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| * ''[[Semicanou]]'' (+14641/14580) → [[Canou family #Semicanou|Canou family]] | | * ''[[Semicanou]]'' (+14641/14580) → [[Canou family #Semicanou|Canou family]] |
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| Considered below are odin, lycoris, van gogh, hnoss, sif, loki, pessoal and rishi. For the rank-4 temperament, see [[Rank-4 temperament #Kalismic (9801/9800)]]. | | Considered below are odin, lycoris, van gogh, sif, loki, pessoal, and linus. For the rank-4 temperament, see [[Rank-4 temperament #Kalismic (9801/9800)]]. |
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| == Odin == | | == Odin == |
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| [[Badness]]: 0.297 × 10<sup>-3</sup> | | [[Badness]]: 0.297 × 10<sup>-3</sup> |
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| == Hnoss ==
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| To the [[wizma]] {{monzo| -6 -8 2 5 }} = 420175/419904, the kalisma is a natural complement, as their product is the [[argyria]], {{monzo| -9 -4 0 3 2 }} = 41503/41472. [[User:VIxen|VIxen]] named it and [[#Gersemi|gersemi]] after Odr's and Freya's daughters since in 11-limit, they both temper out one comma of [[odin]] (9801/9800) and one of [[freya]] (41503/41472). There is also a word play on "semi-" that hints at the split of the octave in half.
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| [[Subgroup]]: 2.3.5.7.11
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| [[Comma list]]: 9801/9800, 41503/41472
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| {{Mapping|legend=1| 2 0 1 2 6 | 0 1 4 0 2 | 0 0 -5 2 -3 }}
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| : mapping generators: ~99/70, ~3, ~144/77
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| {{Optimal ET sequence|legend=1| 22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}
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| [[Badness]]: 0.368 × 10<sup>-3</sup>
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| === 13-limit ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 1716/1715, 2080/2079, 17303/17280
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| Mapping: {{mapping| 2 0 1 2 6 -3 | 0 1 4 0 2 1 | 0 0 -5 2 -3 4 }}
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| {{Optimal ET sequence|legend=1| 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }}
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| Badness: 0.867 × 10<sup>-3</sup>
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| ==== 17-limit ====
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| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 715/714, 1089/1088, 1225/1224, 2025/2023
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| Mapping: {{mapping| 2 0 1 2 6 -3 0 | 0 1 4 0 2 1 6 | 0 0 -5 2 -3 4 -6 }}
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| {{Optimal ET sequence|legend=1| 22f, 54cffgg, 72, 166g, 198g, 270, 364, 436, 634g, 706f }}
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| Badness: 0.862 × 10<sup>-3</sup>
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| ==== 19-limit ====
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| Subgroup: 2.3.5.7.11.13.17.19
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| Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 4200/4199
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| Mapping: {{mapping| 2 0 1 2 6 -3 0 13 | 0 1 4 0 2 1 6 2 | 0 0 -5 2 -3 4 -6 -6 }}
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| {{Optimal ET sequence|legend=1| 72, 94, 166g, 198g, 270, 436, 634g, 706f }}
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| Badness: 0.901 × 10<sup>-3</sup>
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| ==== 23-limit ====
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| Subgroup: 2.3.5.7.11.13.17.19.23
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| Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 2530/2527, 2737/2736
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| Mapping: {{mapping| 2 0 1 2 6 -3 0 13 19 | 0 1 4 0 2 1 6 2 -2 | 0 0 -5 2 -3 4 -6 -6 -2 }}
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| {{Optimal ET sequence|legend=1| 72, 94, 166g, 270, 342f, 436, 706fi }}
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| Badness: 1.14 × 10<sup>-3</sup>
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| === Gersemi ===
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| The extension to 13-limit with [[4225/4224]] is weak but facilitates the use of 18/7 as the equave. [[Fokker block]]s of 128 notes are available for the latter, corresponding to 94edo. 18/7 is split into 4 parts that become ~19/15 in 19-limit. Also, (18/7)<sup>3</sup> ~ 17/1 via the [[5832/5831|chlorisma]]. However, the tones 9/8 and (19/15)/(9/8) = 152/135 have distinct mappings.
