5th-octave temperaments: Difference between revisions

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

[[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.

Other families of 5-limit 5th-octave commas are:

* [[~~Pental~~ family|~~Pental~~ temperaments]] - tempers out the {{monzo|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.

* [[Quintile family|Quintile temperaments]] - tempers out the {{monzo|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.

* [[Quintosec family|Quintosec temperaments]]

* [[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 ~~= 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 = 240.000{{c}}, ~8/7 = 230.992{{c}} (~1029/1024 = 9.008{{c}})

* [[CWE]]: ~147/128 = 240.000{{c}}, ~8/7 = 231.004{{c}} (~1029/1024 = 8.996{{c}})

{{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

== ~~Thunder~~ ==

== Thunderclysmic ==

~~Thunder~~ 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). ~~This temperament~~ gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of ~~this temperament~~. In fact, ~~this temperament~~ 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]] = [[54/47]] = [[85/74]] and 2\5 = [[29/22]] = [[33/25]] = [[62/47]] = [[95/72]] = [[128/97]].

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]]).

~~Thunder~~ can be thought of as the [[125edo|125f]] & [[140edo|140]] temperament in the [[37-limit]] add-47 add-97, with both tunings notable in all corresponding limits.

Thunderclysmic can be thought of as the [[125edo|125f]] & [[140edo|140]] temperament in the [[37-limit]] add-47 add-97, with both tunings notable in all corresponding limits.

It is a complex temperament, however note that it is especially strong and elegant in the no-13's no-17's no-31's [[37-limit]] add-97, on which its mapping is especially efficient, so that the gen range is -3 to 7 for a(n inclusive) span of 11 notes per 1\5 period, so that a 5*11=55 note-per-octave (multiperiod) [[MOS]] may be used.

=== 7-limit ===

7-limit ~~Thunder~~ also tempers out the [[4096000/4084101]] (the [[hemfiness comma]]).

7-limit Thunderclysmic also tempers out the [[4096000/4084101]] (the [[hemfiness comma]]).

[[Subgroup]]: [[7-limit|2.3.5.7]]

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[[Optimal tuning]] ([[CTE]]): 317.059{{~~cent~~}}

[[Optimal tuning]]s:

* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.059{{c}}

* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.046{{c}}

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

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

[[Badness]] (Sintel): 3.009

=== 11-limit ===

~~This temperament~~ extends naturally to the 11-limit by tempering [[385/384]] = ([[147/128]])/([[63/55]]) (or equivalently [[6250/6237]]). Note that ~~this temperament~~ observes the comma [[441/440]] = ([[21/20]])/([[22/21]]) = S21, as if it didn't, we would have 63/55 also equated with [[8/7]], leading to the [[15edo]] tuning which tempers the [[cloudy comma]]. In the 11-limit, the 5 EDO fourth is interpreted as [[33/25]].

Thunderclysmic extends naturally to the 11-limit by tempering [[385/384]] = ([[147/128]])/([[63/55]]) (or equivalently [[6250/6237]]). Note that Thunderclysmic observes the comma [[441/440]] = ([[21/20]])/([[22/21]]) = S21, as if it didn't, we would have 63/55 also equated with [[8/7]], leading to the [[15edo]] tuning which tempers the [[cloudy comma]]. In the 11-limit, the 5 EDO fourth is interpreted as [[33/25]].

~~{{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}~~

[[Subgroup]]: [[11-limit|2.3.5.7.11]]

[[Comma list]]: [[~~15625~~/~~15552~~]], [[~~2100875~~/~~2097152~~]], [[~~385~~/~~384~~]]

[[Comma list]]: [[385/384]], [[1331/1323]], [[6250/6237]]

[[Optimal tuning]] ([[CTE]]): 317.~~136~~{{~~cent~~}}

[[Optimal tuning]]s:

* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.107{{c}}

* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.055{{c}}

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

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

[[Badness]] (Sintel): 1.856

=== 13-limit ===

As ~~this temperament~~ is a weak extension of [[cata]], it is naturally at least 13-limit. Cata admits a fairly simple mapping of prime 13 via ([[6/5]])2 = [[13/9]] so that a gen above that is [[26/15]] as half of [[3/1]]. As cata tempers [[625/624|625/624 = S25]] and [[676/675|676/675 = S26 = S13/S15]] and as the [[kleisma]] is S252 * S26, this replaces the kleisma in the comma list so that we now move it to the end (as both are 13-limit). For simplicity, we show [[325/324|325/324 = S25 * S26]] and the more structurally important aforementioned comma 676/675, omitting 625/624. It also tempers [[1001/1000]] and [[1716/1715]] in the 13-limit.

