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5th-octave temperaments: Difference between revisions

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{{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.
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:
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]]
* [[Quintosec family|Quintosec temperaments]]
* [[Trisedodge family|Trisedodge 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
[[Subgroup]]: 2.3.7
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[[Comma list]]: 68719476736/68641485507
[[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


== Thunderclysmic ==
== 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:


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]]).
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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{{Mapping|legend=1| 5 0 5 18 | 0 6 5 -3 }}
{{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 }}
{{Optimal ET sequence|legend=1| 15, 95bc, 110, 125, 140, 265, 405 }}


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


=== 11-limit ===
=== 11-limit ===
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[[Subgroup]]: [[11-limit|2.3.5.7.11]]
[[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 }}
{{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 }}
{{Optimal ET sequence|legend=1| 15, 95bce, 110e, 125, 140, 265e, 405ee }}


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


=== 13-limit ===
=== 13-limit ===
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[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
[[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 }}
{{Mapping|legend=1| 5 0 5 18 12 0 | 0 6 5 -3 4 14 }}


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


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


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


=== 17-limit ===
=== 17-limit ===
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[[Subgroup]]: [[17-limit|2.3.5.7.11.13.17]]
[[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 }}
{{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, 265ef, 405eef }}
{{Optimal ET sequence|legend=1| 15, 125f, 140, 265ef, 405eef }}


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


=== 19-limit ===
=== 19-limit ===
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[[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]
[[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 }}
{{Mapping|legend=1| 5 0 5 18 12 0 31 12 | 0 6 5 -3 4 14 -8 7 }}
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{{Optimal ET sequence|legend=1| 15, 125f, 140 }}
{{Optimal ET sequence|legend=1| 15, 125f, 140 }}


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


=== 23-limit ===
=== 23-limit ===
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[[Subgroup]]: [[23-limit|2.3.5.7.11.13.17.19.23]]
[[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 }}
{{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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{{Optimal ET sequence|legend=1| 15, 125f, 140 }}
{{Optimal ET sequence|legend=1| 15, 125f, 140 }}


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


=== 29-limit ===
=== 29-limit ===
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[[Subgroup]]: [[29-limit|2.3.5.7.11.13.17.19.23.29]]
[[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 }}
{{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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{{Optimal ET sequence|legend=1| 15, 125f, 140 }}
{{Optimal ET sequence|legend=1| 15, 125f, 140 }}


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


=== 31-limit ===
=== 31-limit ===
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[[Subgroup]]: [[31-limit|2.3.5.7.11.13.17.19.23.29.31]]
[[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 }}
{{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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{{Optimal ET sequence|legend=1| 15k, 125f, 140 }}
{{Optimal ET sequence|legend=1| 15k, 125f, 140 }}


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


=== 37-limit ===
=== 37-limit ===
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[[Subgroup]]: [[37-limit|2.3.5.7.11.13.17.19.23.29.31.37]]
[[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 }}
{{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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{{Optimal ET sequence|legend=1| 15k, 125f, 140 }}
{{Optimal ET sequence|legend=1| 15k, 125f, 140 }}


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


=== 37-limit add-47 add-97 ===
=== 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
[[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 }}
{{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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{{Optimal ET sequence|legend=1| 15ko, 125f, 140 }}
{{Optimal ET sequence|legend=1| 15ko, 125f, 140 }}


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


== Pentonismic (rank-5) ==
== Pentonismic (rank-5) ==
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Supporting ETs: {{Optimal ET sequence|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}
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 harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an indepedent generator instead, which is exactly what [[blacksmith]] does.  
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}}
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