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The '''olympic clan''' of [[rank-3 temperament]]s tempers out the [[olympia]], 131072/130977 = {{monzo| 17 -5 0 -2 -1 }}.  
{{Technical data page}}
The '''olympic clan''' of [[rank-3 temperament|rank-3]] [[temperament]]s [[tempering out|tempers out]] the [[olympia]] ([[ratio]]: 131072/130977, {{monzo|legend=1| 17 -5 0 -2 -1 }}). This has the effect of equating the [[33/32|undecimal quartertone (33/32)]] with a stack of two [[64/63|septimal commas (64/63)]].
 
For the rank-4 olympic temperament, see [[Rank-4 temperament #Olympic (131072/130977)]].
 
== Olympian ==
[[Subgroup]]: 2.3.7.11
 
[[Comma list]]: 131072/130977
 
{{Mapping|legend=3| 1 0 0 17 | 0 1 0 -5 | 0 0 1 -2 }}
: mapping generators: ~2, ~3, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9460{{c}}, ~3/2 = 702.0489{{c}}, ~7/4 = 968.9839{{c}}
: [[error map]]: {{val| -0.054 +0.040 +0.050 +0.038 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0853{{c}}, ~7/4 = 969.0242{{c}}
: error map: {{val| 0.000 +0.130 +0.198 +0.207 }}
 
{{Optimal ET sequence|legend=1| 41, 87, 89, 94, 130, 135, 359, 400, 494, 535, 670, 805, 1164, 1299, 1834, 1969, 5102bde, 5237bde, 7206bddee, 10339bbdddeee }}
 
[[Badness]] (Sintel): 0.267
 
=== Overview to extensions ===
The second comma in the comma list determines how we extend olympian to include prime 5. Akea adds [[385/384]], and finds prime 5 by equating the [[81/80|syntonic comma (81/80)]] with the septimal comma. Orthoschismic adds [[32805/32768]], and finds prime 5 on the chain of fifths. Cassaschismic adds [[19712/19683]], giving prime 5 an independent generator.
 
Pessoal adds [[9801/9800]], splitting the octave into two. Lif adds [[2401/2400]], splitting the perfect fifth into two. Baffin adds [[5632/5625]], splitting the perfect twelfth into two. Lux adds [[3025/3024]], splitting the ~21/16 into two. Hera adds [[6144/6125]] or [[8019/8000]], splitting the ~21/16 into three. Sophia adds [[42875/42768]], splitting the ~8/7 into three. Finally, raglympic adds [[4375/4374]], splitting the ~27/7 into four.
 
All of them have neat extensions to the 13-limit via tempering out both [[2080/2079]] and [[4096/4095]].  


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Akea]]'', {385/384, 2200/2187} → [[Hemifamity family #Akea|Hemifamity family]]
* ''[[Akea]]'' (+385/384) → [[Aberschismic family #Akea|Aberschismic family]]
* ''[[Lux]]'', {3025/3024, 130172/130977} → [[Lehmerismic temperaments #Lux|Lehmerismic temperaments]]
* ''[[Lif]]'' (+2401/2400) → [[Breed family #Lif|Breed family]]
* ''[[Baffin]]'' (+5632/5625) → [[Vishdelismic clan #Baffin|Vishdelismic clan]]
* ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]]
* ''[[Raglympic]]'' (+4375/4374) → [[Ragismic family #Raglympic|Ragismic family]]
* ''[[Orthoschismic]]'' (+540/539) → [[Schismic rank-3 family #Orthoschismic|Schismic rank-3 family]]
* [[Cassaschismic]] (+19712/19683) → [[Septischismic family #Undecimal septischismic (cassaschismic)|Septischismic family]]


Considered below are baffin and sophia, in addition to the no-5 subgroup temperament, but we can begin by looking at the rank-4 temperament.  
Considered below are [[#Pessoal|pessoal]], [[#Lux|lux]], and [[#Sophia|sophia]].
 
== Pessoal ==
{{See also| Pessoalisma }}
 
Pessoal tempers out the [[kalisma]]. It was named by [[Aura]] in 2023, meaning [[Wiktionary: pessoal #Portuguese|"personal"]], for the fact that it is associated with [[abigail]], which is in turn a person's name.  


== Baffin ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 5632/5625, 131072/130977
[[Comma list]]: 9801/9800, 131072/130977
 
{{Mapping|legend=1| 2 0 1 10 14 | 0 1 0 -1 -3 | 0 0 3 -1 2 }}
: mapping generators: ~99/70, ~3, ~32/21
 
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 599.9711{{c}}, ~3/2 = 702.0265{{c}}, ~32/21 = 728.7942{{c}}
: [[error map]]: {{val| -0.058 +0.014 +0.040 +0.122 -0.040 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0635{{c}}, ~32/21 = 728.8214{{c}}
: error map: {{val| 0.000 +0.109 +0.150 +0.289 +0.134 }}
 
{{Optimal ET sequence|legend=1| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde }}
 
[[Badness]] (Sintel): 0.599
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 4096/4095
 
{{Mapping|legend=0| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9835{{c}}, ~3/2 = 702.0253{{c}}, ~32/21 = 728.7700{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0477{{c}}, ~32/21 = 728.7882{{c}}
 
{{Optimal ET sequence|legend=0| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d }}
 
Badness (Sintel): 0.366
 
== Lux ==
The last generator of lux, represented by [[55/48]], exceeds [[8/7]] by [[385/384]], which is equated with a number of important [[superparticular ratio]]s in the [[13-limit]]: [[325/324]], [[352/351]], [[364/363]], and [[441/440]].