As Thunderclysmic is a weak extension of [[cata]], it is naturally at least 13-limit. Cata admits a fairly simple mapping of prime 13 via ([[6/5]])2 = [[13/9]] so that a gen above that is [[26/15]] as half of [[3/1]]. As cata tempers [[625/624|625/624 = S25]] and [[676/675|676/675 = S26 = S13/S15]] and as the [[kleisma]] is S252 * S26, this replaces the kleisma in the comma list so that we now move it to the end (as both are 13-limit). For simplicity, we show [[325/324|325/324 = S25 * S26]] and the more structurally important aforementioned comma 676/675, omitting 625/624. It also tempers [[1001/1000]] and [[1716/1715]] in the 13-limit.

~~{{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}~~

[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]

[[Comma list]]: [[~~2100875~~/~~2097152~~]], [[385/384]], [[~~325~~/~~324~~]], [[~~676~~/~~675~~]]

[[Comma list]]: [[325/324]], [[385/384]], [[625/624]], [[1331/1323]]

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[[Optimal tuning]] ([[CTE]]): 317.136{{cent}}

[[Optimal tuning]]s:

* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.136{{c}}

* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.086{{c}}

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

[[Badness]] (Sintel): 1.458

=== 17-limit ===

In the 17-limit, the 5 EDO fifth is interpreted as [[85/56]] = [[561/560]] * [[50/33]], so that [[17/16]] is reached at 11 periods minus 8 gens at approx. 103{{cent}}; equating it with [[16/15]] might seem natural but is not the route taken due to the precision affording observing their difference, [[256/255]]. (If you do want to equate 17/16 with 16/15, you get the 15 & 155 temperament instead, for which the main tuning is [[155edo]], but you very much do pay for it; notice the errors!) Interestingly, the CTE tuning of the 17-limit of ~~this temperament~~ is practically the same as that of the 29-limit (up to a thousandth of a cent), which is also notable as being where this temperament exhibits the lowest Dirichlet badness. It's also the smallest prime limit where the vals larger than [[140edo]] haven't disappeared from the [[optimal ET sequence]], as from the 19-limit and onwards the optimal ET sequence is always [[15edo|15(ko)]], [[125edo|125f]], [[140edo|140]].

In the 17-limit, the 5 EDO fifth is interpreted as [[85/56]] = [[561/560]] * [[50/33]], so that [[17/16]] is reached at 11 periods minus 8 gens at approx. 103{{cent}}; equating it with [[16/15]] might seem natural but is not the route taken due to the precision affording observing their difference, [[256/255]]. (If you do want to equate 17/16 with 16/15, you get the 15 & 155 temperament instead, for which the main tuning is [[155edo]], but you very much do pay for it; notice the errors!) Interestingly, the CTE tuning of the 17-limit of Thunderclysmic is practically the same as that of the 29-limit (up to a thousandth of a cent), which is also notable as being where this temperament exhibits the lowest Dirichlet badness. It's also the smallest prime limit where the vals larger than [[140edo]] haven't disappeared from the [[optimal ET sequence]], as from the 19-limit and onwards the optimal ET sequence is always [[15edo|15(ko)]], [[125edo|125f]], [[140edo|140]].

~~{{Optimal ET sequence|legend=1| 15, 125f, 140, 265ef, 405eef }}~~

[[Subgroup]]: [[17-limit|2.3.5.7.11.13.17]]

[[Comma list]]: [[~~2100875~~/~~2097152~~]], [[385/384]], [[~~325~~/~~324~~]], [[~~676~~/~~675~~]], [[~~561~~/~~560~~]]

[[Comma list]]: [[325/324]], [[385/384]], [[442/441]], [[625/624]], [[1331/1323]]

[[Optimal tuning]] ([[CTE]]): 317.111{{~~cent~~}}

[[Optimal tuning]]s:

* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.111{{c}}

* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.066{{c}}

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

[[Badness]] (Sintel): 1.493

=== 19-limit ===

As [[33/25]] and [[95/72]] are both close to the 5 EDO fourth, ~~this temperament~~ extends naturally to the 19-limit by tempering [[2376/2375]] = ([[33/25]])/([[95/72]]) = ([[6/5]])3 / ([[19/11]]) and thus equivalently by tempering ([[26/15]])/([[19/11]]) = [[286/285]]. This is equivalent to tempering [[400/399|400/399 = S20]] = ([[20/19]])/([[21/20]]), which is natural to temper given that we observe [[441/440|441/440 = S21]] as aforementioned.

