Olympic clan: Difference between revisions

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: 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 }}

[[Badness]]: 0.~~0183 × 10-3~~

[[Badness]] (Sintel): 0.267

=== Overview to extensions ===

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Temperaments discussed elsewhere are:

* ''[[Akea]]'' (+385/384) → [[Hemifamity family #Akea|Hemifamity family]]

* ''[[Cassaschismic]]'' (+19712/19683) → [[Garischismic family #Cassaschismic|Garischismic 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]]

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== 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

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[[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 }}

[[Badness]]: 1.~~18 × 10-3~~

[[Badness]] (Sintel): 1.41

=== 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 }}

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}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7264{{c}}, ~7/4 = 969.6672{{c}}

{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}

{{Optimal ET sequence|legend=0| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}

Badness: 0.~~833 × 10-3~~

Badness (Sintel): 0.779

== 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

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.

[[~~Optimal tuning~~]] ([[~~POTE~~]]~~): ~2 = 1\1, ~8192/4725 = 951.0868, ~5~~/~~4 = 386~~.~~6183~~

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>.

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

~~[[Badness]]: 0.865 × 10-3~~

~~=== 11-limit ===~~

[[Subgroup]]: 2.3.5.7.11

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: 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 }}

[[Badness]]: 0.~~976 × 10-3~~

[[Badness]] (Sintel): 1.17

=== 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 }}

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 × 10-3~~

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

== 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

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: 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 }}

[[Badness]]: 3.~~78 × 10-3~~

[[Badness]] (Sintel): 4.54

=== 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 }}

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 × 10-3~~

Badness (Sintel): 1.56

=== 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 }}

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 × 10-3~~

== References ==

[[Category:Temperament clans]]

[[Category:Olympic clan| ]]

~~[[Category:Olympic| ]] ~~

[[Category:Rank 3]]

@@ Line 10: / Line 10: @@
 {{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 }}
-[[Badness]]: 0.0183 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.267
 === Overview to extensions ===
@@ Line 24: / Line 27: @@
 Temperaments discussed elsewhere are:
 * ''[[Akea]]'' (+385/384) → [[Hemifamity family #Akea|Hemifamity family]]
-* ''[[Cassaschismic]]'' (+19712/19683) → [[Garischismic family #Cassaschismic|Garischismic 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]]
@@ Line 33: / Line 36: @@
 == 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
@@ Line 39: / Line 48: @@
 {{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 }}
-[[Badness]]: 1.18 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 1.41
 === 13-limit ===
@@ Line 52: / Line 65: @@
 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}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7264{{c}}, ~7/4 = 969.6672{{c}}
-{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}
+{{Optimal ET sequence|legend=0| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}
-Badness: 0.833 × 10<sup>-3</sup>
+Badness (Sintel): 0.779
 == 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
+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.
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8192/4725 = 951.0868, ~5/4 = 386.6183
+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>.
-{{Optimal ET sequence|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }}
-[[Badness]]: 0.865 × 10<sup>-3</sup>
-=== 11-limit ===
 [[Subgroup]]: 2.3.5.7.11
@@ Line 80: / Line 85: @@
 {{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 }}
-[[Badness]]: 0.976 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 1.17
 === 13-limit ===
@@ Line 94: / Line 104: @@
 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
 == 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
@@ Line 108: / Line 122: @@
 {{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 }}
-[[Badness]]: 3.78 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 4.54
 === 13-limit ===
@@ Line 122: / Line 141: @@
 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 ===
@@ Line 135: / Line 156: @@
 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:Olympic clan| ]] <!-- main article -->
-[[Category:Olympic| ]] <!-- key article -->
 [[Category:Rank 3]]