Olympic clan: Difference between revisions

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The '''olympic clan''' of [[rank-3 temperament]]s tempers out the [[olympia]], 131072/130977 = {{monzo| 17 -5 0 -2 -1 }}.

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 olympian, no-5 subgroup temperament, in addition to baffin and sophia~~.

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

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[[Comma list]]: 131072/130977

~~[[Sval]] [[mapping]]: [~~{{~~val~~| 1 0 0 17 ~~}}, {{val~~| 0 1 0 -5 ~~}}, {{val~~| 0 0 1 -2 }}]

: sval mapping generators: ~2, ~3, ~7

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[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.0805, ~7/4 = 969.0275

{{~~Val list~~|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 × 10-3

=== 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]]

Considered below are orthoschismic, baffin and sophia.

== Orthoschismic ==

[[Subgroup]]: 2.3.5.7.11

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7405, ~7/4 = 969.6950

{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }}

[[Badness]]: 1.18 × 10-3

=== 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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.7333, ~7/4 = 969.7085

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

Badness: 0.833 × 10-3

== Baffin ==

=== 7-limit (decovulture) ===

[[Subgroup]]: 2.3.5.7~~.11~~

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 67108864/66976875

~~[[Mapping]]: [~~{{~~val~~| 1 0 0 13 ~~}}, {{val~~| 0 2 0 -7 ~~}}, {{val~~| 0 0 1 -2 }}]

: mapping generators: ~2, ~8192/4725, ~5

[[Optimal tuning]] ([[POTE]]): ~8192/4725 = 951.0868, ~5/4 = 386.6183

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8192/4725 = 951.0868, ~5/4 = 386.6183

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

{{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

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[[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 }}]

[[Optimal tuning]] ([[POTE]]): ~400/231 = 951.0585, ~5/4 = 386.7912

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~400/231 = 951.0585, ~5/4 = 386.7912

{{~~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 × 10-3

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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 tuning (POTE): ~2 = 1\1, ~26/15 = 951.0882, ~5/4 = 386.7507

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

{{Optimal ET sequence|legend=1| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }}

Badness: 0.604 × 10-3

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[[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 }}]

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.3024, ~256/245 = 77.1952

{{~~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 × 10-3

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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 tuning (POTE): ~2 = 1\1, ~3/2 = 702.3319, ~117/112 = 77.2152

Optimal ~~GPV~~ sequence~~: {{Val list~~| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}

{{Optimal ET sequence|legend=1| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}

Badness: 1.67 × 10-3

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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 tuning (POTE): ~3/2 = 702.3205, ~68/65 = 77.2255

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.3205, ~68/65 = 77.2255

Optimal ~~GPV~~ sequence~~: {{Val list~~| 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: 0.989 × 10-3

[[Category:Temperament clans]]

[[Category:Pages with mostly numerical content]]

[[Category:Olympic clan| ]]

[[Category:Olympic| ]]

[[Category:Rank 3]]

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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 olympian, no-5 subgroup temperament, in addition to baffin and sophia.
 For the rank-4 olympic temperament, see [[Rank-4 temperament #Olympic (131072/130977)]].
@@ Line 15: / Line 9: @@
 [[Comma list]]: 131072/130977
-[[Sval]] [[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 }}
 : sval mapping generators: ~2, ~3, ~7
@@ Line 21: / Line 15: @@
 [[Optimal tuning]] ([[POTE]]): ~3/2 = 702.0805, ~7/4 = 969.0275
-{{Val list|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>
+=== 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]]
+Considered below are orthoschismic, baffin and sophia.
+== Orthoschismic ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 540/539, 32805/32768
+{{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 ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }}
+[[Badness]]: 1.18 × 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 tuning (CTE): ~2 = 1\1, ~3/2 = 701.7333, ~7/4 = 969.7085
+{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}
+Badness: 0.833 × 10<sup>-3</sup>
 == Baffin ==
 === 7-limit (decovulture) ===
-[[Subgroup]]: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 67108864/66976875
-[[Mapping]]: [{{val| 1 0 0 13 }}, {{val| 0 2 0 -7 }}, {{val| 0 0 1 -2 }}]
+{{Mapping|legend=1| 1 0 0 13 | 0 2 0 -7 | 0 0 1 -2 }}
 : mapping generators: ~2, ~8192/4725, ~5
-[[Optimal tuning]] ([[POTE]]): ~8192/4725 = 951.0868, ~5/4 = 386.6183
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8192/4725 = 951.0868, ~5/4 = 386.6183
-{{Val list|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }}
+{{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>
@@ Line 46: / Line 79: @@
 [[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 }}
-[[Optimal tuning]] ([[POTE]]): ~400/231 = 951.0585, ~5/4 = 386.7912
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~400/231 = 951.0585, ~5/4 = 386.7912
-{{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>
@@ Line 59: / Line 92: @@
 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 tuning (POTE): ~2 = 1\1, ~26/15 = 951.0882, ~5/4 = 386.7507
-Optimal GPV sequence: {{Val list| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }}
+{{Optimal ET sequence|legend=1| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }}
 Badness: 0.604 × 10<sup>-3</sup>
@@ Line 74: / Line 107: @@
 [[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 }}
-[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.3024, ~256/245 = 77.1952
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.3024, ~256/245 = 77.1952
-{{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>
@@ Line 87: / Line 120: @@
 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 tuning (POTE): ~2 = 1\1, ~3/2 = 702.3319, ~117/112 = 77.2152
-Optimal GPV sequence: {{Val list| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}
+{{Optimal ET sequence|legend=1| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}
 Badness: 1.67 × 10<sup>-3</sup>
@@ Line 100: / Line 133: @@
 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 tuning (POTE): ~3/2 = 702.3205, ~68/65 = 77.2255
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.3205, ~68/65 = 77.2255
-Optimal GPV sequence: {{Val list| 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: 0.989 × 10<sup>-3</sup>
 [[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Olympic clan| ]] <!-- main article -->
+[[Category:Olympic| ]] <!-- key article -->
 [[Category:Rank 3]]