As [[33/25]] and [[95/72]] are both close to the 5 EDO fourth, Thunderclysmic extends naturally to the 19-limit by tempering [[2376/2375]] = ([[33/25]])/([[95/72]]) = ([[6/5]])3 / ([[19/11]]) and thus equivalently by tempering ([[26/15]])/([[19/11]]) = [[286/285]]. This is equivalent to tempering [[400/399|400/399 = S20]] = ([[20/19]])/([[21/20]]), which is natural to temper given that we observe [[441/440|441/440 = S21]] as aforementioned.

[[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]

[[Comma list]]: [[~~2100875~~/~~2097152~~]], [[~~385~~/~~384~~]], [[~~325~~/~~324~~]], [[~~676~~/~~675~~]], [[~~561~~/~~560~~]], [[~~286~~/~~285~~]]

[[Comma list]]: [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[442/441]], [[1331/1323]]

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

[[Badness]] (Sintel): 1.507

=== 23-limit ===

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[[Subgroup]]: [[23-limit|2.3.5.7.11.13.17.19.23]]

[[Comma list]]: [[~~2100875~~/~~2097152~~]], [[~~385~~/~~384~~]], [[325/324]], [[~~676~~/~~675~~]], [[~~561~~/~~560~~]], [[~~286~~/~~285~~]], [[~~736~~/~~735~~]]

[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[442/441]], [[484/483]]

{{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 | 0 6 5 -3 4 14 -8 7 5 }}

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

[[Badness]] (Sintel): 1.424

=== 29-limit ===

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[[Subgroup]]: [[29-limit|2.3.5.7.11.13.17.19.23.29]]

[[Comma list]]: [[~~2100875~~/~~2097152~~]], [[~~385~~/~~384~~]], [[325/324]], [[~~676~~/~~675~~]], [[~~561~~/~~560~~]], [[~~286~~/~~285~~]], [[~~736~~/~~735~~]], [[~~726~~/~~725~~]]

[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[442/441]], [[484/483]], [[552/551]]

{{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 | 0 6 5 -3 4 14 -8 7 5 4 }}

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

[[Badness]] (Sintel): 1.318

=== 31-limit ===

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[[Subgroup]]: [[31-limit|2.3.5.7.11.13.17.19.23.29.31]]

[[Comma list]]: [[~~2100875~~/~~2097152~~]], [[~~385~~/~~384~~]], [[325/324]], [[~~676~~/~~675~~]], [[~~561~~/~~560~~]], [[~~286~~/~~285~~]], [[~~736~~/~~735~~]], [[~~726~~/~~725~~]], [[~~3969~~/~~3968~~]]

[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[435/434]], [[442/441]], [[484/483]], [[528/527]]

{{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 1 | 0 6 5 -3 4 14 -8 7 5 4 18 }}

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

[[Badness]] (Sintel): 1.501

=== 37-limit ===

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[[Subgroup]]: [[37-limit|2.3.5.7.11.13.17.19.23.29.31.37]]

[[Comma list]]: [[~~2100875~~/~~2097152~~]], [[~~385~~/~~384~~]], [[325/324]], [[~~676~~/~~675~~]], [[~~561~~/~~560~~]], [[~~286~~/~~285~~]], [[~~736~~/~~735~~]], [[~~726~~/~~725~~]], [[~~3969~~/~~3968~~]], [[~~481~~/~~480~~]]

[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[435/434]], [[442/441]], [[481/480]], [[484/483]], [[528/527]]

{{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 1 30 | 0 6 5 -3 4 14 -8 7 5 4 18 -3 }}

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

[[Badness]] (Sintel): 1.537

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

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[[Subgroup]]: 2.3.5.7.11.13.17.19.23.29.31.37.47.97

