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5th-octave temperaments: Difference between revisions

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Navbox fractional octave is supposed to exist alongside fractional-octave navigation, not replace it. One is a detailed navbox at the bottom, the other a quick minimalist infobox in the top right. Navigation also automatically grants categories which is important, without it the page becomes uncategorised (yes, the navbox could be made to do that, but that’s not the intention, it’s not meant as a replacement).
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→Quint: move to very low accuracy temperaments
 
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{{Fractional-octave navigation|5}}[[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.
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 independent 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
[[Comma list]]: 4194304/4084101


{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
{{Mapping|legend=1| 5 0 22 | 0 1 -1 }}


: Mapping generators: ~9/8, ~7
: Mapping generators: ~512/441, ~3


{{Multival|legend=1| 0 0 5 0 8 12 }}
[[Support]]ing [[ET]]s: {{EDOs|5, 65d, 70, 75, 80, 85, 90, 95}}
 
[[Optimal tuning]]s:
* [[POTE]]: ~9/8 = 1\5, ~7/4 = 1017.903 (~21/20 = 57.903)
* [[CTE]]: ~9/8 = 1\5, ~7/4 = 968.8259 (~63/64 = 8.8259)
 
{{Optimal ET sequence|legend=1| 5, 15ccd }}


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}
[[Badness]]: 0.048312
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