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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)]].  
Temperaments discussed elsewhere are:
* ''[[Akea]]'', {385/384, 2200/2187} → [[Hemifamity family #Akea|Hemifamity family]]
* ''[[Lux]]'', {3025/3024, 131072/130977} → [[Lehmerismic temperaments #Lux|Lehmerismic temperaments]]
* ''[[Hera]]'', {6144/6125, 8019/8000} → [[Porwell family #Hera|Porwell family]]
 
Considered below are baffin and sophia, in addition to the no-5 subgroup temperament.  


For the rank-4 olympic temperament, see [[Rank-4 temperament #Olympic (131072/130977)]].  
For the rank-4 olympic temperament, see [[Rank-4 temperament #Olympic (131072/130977)]].  


== No-5 olympic ==
== Olympian ==
[[Subgroup]]: 2.3.7.11
[[Subgroup]]: 2.3.7.11


[[Comma list]]: 131072/130977
[[Comma list]]: 131072/130977


[[Mapping]]: [{{val| 1 0 0 17 }}, {{val| 0 1 0 -5 }}, {{val| 0 0 1 -2 }}]
{{Mapping|legend=2| 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 the harmonic 5. Akea adds [[385/384]], and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the septimal comma. Orthoschismic adds [[32805/32768]], and finds the harmonic 5 on the chain of fifths. Cassaschismic adds [[19712/19683]] with an independent generator for harmonic 5. 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. Finally, sophia adds 42875/42768, splitting the ~8/7 into three. These all have neat extensions to the 13-limit via tempering out both [[2080/2079]] and [[4096/4095]].
 
Temperaments discussed elsewhere are:
* ''[[Akea]]'' (+385/384) → [[Hemifamity family #Akea|Hemifamity family]]
* [[Cassaschismic]] (+19712/19683) → [[Garischismic family #Cassaschismic|Garischismic family]]
* ''[[Pessoal]]'' (+9801/9800) → [[Kalismic temperaments #Pessoal|Kalismic temperaments]]
* ''[[Lif]]'' (+2401/2400) → [[Breed family #Lif|Breed family]]
* ''[[Lux]]'' (+3025/3024) → [[Lehmerismic temperaments #Lux|Lehmerismic temperaments]]
* ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]]


Mapping generators: ~2, ~3, ~7
Considered below are orthoschismic, baffin and sophia.


Optimal tuning ([[POTE]]): ~3/2 = 702.0805, ~7/4 = 969.0275
== Orthoschismic ==
Orthoschismic is related to [[schismic]], but with an additional generator for [[prime interval|prime]] [[7/1|7]], extended to the 11-limit in one of the most efficient ways possible.  


{{Val list|legend=1| 41, 87, 89, 94, 130, 135, 359, 400, 494, 535, 670, 805, 1164, 1299, 1834, 1969, 5102bde, 5237bde, 7206bddee, 10339bbdddeee }}
Orthoschismic can be notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in [[olympic]]), and the other for the generic aberschisma step which stands in for the [[garischisma]] and the [[aberschisma]].


[[Badness]]: 0.0183 × 10<sup>-3</sup>
It was named by [[Flora Canou]] in 2023, using the Greek prefix [[wiktionary: ortho- #English|ortho-]] to signify the additional rank.  


== Baffin ==
=== 7-limit (decovulture) ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 67108864/66976875
[[Comma list]]: 540/539, 32805/32768


[[Mapping]]: [{{val| 1 0 0 13 }}, {{val| 0 2 0 -7 }}, {{val| 0 0 1 -2 }}]
{{Mapping|legend=1| 1 0 15 0 17 | 0 1 -8 0 -5 | 0 0 0 1 -2 }}


Mapping generators: ~2, ~8192/4725, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9335{{c}}, ~3/2 = 701.6817{{c}}, ~7/4 = 969.6199{{c}}
: [[error map]]: {{val| -0.067 -0.340 -0.233 +0.661 +0.502 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7248{{c}}, ~7/4 = 969.6728{{c}}
: error map: {{val| 0.000 -0.230 -0.112 +0.847 +0.713 }}


Optimal tuning ([[POTE]]): ~8192/4725 = 951.0868, ~5/4 = 386.6183
{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }}


{{Val list|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }}
[[Badness]] (Sintel): 1.41


[[Badness]]: 0.865 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 4096/4095
 
Mapping: {{mapping| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}
 
Optimal tunings:
* WE: ~2 = 1199.9399{{c}}, ~3/2 = 701.6899{{c}}, ~7/4 = 969.6107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7264{{c}}, ~7/4 = 969.6672{{c}}
 
{{Optimal ET sequence|legend=0| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}
 
Badness (Sintel): 0.779
 
== Baffin ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Decovulture]].''
 