[[Comma list]]: [[~~2100875~~/~~2097152~~]], [[~~385~~/~~384~~]], [[325/324]], [[~~676~~/~~675~~]], [[~~561~~/~~560~~]], [[~~286~~/~~285~~]], [[~~736~~/~~735~~]], [[~~726~~/~~725~~]], [[~~3969~~/~~3968~~]], [[~~481~~/~~480~~]], [[2304/2303]], [[9216/9215]]

[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[435/434]], [[442/441]], [[481/480]], [[484/483]], [[528/527]], [[2304/2303]], [[9216/9215]]

{{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 1 30 4 33 | 0 6 5 -3 4 14 -8 7 5 4 18 -3 18 0 }}

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

[[Badness]] (Sintel): 1.715

== Pentonismic (rank-5) ==

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Supporting ETs: {{Optimal ET sequence|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}

== ~~Quint~~ ==

== Obscenity ==

~~''Quint'' preserves the~~ 5-limit ~~mapping~~ of ~~5edo, and the harmonic~~ 7 ~~is mapped~~ to ~~an independent generator~~. ~~In what way is this useful is unexplained~~.

Obscenity maps [[32/21]] to 3\5 and was named in subtle reference to [[Syntonic–chromatic equivalence continuum#Absurdity (5-limit)|absurdity]] as a kind of septal (2.3.7) analog to it. It tempers out the obsceniton, 4194304/4084101.

