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]]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)]]. | ||
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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[[Badness]]: 0.0183 × 10<sup>-3</sup> | [[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 harmonics 5 by equating the [[81/80|syntonic comma (81/80)]] with the septimal comma. 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]]'' → [[Hemifamity family #Akea|Hemifamity family]] | |||
* ''[[Lux]]'' → [[Lehmerismic temperaments #Lux|Lehmerismic temperaments]] | |||
* ''[[Hera]]'' → [[Porwell family #Hera|Porwell family]] | |||
Considered below are baffin and sophia. | |||
== Baffin == | == Baffin == | ||
=== 7-limit (decovulture) === | === 7-limit (decovulture) === | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 67108864/66976875 | [[Comma list]]: 67108864/66976875 | ||
Revision as of 11:52, 3 May 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
Sval 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
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 harmonics 5 by equating the syntonic comma (81/80) with the septimal comma. 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
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
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 GPV sequence: Template:Val list
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
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 GPV sequence: Template:Val list
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 GPV sequence: Template:Val list
Badness: 0.989 × 10-3