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 4225/4224, 9801/9800, 41503/41472
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| Mapping: {{mapping| 2 0 1 2 6 9 | 0 1 9 -2 5 -6 | 0 0 -10 4 -6 7 }}
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| : mapping generators: ~99/70, ~3, ~154/65
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| {{Optimal ET sequence|legend=1| 44, 50, 94, 144, 176, 220, 270, 590, 684, 954 }}
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| Badness: 1.06 × 10<sup>-3</sup>
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| ==== 17-limit ====
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| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 1089/1088, 1225/1224, 2025/2023, 4225/4224
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| Mapping: {{mapping| 2 0 1 2 6 9 0 | 0 1 9 -2 5 -6 12 | 0 0 -10 4 -6 7 -12 }}
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| {{Optimal ET sequence|legend=1| 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684 }}
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| Badness: 1.46 × 10<sup>-3</sup>
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| ==== 19-limit ====
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| Subgroup: 2.3.5.7.11.13.17.19
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| Comma list: 1089/1088, 1225/1224, 1729/1728, 2926/2925, 3762/3757
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| Mapping: {{mapping| 2 0 1 2 6 9 0 1 | 0 1 9 -2 5 -6 12 11 | 0 0 -10 4 -6 7 -12 -11 }}
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| {{Optimal ET sequence|legend=1| 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h }}
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| Badness: 1.11 × 10<sup>-3</sup>
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| ==== 23-limit ====
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| Subgroup: 2.3.5.7.11.13.17.19.23
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| Comma list: 897/896, 1089/1088, 1225/1224, 1729/1728, 2737/2736, 2926/2925
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| Mapping: {{mapping| 2 0 1 2 6 9 0 1 7 | 0 1 9 -2 5 -6 12 11 3 | 0 0 -10 4 -6 7 -12 -11 -3 }}
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| {{Optimal ET sequence|legend=1| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }}
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| Badness: 1.23 × 10<sup>-3</sup>
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| == Sif == | | == Sif == |
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| Badness: 0.771 × 10<sup>-3</sup> | | Badness: 0.771 × 10<sup>-3</sup> |
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| == Rishi ==
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| The 7-limit comma {{monzo| 65 -84 10 16 }} ~ 0.13¢ has the ratio of the exponents of 3 and 2 that is close to the one in 81/8. The square root of the latter is close to 35/11. This suggests tempering out (81/8)(35/11)<sup>-2</sup>, which is the kalisma.
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| Apart from 35/11, 35/33, and the equivalents of their squares, 81/8 and 9/8, another equave that comes to mind is 3/2, especially after tempering out the [[chalmersia]]. When 3/2 is chosen as the equave, Fokker blocks of 34 notes can be used that are close to [[34edf]] and 58edo.
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| When [[User:VIxen|VIxen]] was naming this temperament, [[sruti]] was its only named rank-2 subtemperament. Rishis (sages) are believed to have created shruti texts.
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| [[Subgroup]]: 2.3.5.7.11
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| [[Comma list]]: 9801/9800, 572145834917888/571919811374025
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| {{Mapping|legend=1| 2 0 3 -10 -4 | 0 1 2 4 4 | 0 0 8 -5 3 }}
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| : mapping generators: ~99/70, ~3, ~17364375/14172488
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| {{Optimal ET sequence|legend=1| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 7414, 9874, 12828e }}
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| [[Badness]]: 2.10 × 10<sup>-3</sup>
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| === 13-limit ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 9801/9800, 10648/10647, 371293/371250
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| Mapping: {{mapping| 2 0 3 -10 -4 2 | 0 1 2 4 4 3 | 0 0 8 -5 3 7 }}
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| {{Optimal ET sequence|legend=1| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e }}
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| [[Badness]]: 0.505 × 10<sup>-3</sup>
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| [[Category:Temperament collections]] | | [[Category:Temperament collections]] |