[[Subgroup]]: 2.3.5.7

[[Subgroup]]: 2.3.7

~~[[Comma list]]: 16/15, 27/25~~

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

: ~~Mapping generators: ~9~~/~~8, ~7~~

[[Comma list]]: 4194304/4084101

~~[[Optimal tuning]] ([[POTE]])~~: ~9/~~8 = 1＼5~~, ~~~7/4 = 1017.903~~

: 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: @@
-{{Fractional-octave navigation|5}}
+{{Infobox fractional-octave|5}}[[5edo]] is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of [[12edo]].
-[[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.
 Other families of 5-limit 5th-octave commas are:
-* [[Pental family|Pental temperaments]] - tempers out the {{monzo|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.
+* [[Quintile family|Quintile temperaments]] - tempers out the {{monzo|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.
 * [[Quintosec family|Quintosec temperaments]]
 * [[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
@@ Line 18: / 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 = 240.000{{c}}, ~8/7 = 230.992{{c}} (~1029/1024 = 9.008{{c}})
+* [[CWE]]: ~147/128 = 240.000{{c}}, ~8/7 = 231.004{{c}} (~1029/1024 = 8.996{{c}})
-{{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
-== Thunder ==
+== Thunderclysmic ==
-Thunder 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). This temperament gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of this temperament. In fact, this temperament 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]] = [[54/47]] = [[85/74]] and 2\5 = [[29/22]] = [[33/25]] = [[62/47]] = [[95/72]] = [[128/97]].
+\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]]).
-Thunder can be thought of as the [[125edo|125f]] & [[140edo|140]] temperament in the [[37-limit]] add-47 add-97, with both tunings notable in all corresponding limits.
+Thunderclysmic can be thought of as the [[125edo|125f]] & [[140edo|140]] temperament in the [[37-limit]] add-47 add-97, with both tunings notable in all corresponding limits.
+It is a complex temperament, however note that it is especially strong and elegant in the no-13's no-17's no-31's [[37-limit]] add-97, on which its mapping is especially efficient, so that the gen range is -3 to 7 for a(n inclusive) span of 11 notes per 1\5 period, so that a 5*11=55 note-per-octave (multiperiod) [[MOS]] may be used.
 === 7-limit ===
--limit Thunder also tempers out the [[4096000/4084101]] (the [[hemfiness comma]]).
+-limit Thunderclysmic also tempers out the [[4096000/4084101]] (the [[hemfiness comma]]).
 [[Subgroup]]: [[7-limit|2.3.5.7]]
@@ Line 44: / Line 49: @@
 {{Mapping|legend=1| 5 0 5 18 | 0 6 5 -3 }}
-[[Optimal tuning]] ([[CTE]]): 317.059{{cent}}
+[[Optimal tuning]]s:
+* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.059{{c}}
+* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.046{{c}}
 {{Optimal ET sequence|legend=1| 15, 95bc, 110, 125, 140, 265, 405 }}
-[[Badness]] (Dirichlet): 3.009
+[[Badness]] (Sintel): 3.009
 === 11-limit ===
-This temperament extends naturally to the 11-limit by tempering [[385/384]] = ([[147/128]])/([[63/55]]) (or equivalently [[6250/6237]]). Note that this temperament observes the comma [[441/440]] = ([[21/20]])/([[22/21]]) = S21, as if it didn't, we would have 63/55 also equated with [[8/7]], leading to the [[15edo]] tuning which tempers the [[cloudy comma]]. In the 11-limit, the 5 EDO fourth is interpreted as [[33/25]].
+Thunderclysmic extends naturally to the 11-limit by tempering [[385/384]] = ([[147/128]])/([[63/55]]) (or equivalently [[6250/6237]]). Note that Thunderclysmic observes the comma [[441/440]] = ([[21/20]])/([[22/21]]) = S21, as if it didn't, we would have 63/55 also equated with [[8/7]], leading to the [[15edo]] tuning which tempers the [[cloudy comma]]. In the 11-limit, the 5 EDO fourth is interpreted as [[33/25]].
-{{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}
 [[Subgroup]]: [[11-limit|2.3.5.7.11]]
-[[Comma list]]: [[15625/15552]], [[2100875/2097152]], [[385/384]]
+[[Comma list]]: [[385/384]], [[1331/1323]], [[6250/6237]]
 {{Mapping|legend=1| 5 0 5 18 12 | 0 6 5 -3 4 }}
-[[Optimal tuning]] ([[CTE]]): 317.136{{cent}}
+[[Optimal tuning]]s:
+* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.107{{c}}
+* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.055{{c}}
 {{Optimal ET sequence|legend=1| 15, 95bce, 110e, 125, 140, 265e, 405ee }}
-[[Badness]] (Dirichlet): 1.856
+[[Badness]] (Sintel): 1.856
 === 13-limit ===
-As this temperament is a weak extension of [[cata]], it is naturally at least 13-limit. Cata admits a fairly simple mapping of prime 13 via ([[6/5]])<sup>2</sup> = [[13/9]] so that a gen above that is [[26/15]] as half of [[3/1]]. As cata tempers [[625/624|625/624 = S25]] and [[676/675|676/675 = S26 = S13/S15]] and as the [[kleisma]] is S25<sup>2</sup> * S26, this replaces the kleisma in the comma list so that we now move it to the end (as both are 13-limit). For simplicity, we show [[325/324|325/324 = S25 * S26]] and the more structurally important aforementioned comma 676/675, omitting 625/624. It also tempers [[1001/1000]] and [[1716/1715]] in the 13-limit.
