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


{{Mapping|legend=2| 1 0 0 17 | 0 1 0 -5 | 0 0 1 -2 }}
{{Mapping|legend=3| 1 0 0 17 | 0 1 0 -5 | 0 0 1 -2 }}
: mapping generators: ~2, ~3, ~7
: mapping generators: ~2, ~3, ~7


Line 23: Line 23:


=== Overview to extensions ===
=== 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]].  
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) → [[Hemifamity family #Akea|Hemifamity family]]
* ''[[Akea]]'' (+385/384) → [[Aberschismic family #Akea|Aberschismic 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]]
* ''[[Lif]]'' (+2401/2400) → [[Breed family #Lif|Breed family]]
* ''[[Lux]]'' (+3025/3024) → [[Lehmerismic temperaments #Lux|Lehmerismic temperaments]]
* ''[[Baffin]]'' (+5632/5625) → [[Vishdelismic clan #Baffin|Vishdelismic clan]]
* ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]]
* ''[[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 orthoschismic, baffin and sophia.
Considered below are [[#Pessoal|pessoal]], [[#Lux|lux]], and [[#Sophia|sophia]].
 
== 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]].
== Pessoal ==
{{See also| Pessoalisma }}


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


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 32805/32768
[[Comma list]]: 9801/9800, 131072/130977


{{Mapping|legend=1| 1 0 15 0 17 | 0 1 -8 0 -5 | 0 0 0 1 -2 }}
{{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:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9335{{c}}, ~3/2 = 701.6817{{c}}, ~7/4 = 969.6199{{c}}
* [[WE]]: ~99/70 = 599.9711{{c}}, ~3/2 = 702.0265{{c}}, ~32/21 = 728.7942{{c}}
: [[error map]]: {{val| -0.067 -0.340 -0.233 +0.661 +0.502 }}
: [[error map]]: {{val| -0.058 +0.014 +0.040 +0.122 -0.040 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7248{{c}}, ~7/4 = 969.6728{{c}}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0635{{c}}, ~32/21 = 728.8214{{c}}
: error map: {{val| 0.000 -0.230 -0.112 +0.847 +0.713 }}
: error map: {{val| 0.000 +0.109 +0.150 +0.289 +0.134 }}


{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }}
{{Optimal ET sequence|legend=1| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde }}


[[Badness]] (Sintel): 1.41
[[Badness]] (Sintel): 0.599


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


Comma list: 540/539, 729/728, 4096/4095
Comma list: 1716/1715, 2080/2079, 4096/4095


Mapping: {{mapping| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}
{{Mapping|legend=0| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9399{{c}}, ~3/2 = 701.6899{{c}}, ~7/4 = 969.6107{{c}}
* WE: ~99/70 = 599.9835{{c}}, ~3/2 = 702.0253{{c}}, ~32/21 = 728.7700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7264{{c}}, ~7/4 = 969.6672{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0477{{c}}, ~32/21 = 728.7882{{c}}
 
{{Optimal ET sequence|legend=0| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}


Badness (Sintel): 0.779
{{Optimal ET sequence|legend=0| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d }}


== Baffin ==
Badness (Sintel): 0.366
: ''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.  
== 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]].  


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>.  
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.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 5632/5625, 131072/130977
[[Comma list]]: 3025/3024, 131072/130977


{{Mapping|legend=1| 1 0 0 13 -9 | 0 2 0 -7 4 | 0 0 1 -2 4 }}
{{Mapping|legend=1| 1 0 -5 4 9 | 0 1 4 -1 -3 | 0 0 5 2 -4 }}
: mapping generators: ~2, ~400/231, ~5
: mapping generators: ~2, ~3, ~55/48


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9208{{c}}, ~400/231 = 950.9957{{c}}, ~5/4 = 386.7656{{c}}
* [[WE]]: ~2 = 1199.9605{{c}}, ~3/2 = 702.0886{{c}}, ~55/48 = 235.5706{{c}}
: [[error map]]: {{val| -0.079 +0.036 +0.293 -0.040 -0.193 }}
: [[error map]]: {{val| -0.039 +0.094 -0.067 +0.108 -0.103 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.0635{{c}}, ~5/4 = 386.7769{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1120{{c}}, ~55/48 = 235.5763{{c}}
: error map: {{val| 0.000 +0.172 +0.463 +0.176 +0.044 }}
: error map: {{val| 0.000 +0.157 +0.016 +0.215 +0.041 }}


{{Optimal ET sequence|legend=1| 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c }}
{{Optimal ET sequence|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}


[[Badness]] (Sintel): 1.17
[[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: {{mapping| 1 0 0 13 -9 1 | 0 2 0 -7 4 3 | 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 tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9633{{c}}, ~400/231 = 951.0591{{c}}, ~5/4 = 386.7388{{c}}
* WE: ~2 = 1199.9727{{c}}, ~3/2 = 702.0946{{c}}, ~55/48 = 235.5597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.0896{{c}}, ~5/4 = 386.7451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1109{{c}}, ~55/48 = 235.5649{{c}}


{{Optimal ET sequence|legend=0| 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 (Sintel): 0.565
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 ==
== Sophia ==
Line 139: Line 139:
Comma list: 2080/2079, 4096/4095, 13720/13689
Comma list: 2080/2079, 4096/4095, 13720/13689


Mapping: {{mapping| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 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 }}


Optimal tunings:  
Optimal tunings:  
Line 154: Line 154:
Comma list: 595/594, 833/832, 1156/1155, 4096/4095
Comma list: 595/594, 833/832, 1156/1155, 4096/4095


Mapping: {{mapping| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }}
{{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:  
Optimal tunings:  
Line 163: Line 163:


Badness (Sintel): 0.940
Badness (Sintel): 0.940
== References ==


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