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Olympic clan: Difference between revisions

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The '''olympic clan''' of [[rank-3 temperament|rank-3]] [[temperament]]s [[tempering out|tempers out]] the [[olympia]], 131072/130977 = {{monzo| 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)]].  
{{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)]].  
For the rank-4 olympic temperament, see [[Rank-4 temperament #Olympic (131072/130977)]].  
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{{Mapping|legend=2| 1 0 0 17 | 0 1 0 -5 | 0 0 1 -2 }}
{{Mapping|legend=2| 1 0 0 17 | 0 1 0 -5 | 0 0 1 -2 }}
: mapping generators: ~2, ~3, ~7


: sval mapping generators: ~2, ~3, ~7
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.9460{{c}}, ~3/2 = 702.0489{{c}}, ~7/4 = 968.9839{{c}}
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.0805, ~7/4 = 969.0275
: [[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 }}
{{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]]: 0.0183 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.267


=== Overview to extensions ===
=== Overview to extensions ===
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Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Akea]]'' (+385/384) → [[Hemifamity family #Akea|Hemifamity family]]
* ''[[Akea]]'' (+385/384) → [[Hemifamity family #Akea|Hemifamity family]]
* [[Cassaschismic]] (+19712/19683) → [[Garischismic family #Cassaschismic|Garischismic family]]
* ''[[Pessoal]]'' (+9801/9800) → [[Kalismic temperaments #Pessoal|Kalismic temperaments]]
* ''[[Pessoal]]'' (+9801/9800) → [[Kalismic temperaments #Pessoal|Kalismic temperaments]]
* ''[[Lif]]'' (+2401/2400) → [[Breed family #Lif|Breed family]]
* ''[[Lif]]'' (+2401/2400) → [[Breed family #Lif|Breed family]]
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* ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]]
* ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]]


Considered below are orthoschismic, cassaschismic, baffin and sophia.
Considered below are orthoschismic, baffin and sophia.


== Orthoschismic ==
== 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.
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]].
It was named by [[Flora Canou]] in 2023, using the Greek prefix [[wiktionary: ortho- #English|ortho-]] to signify the additional rank.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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{{Mapping|legend=1| 1 0 15 0 17 | 0 1 -8 0 -5 | 0 0 0 1 -2 }}
{{Mapping|legend=1| 1 0 15 0 17 | 0 1 -8 0 -5 | 0 0 0 1 -2 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7405, ~7/4 = 969.6950
[[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 ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }}
{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }}


[[Badness]]: 1.18 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.41


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}
Mapping: {{mapping| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7333, ~7/4 = 969.7085
Optimal tunings:
 
* WE: ~2 = 1199.9399{{c}}, ~3/2 = 701.6899{{c}}, ~7/4 = 969.6107{{c}}
{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7264{{c}}, ~7/4 = 969.6672{{c}}
 
Badness: 0.833 × 10<sup>-3</sup>
 
== Cassaschismic ==
{{See also| Garischismic family #Cassaschismic }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 19712/19683, 41503/41472
 
{{Mapping|legend=1| 1 0 0 25 -33 | 0 1 0 -14 23 | 0 0 1 0 0 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2280, ~5/4 = 386.3137
 
{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 1121 }}
 
[[Badness]]: 1.41 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 4096/4095, 19712/19683
 
Mapping: {{mapping| 1 0 0 25 -33 -13 | 0 1 0 -14 23 12 | 0 0 1 0 0 -1 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2289, ~5/4 = 386.2869
 
{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851, 2283b }}
 
Badness: 0.872 × 10<sup>-3</sup>
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079
 
Sval mapping: {{mapping| 1 0 0 25 -33 -13 -6 | 0 1 0 -14 23 12 5 | 0 0 1 0 0 -1 1 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2293, ~5/4 = 386.3021


{{Optimal ET sequence|legend=1| 41, 53, 94, 176, 217, 270, 581, 851 }}
{{Optimal ET sequence|legend=0| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}


Badness: 0.496 × 10<sup>-3</sup>
Badness (Sintel): 0.779


== Baffin ==
== Baffin ==
=== 7-limit (decovulture) ===
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Decovulture]].''
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 67108864/66976875
 
{{Mapping|legend=1| 1 0 0 13 | 0 2 0 -7 | 0 0 1 -2 }}
 
: mapping generators: ~2, ~8192/4725, ~5
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8192/4725 = 951.0868, ~5/4 = 386.6183


{{Optimal ET sequence|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }}
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.


[[Badness]]: 0.865 × 10<sup>-3</sup>
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


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{{Mapping|legend=1| 1 0 0 13 -9 | 0 2 0 -7 4 | 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]]): ~2 = 1\1, ~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 }}


{{Optimal ET sequence|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 ===
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Mapping: {{mapping| 1 0 0 13 -9 1 | 0 2 0 -7 4 3 | 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): ~2 = 1\1, ~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 ET sequence|legend=1| 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


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{{Mapping|legend=1| 1 0 2 3 11 | 0 1 0 0 -5 | 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]]): ~2 = 1\1, ~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 }}


{{Optimal ET sequence|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 ===
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Mapping: {{mapping| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 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): ~2 = 1\1, ~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 ET sequence|legend=1| 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 ===
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Mapping: {{mapping| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 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 tuning (POTE): ~2 = 1\1, ~3/2 = 702.3205, ~68/65 = 77.2255
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 }}


{{Optimal ET sequence|legend=1| 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:Olympic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Olympic_clan"