+As Thunderclysmic is a weak extension of [[cata]], it is naturally at least 13-limit. Cata admits a fairly simple mapping of prime 13 via ([[6/5]])<sup>2</sup> = [[13/9]] so that a gen above that is [[26/15]] as half of [[3/1]]. As cata tempers [[625/624|625/624 = S25]] and [[676/675|676/675 = S26 = S13/S15]] and as the [[kleisma]] is S25<sup>2</sup> * S26, this replaces the kleisma in the comma list so that we now move it to the end (as both are 13-limit). For simplicity, we show [[325/324|325/324 = S25 * S26]] and the more structurally important aforementioned comma 676/675, omitting 625/624. It also tempers [[1001/1000]] and [[1716/1715]] in the 13-limit.
-{{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}
 [[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
-[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]]
+[[Comma list]]: [[325/324]], [[385/384]], [[625/624]], [[1331/1323]]
 {{Mapping|legend=1| 5 0 5 18 12 0 | 0 6 5 -3 4 14 }}
@@ Line 80: / Line 85: @@
 [[Optimal tuning]] ([[CTE]]): 317.136{{cent}}
-{{Optimal ET sequence|legend=1| 15, 125f, 140 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.136{{c}}
+* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.086{{c}}
+{{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}
-[[Badness]] (Dirichlet): 1.458
+[[Badness]] (Sintel): 1.458
 === 17-limit ===
-In the 17-limit, the 5 EDO fifth is interpreted as [[85/56]] = [[561/560]] * [[50/33]], so that [[17/16]] is reached at 11 periods minus 8 gens at approx. 103{{cent}}; equating it with [[16/15]] might seem natural but is not the route taken due to the precision affording observing their difference, [[256/255]]. (If you do want to equate 17/16 with 16/15, you get the 15 & 155 temperament instead, for which the main tuning is [[155edo]], but you very much do pay for it; notice the errors!) Interestingly, the CTE tuning of the 17-limit of this temperament is practically the same as that of the 29-limit (up to a thousandth of a cent), which is also notable as being where this temperament exhibits the lowest Dirichlet badness. It's also the smallest prime limit where the vals larger than [[140edo]] haven't disappeared from the [[optimal ET sequence]], as from the 19-limit and onwards the optimal ET sequence is always [[15edo|15(ko)]], [[125edo|125f]], [[140edo|140]].
+In the 17-limit, the 5 EDO fifth is interpreted as [[85/56]] = [[561/560]] * [[50/33]], so that [[17/16]] is reached at 11 periods minus 8 gens at approx. 103{{cent}}; equating it with [[16/15]] might seem natural but is not the route taken due to the precision affording observing their difference, [[256/255]]. (If you do want to equate 17/16 with 16/15, you get the 15 & 155 temperament instead, for which the main tuning is [[155edo]], but you very much do pay for it; notice the errors!) Interestingly, the CTE tuning of the 17-limit of Thunderclysmic is practically the same as that of the 29-limit (up to a thousandth of a cent), which is also notable as being where this temperament exhibits the lowest Dirichlet badness. It's also the smallest prime limit where the vals larger than [[140edo]] haven't disappeared from the [[optimal ET sequence]], as from the 19-limit and onwards the optimal ET sequence is always [[15edo|15(ko)]], [[125edo|125f]], [[140edo|140]].
-{{Optimal ET sequence|legend=1| 15, 125f, 140, 265ef, 405eef }}
 [[Subgroup]]: [[17-limit|2.3.5.7.11.13.17]]
-[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]]
+[[Comma list]]: [[325/324]], [[385/384]], [[442/441]], [[625/624]], [[1331/1323]]
 {{Mapping|legend=1| 5 0 5 18 12 0 31 | 0 6 5 -3 4 14 -8 }}
-[[Optimal tuning]] ([[CTE]]): 317.111{{cent}}
+[[Optimal tuning]]s:
+* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.111{{c}}
+* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.066{{c}}
-{{Optimal ET sequence|legend=1| 15, 125f, 140 }}
+{{Optimal ET sequence|legend=1| 15, 125f, 140, 265ef, 405eef }}
-[[Badness]] (Dirichlet): 1.493
+[[Badness]] (Sintel): 1.493
 === 19-limit ===
-As [[33/25]] and [[95/72]] are both close to the 5 EDO fourth, this temperament extends naturally to the 19-limit by tempering [[2376/2375]] = ([[33/25]])/([[95/72]]) = ([[6/5]])<sup>3</sup> / ([[19/11]]) and thus equivalently by tempering ([[26/15]])/([[19/11]]) = [[286/285]]. This is equivalent to tempering [[400/399|400/399 = S20]] = ([[20/19]])/([[21/20]]), which is natural to temper given that we observe [[441/440|441/440 = S21]] as aforementioned.
