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 }}. This has the effect of equating the [[33/32|undecimal quartertone (33/32)]] with a stack of two [[64/63|septimal commas (64/63)]].  
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)]].  


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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[[Comma list]]: 131072/130977
[[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
: sval mapping generators: ~2, ~3, ~7
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Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Akea]]'' → [[Hemifamity family #Akea|Hemifamity family]]
* ''[[Akea]]'' (+385/384) → [[Hemifamity family #Akea|Hemifamity family]]
* ''[[Pessoal]]'' → [[Kalismic temperaments #Pessoal|Kalismic temperaments]]
* ''[[Pessoal]]'' (+9801/9800) → [[Kalismic temperaments #Pessoal|Kalismic temperaments]]
* ''[[Lux]]'' → [[Lehmerismic temperaments #Lux|Lehmerismic temperaments]]
* ''[[Lux]]'' (+3025/3024) → [[Lehmerismic temperaments #Lux|Lehmerismic temperaments]]
* ''[[Hera]]'' → [[Porwell family #Hera|Porwell family]]
* ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]]


Considered below are baffin and sophia.
Considered below are baffin and sophia.
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[[Comma list]]: 67108864/66976875
[[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
: mapping generators: ~2, ~8192/4725, ~5
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[[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 }}


[[Optimal tuning]] ([[POTE]]): ~400/231 = 951.0585, ~5/4 = 386.7912
[[Optimal tuning]] ([[POTE]]): ~400/231 = 951.0585, ~5/4 = 386.7912
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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 tuning (POTE): ~26/15 = 951.0882, ~5/4 = 386.7507
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[[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 }}


[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.3024, ~256/245 = 77.1952
[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.3024, ~256/245 = 77.1952
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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 tuning (POTE): ~3/2 = 702.3319, ~117/112 = 77.2152
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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 tuning (POTE): ~3/2 = 702.3205, ~68/65 = 77.2255
Optimal tuning (POTE): ~3/2 = 702.3205, ~68/65 = 77.2255
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[[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]]

Revision as of 08:08, 14 September 2023

The olympic clan of rank-3 temperaments tempers out the olympia, 131072/130977 = [17 -5 0 -2 -1⟩. This has the effect of equating the undecimal quartertone (33/32) with a stack of two septimal commas (64/63).

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

Olympian

Subgroup: 2.3.7.11

Comma list: 131072/130977

Subgroup-val mapping: [⟨1 0 0 17], ⟨0 1 0 -5], ⟨0 0 1 -2]]

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

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

Optimal ET sequence: 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 syntonic comma (81/80) with the septimal comma. Pessoal adds 9801/9800, splitting the octave 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:

Considered below are baffin and sophia.

Baffin

7-limit (decovulture)

Subgroup: 2.3.5.7

Comma list: 67108864/66976875

Mapping: [⟨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 ET sequence: 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

Comma list: 5632/5625, 131072/130977

Mapping: [⟨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 ET sequence: 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c

Badness: 0.976 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4096/4095

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 ET sequence: 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f

Badness: 0.604 × 10-3

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

Subgroup: 2.3.5.7.11

Comma list: 42875/42768, 131072/130977

Mapping: [⟨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 ET sequence: 46, 94, 140, 171, 217, 311, 979, 1290

Badness: 3.78 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 13720/13689

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 ET sequence: 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290

Badness: 1.67 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 833/832, 1156/1155, 4096/4095

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 ET sequence: 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g

Badness: 0.989 × 10-3

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