[[Mapping]]: [{{val| 1 0 0 13 -9 }}, {{val| 0 2 0 -7 4 }}, {{val| 0 0 1 -2 4 }}]
This ultra-efficient full 13-limit temperament was first considered by [[Flora Canou]] in 2021. After a few attempts to come up with a memorable name, ''lux'', suggested by [[Godtone]], who associated certain intervals of 13 with light, was adopted.


Mapping generators: ~2, ~400/231, ~5
[[Subgroup]]: 2.3.5.7.11


Optimal tuning ([[POTE]]): ~400/231 = 951.0585, ~5/4 = 386.7912
[[Comma list]]: 3025/3024, 131072/130977


{{Val list|legend=1| 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210 }}
{{Mapping|legend=1| 1 0 -5 4 9 | 0 1 4 -1 -3 | 0 0 5 2 -4 }}
: mapping generators: ~2, ~3, ~55/48


[[Badness]]: 0.976 × 10<sup>-3</sup>
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9605{{c}}, ~3/2 = 702.0886{{c}}, ~55/48 = 235.5706{{c}}
: [[error map]]: {{val| -0.039 +0.094 -0.067 +0.108 -0.103 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1120{{c}}, ~55/48 = 235.5763{{c}}
: error map: {{val| 0.000 +0.157 +0.016 +0.215 +0.041 }}
 
{{Optimal ET sequence|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}
 
[[Badness]] (Sintel): 0.611


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 4096/4095
Comma list: 2080/2079, 3025/3024, 4096/4095


Mapping: [{{val| 1 0 0 13 -9 1 }}, {{val| 0 2 0 -7 4 3 }}, {{val| 0 0 1 -2 4 1 }}]
{{Mapping|legend=0| 1 0 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}


Optimal tuning (POTE): ~26/15 = 951.0882, ~5/4 = 386.7507
Optimal tunings:  
* WE: ~2 = 1199.9727{{c}}, ~3/2 = 702.0946{{c}}, ~55/48 = 235.5597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1109{{c}}, ~55/48 = 235.5649{{c}}


Optimal GPV sequence: {{Val list| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }}
{{Optimal ET sequence|legend=0| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}


Badness: 0.604 × 10<sup>-3</sup>
Badness (Sintel): 0.337


Complexity spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11
== Sophia ==
Named by [[Scott Dakota]] in 2022, sophia tempers out [[42875/42768]], and by virtue of the identity 42875/42768 = ([[595/594]])⋅([[1225/1224]]), it is naturally a [[17-limit]] temperament.


== Sophia ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 42875/42768, 131072/130977
[[Comma list]]: 42875/42768, 131072/130977


[[Mapping]]: [{{val| 1 0 2 3 11 }}, {{val| 0 1 0 0 -5 }}, {{val| 0 0 5 -3 6 }}]
{{Mapping|legend=1| 1 0 2 3 11 | 0 1 0 0 -5 | 0 0 5 -3 6 }}
: mapping generators: ~2, ~3, ~256/245


Mapping generators: ~2, ~3, ~256/245
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9947{{c}}, ~3/2 = 702.2994{{c}}, ~256/245 = 77.1949{{c}}
: [[error map]]: {{val| -0.005 +0.339 -0.350 -0.426 +0.323 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3025{{c}}, ~256/245 = 77.1949{{c}}
: error map: {{val| 0.000 +0.347 -0.339 -0.411 +0.339 }}


Optimal tuning ([[POTE]]): ~3/2 = 702.3024, ~256/245 = 77.1952
{{Optimal ET sequence|legend=1| 46, 94, 140, 171, 217, 311, 979, 1290 }}


{{Val list|legend=1| 46, 94, 140, 171, 217, 311, 979, 1290 }}
[[Badness]] (Sintel): 4.54
 
[[Badness]]: 3.78 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 57: Line 139:
Comma list: 2080/2079, 4096/4095, 13720/13689
Comma list: 2080/2079, 4096/4095, 13720/13689


Mapping: [{{val| 1 0 2 3 11 7 }}, {{val| 0 1 0 0 -5 -2 }}, {{val| 0 0 5 -3 6 -2 }}]
{{Mapping|legend=0| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }}


Mapping generators: ~2, ~3, ~256/245
Optimal tunings:
* WE: ~2 = 1199.9732{{c}}, ~3/2 = 702.3162{{c}}, ~117/112 = 77.2135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3335{{c}}, ~117/112 = 77.2145{{c}}


Optimal tuning (POTE): ~3/2 = 702.3319, ~256/245 = 77.2152
{{Optimal ET sequence|legend=0| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}


Optimal GPV sequence: {{Val list| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}
Badness (Sintel): 1.56


Badness: 1.67 × 10<sup>-3</sup>
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Comma list: 595/594, 833/832, 1156/1155, 4096/4095
{{Mapping|legend=0| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }}
Optimal tunings:
* WE: ~2 = 1199.9956{{c}}, ~3/2 = 702.3179{{c}}, ~68/65 = 77.2252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3207{{c}}, ~68/65 = 77.2254{{c}}
{{Optimal ET sequence|legend=0| 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g }}
Badness (Sintel): 0.940
[[Category:Olympic clan| ]] <!-- main article -->
[[Category:Olympic| ]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Olympic clan| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Olympic_clan"