Baffin tempers out [[5632/5625]] and [[160083/160000]] in addition to the olympia, so that the [[3/1|3rd]] [[harmonic]] is split in halves. This leads naturally to a [[13-limit]] temperament where [[676/675]], [[1001/1000]], [[2080/2079]] and [[4096/4095]] are all tempered out.
 
It was named by [[Gene Ward Smith]] some time before 2011 for {{w|Baffin Island}}, as part of a series of island-themed names<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96550.html#96556 Yahoo! Tuning Group | ''Madagascar<nowiki>[</nowiki>19<nowiki>]</nowiki>'']</ref>.


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


[[Comma list]]: 5632/5625, 131072/130977
[[Comma list]]: 5632/5625, 131072/130977


[[Mapping]]: [{{val| 1 0 0 13 -9 }}, {{val| 0 2 0 -7 4 }}, {{val| 0 0 1 -2 4 }}]
{{Mapping|legend=1| 1 0 0 13 -9 | 0 2 0 -7 4 | 0 0 1 -2 4 }}
: mapping generators: ~2, ~400/231, ~5


Optimal tuning ([[POTE]]): ~400/231 = 951.0585, ~5/4 = 386.7912
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9208{{c}}, ~400/231 = 950.9957{{c}}, ~5/4 = 386.7656{{c}}
: [[error map]]: {{val| -0.079 +0.036 +0.293 -0.040 -0.193 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.0635{{c}}, ~5/4 = 386.7769{{c}}
: error map: {{val| 0.000 +0.172 +0.463 +0.176 +0.044 }}


{{Val list|legend=1| 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c }}
{{Optimal ET sequence|legend=1| 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c }}


[[Badness]]: 0.976 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.17


=== 13-limit ===
=== 13-limit ===
Line 59: Line 102:
Comma list: 676/675, 1001/1000, 4096/4095
Comma list: 676/675, 1001/1000, 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: {{mapping| 1 0 0 13 -9 1 | 0 2 0 -7 4 3 | 0 0 1 -2 4 1 }}


Optimal tuning (POTE): ~26/15 = 951.0882, ~5/4 = 386.7507
Optimal tunings:  
* WE: ~2 = 1199.9633{{c}}, ~400/231 = 951.0591{{c}}, ~5/4 = 386.7388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.0896{{c}}, ~5/4 = 386.7451{{c}}


Optimal GPV sequence: {{Val list| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }}
{{Optimal ET sequence|legend=0| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }}


Badness: 0.604 × 10<sup>-3</sup>
Badness (Sintel): 0.565


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
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 ==
== 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.
[[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


Optimal tuning ([[POTE]]): ~3/2 = 702.3024, ~256/245 = 77.1952
[[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 }}


{{Val list|legend=1| 46, 94, 140, 171, 217, 311, 979, 1290 }}
{{Optimal ET sequence|legend=1| 46, 94, 140, 171, 217, 311, 979, 1290 }}


[[Badness]]: 3.78 × 10<sup>-3</sup>
[[Badness]] (Sintel): 4.54


=== 13-limit ===
=== 13-limit ===
Line 87: 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: {{mapping| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }}


Optimal tuning (POTE): ~3/2 = 702.3319, ~117/112 = 77.2152
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 GPV sequence: {{Val list| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}
{{Optimal ET sequence|legend=0| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}


Badness: 1.67 × 10<sup>-3</sup>
Badness (Sintel): 1.56


=== 17-limit ===
=== 17-limit ===
Line 100: Line 154:
Comma list: 595/594, 833/832, 1156/1155, 4096/4095
Comma list: 595/594, 833/832, 1156/1155, 4096/4095


Mapping: [{{val| 1 0 2 3 11 7 7 }}, {{val| 0 1 0 0 -5 -2 -2 }}, {{val| 0 0 5 -3 6 -2 4 }}]
Mapping: {{mapping| 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 tuning (POTE): ~3/2 = 702.3205, ~68/65 = 77.2255
{{Optimal ET sequence|legend=0| 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g }}


Optimal GPV sequence: {{Val list| 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g }}
Badness (Sintel): 0.940


Badness: 0.989 × 10<sup>-3</sup>
== References ==


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Olympic clan| ]] <!-- main article -->
[[Category:Olympic clan| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Olympic_clan"