 1 -1 }}
+: Mapping generators: ~512/441, ~3
+[[Support]]ing [[ET]]s: {{EDOs|5, 65d, 70, 75, 80, 85, 90, 95}}
 {{Navbox fractional-octave}}
-[[Badness]]: 0.048312