+As [[33/25]] and [[95/72]] are both close to the 5 EDO fourth, Thunderclysmic extends naturally to the 19-limit by tempering [[2376/2375]] = ([[33/25]])/([[95/72]]) = ([[6/5]])<sup>3</sup> / ([[19/11]]) and thus equivalently by tempering ([[26/15]])/([[19/11]]) = [[286/285]]. This is equivalent to tempering [[400/399|400/399 = S20]] = ([[20/19]])/([[21/20]]), which is natural to temper given that we observe [[441/440|441/440 = S21]] as aforementioned.
 [[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]
-[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]], [[286/285]]
+[[Comma list]]: [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[442/441]], [[1331/1323]]
 {{Mapping|legend=1| 5 0 5 18 12 0 31 12 | 0 6 5 -3 4 14 -8 7 }}
@@ Line 114: / Line 123: @@
 {{Optimal ET sequence|legend=1| 15, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.507
+[[Badness]] (Sintel): 1.507
 === 23-limit ===
@@ Line 121: / Line 130: @@
 [[Subgroup]]: [[23-limit|2.3.5.7.11.13.17.19.23]]
-[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]], [[286/285]], [[736/735]]
+[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[442/441]], [[484/483]]
 {{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 | 0 6 5 -3 4 14 -8 7 5 }}
@@ Line 129: / Line 138: @@
 {{Optimal ET sequence|legend=1| 15, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.424
+[[Badness]] (Sintel): 1.424
 === 29-limit ===
@@ Line 136: / Line 145: @@
 [[Subgroup]]: [[29-limit|2.3.5.7.11.13.17.19.23.29]]
-[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]], [[286/285]], [[736/735]], [[726/725]]
+[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[442/441]], [[484/483]], [[552/551]]
 {{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 | 0 6 5 -3 4 14 -8 7 5 4 }}
@@ Line 144: / Line 153: @@
 {{Optimal ET sequence|legend=1| 15, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.318
+[[Badness]] (Sintel): 1.318
 === 31-limit ===
@@ Line 151: / Line 160: @@
 [[Subgroup]]: [[31-limit|2.3.5.7.11.13.17.19.23.29.31]]
-[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]], [[286/285]], [[736/735]], [[726/725]], [[3969/3968]]
+[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[435/434]], [[442/441]], [[484/483]], [[528/527]]
 {{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 1 | 0 6 5 -3 4 14 -8 7 5 4 18 }}
@@ Line 159: / Line 168: @@
 {{Optimal ET sequence|legend=1| 15k, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.501
+[[Badness]] (Sintel): 1.501
 === 37-limit ===
@@ Line 166: / Line 175: @@
 [[Subgroup]]: [[37-limit|2.3.5.7.11.13.17.19.23.29.31.37]]
-[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]], [[286/285]], [[736/735]], [[726/725]], [[3969/3968]], [[481/480]]
+[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[435/434]], [[442/441]], [[481/480]], [[484/483]], [[528/527]]
 {{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 1 30 | 0 6 5 -3 4 14 -8 7 5 4 18 -3 }}
@@ Line 174: / Line 183: @@
 {{Optimal ET sequence|legend=1| 15k, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.537
+[[Badness]] (Sintel): 1.537
 === 37-limit add-47 add-97 ===
@@ Line 181: / Line 190: @@
 [[Subgroup]]: 2.3.5.7.11.13.17.19.23.29.31.37.47.97
-[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]], [[286/285]], [[736/735]], [[726/725]], [[3969/3968]], [[481/480]], [[2304/2303]], [[9216/9215]]
+[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[435/434]], [[442/441]], [[481/480]], [[484/483]], [[528/527]], [[2304/2303]], [[9216/9215]]
 {{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 1 30 4 33 | 0 6 5 -3 4 14 -8 7 5 4 18 -3 18 0 }}
@@ Line 189: / Line 198: @@
 {{Optimal ET sequence|legend=1| 15ko, 125f, 140 }}
-[[Badness]] (Dirichlet): 1.715
+[[Badness]] (Sintel): 1.715
 == Pentonismic (rank-5) ==
@@ Line 206: / Line 213: @@
 Supporting ETs: {{Optimal ET sequence|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}
-== Quint ==
+== Obscenity ==
-''Quint'' preserves the 5-limit mapping of 5edo, and the harmonic 7 is mapped to an independent generator. In what way is this useful is unexplained.
+Obscenity maps [[32/21]] to 3\5 and was named in subtle reference to [[Syntonic–chromatic equivalence continuum#Absurdity (5-limit)|absurdity]] as a kind of septal (2.3.7) analog to it. It tempers out the obsceniton, 4194304/4084101.
-[[Subgroup]]: 2.3.5.7
+[[Subgroup]]: 2.3.7
-[[Comma list]]: 16/15, 27/25
-{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
-: Mapping generators: ~9/8, ~7
+[[Comma list]]: 4194304/4084101
-{{Multival|legend=1| 0 0 5 0 8 12 }}
+{{Mapping|legend=1| 5 0 22 | 0 1 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~9/8 = 1＼5, ~7/4 = 1017.903
+: Mapping generators: ~512/441, ~3
-{{Optimal ET sequence|legend=1| 5, 15ccd }}
+[[Support]]ing [[ET]]s: {{EDOs|5, 65d, 70, 75, 80, 85, 90, 95}}
-[[Badness]]: 0.048312
+{{Navbox fractional-octave}}