User:VIxen/Sandbox: Difference between revisions
m →Gersemi: confused the badnesses of the 13-limit extensions |
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== Gersemi == | == Gersemi == | ||
To the wizma [-6 -8 2 5⟩ = 420175/419904, the kalisma is a natural complement, as their product is the [[tinge]]. | |||
18/7 is a possible [[equave]]. [[Fokker block]]s of 128 notes are available for it, 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. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
[[Comma list]]: 4225/4224, 9801/9800, 41503/41472 | |||
[[Mapping]]: [{{val| 2 0 1 2 6 9 }}, {{val| 0 1 9 -2 5 -6 }}, {{val| 0 0 -10 4 -6 7 }}] | |||
Mapping generators: ~99/70, ~3, ~154/65 | |||
{{Val list|legend=1| 44, 50, 94, 144, 176, 220, 270, 590, 684, 954 }} | |||
[[Badness]]: 1.06 × 10<sup>-3</sup> | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
[[Comma list]]: 1089/1088, 1225/1224, 2025/2023, 4225/4224 | |||
[[Mapping]]: [{{val| 2 0 1 2 6 9 0 }}, {{val| 0 1 9 -2 5 -6 12 }}, {{val| 0 0 -10 4 -6 7 -12 }}] | |||
Mapping generators: ~99/70, ~3, ~154/65 | |||
{{Val list|legend=1| 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684 }} | |||
[[Badness]]: 1.46 × 10<sup>-3</sup> | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.19 | |||
[[Comma list]]: 1089/1088, 1225/1224, 1729/1728, 2926/2925, 3762/3757 | |||
[[Mapping]]: [{{val| 2 0 1 2 6 9 0 1 }}, {{val| 0 1 9 -2 5 -6 12 11 }}, {{val| 0 0 -10 4 -6 7 -12 -11 }}] | |||
Mapping generators: ~99/70, ~3, ~45/19 | |||
{{Val list|legend=1| 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h }} | |||
[[Badness]]: 1.11 × 10<sup>-3</sup> | |||
=== 23-limit === | |||
Subgroup: 2.3.5.7.11.13.19.23 | |||
[[Comma list]]: 897/896, 1089/1088, 1225/1224, 1729/1728, 2737/2736, 2926/2925 | |||
[[Mapping]]: [{{val| 2 0 1 2 6 9 0 1 7 }}, {{val| 0 1 9 -2 5 -6 12 11 3 }}, {{val| 0 0 -10 4 -6 7 -12 -11 -3 }}] | |||
Mapping generators: ~99/70, ~3, ~45/19 | |||
{{Val list|legend=1| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }} | |||
[[Badness]]: 1.23 × 10<sup>-3</sup> | |||
=== 13-limit (ibnsinmic) === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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[[Badness]]: 0.867 × 10<sup>-3</sup> | [[Badness]]: 0.867 × 10<sup>-3</sup> | ||
=== 13-limit (schisminic | === 13-limit (schisminic) === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Revision as of 16:58, 23 June 2022
Wizmic family
These are rank-3 temperaments where the wizma 420175/419904 is tempered out. For the clan of rank-2 temperaments with this comma, see Wizmic microtemperaments.
Wizmic
Subgroup: 2.3.5.7
Comma list: 420175/419904
Mapping: [⟨1 0 3 0], ⟨0 1 4 0], ⟨0 0 -5 2]]
Mapping generators: ~2, ~3, ~648/245
Badness: 0.0864 × 10-3
Gersemi
To the wizma [-6 -8 2 5⟩ = 420175/419904, the kalisma is a natural complement, as their product is the tinge.
18/7 is a possible equave. Fokker blocks of 128 notes are available for it, corresponding to 94edo. 18/7 is split into 4 parts that become ~19/15 in 19-limit. Also, (18/7)3 ~ 17/1 via the chlorisma. However, the tones 9/8 and (19/15)/(9/8) = 152/135 have distinct mappings.
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 41503/41472
Mapping: [⟨2 0 1 2 6], ⟨0 1 4 0 2], ⟨0 0 -5 2 -3]]
Mapping generators: ~99/70, ~3, ~144/77
Badness: 0.368 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 9801/9800, 41503/41472
Mapping: [⟨2 0 1 2 6 9], ⟨0 1 9 -2 5 -6], ⟨0 0 -10 4 -6 7]]
Mapping generators: ~99/70, ~3, ~154/65
Badness: 1.06 × 10-3
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 1089/1088, 1225/1224, 2025/2023, 4225/4224
Mapping: [⟨2 0 1 2 6 9 0], ⟨0 1 9 -2 5 -6 12], ⟨0 0 -10 4 -6 7 -12]]
Mapping generators: ~99/70, ~3, ~154/65
Badness: 1.46 × 10-3
19-limit
Subgroup: 2.3.5.7.11.13.19
Comma list: 1089/1088, 1225/1224, 1729/1728, 2926/2925, 3762/3757
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]]
Mapping generators: ~99/70, ~3, ~45/19
Badness: 1.11 × 10-3
23-limit
Subgroup: 2.3.5.7.11.13.19.23
Comma list: 897/896, 1089/1088, 1225/1224, 1729/1728, 2737/2736, 2926/2925
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]]
Mapping generators: ~99/70, ~3, ~45/19
Badness: 1.23 × 10-3
13-limit (ibnsinmic)
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 17303/17280
Mapping: [⟨2 0 1 2 6 -3], ⟨0 1 4 0 2 1], ⟨0 0 -5 2 -3 4]]
Mapping generators: ~99/70, ~3, ~144/77
Badness: 0.867 × 10-3
13-limit (schisminic)
Subgroup: 2.3.5.7.11.13
Comma list: 4096/4095, 9801/9800, 41503/41472
Mapping: [⟨2 0 1 2 6 21], ⟨0 1 4 0 2 -6], ⟨0 0 -5 2 -3 3]]
Mapping generators: ~99/70, ~3, ~144/77
Badness: 1.56 × 10-3
Skeetsmic family
These are rank-3 temperaments where the skeetsma is tempered out.
Skeetsmic
Subgroup: 2.3.5.7
Comma list: 14348907/14336000
Mapping: [⟨1 0 0 -14], ⟨0 1 0 15], ⟨0 0 1 -3]]
Mapping generators: ~2, ~3, ~5
Badness: 0.333 × 10-3
Skald
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 1240029/1239040
Mapping: [⟨2 0 1 -28 -25], ⟨0 1 0 15 13], ⟨0 0 1 -3 -2]]
Mapping generators: ~99/70, ~3, ~5
Badness: 3.20 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 59535/59488
Mapping: [⟨2 0 1 -31 -27 -22], ⟨0 1 0 15 13 11], ⟨0 0 2 -6 -4 -3]]
Mapping generators: ~99/70, ~3, ~220/117
Badness: 1.31 × 10-3
Other from the kalismic family
Rishi
The 7-limit comma [65 -84 10 16⟩ ~ 0.13c 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)-2, which is the kalisma.
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 pitches per equave can be used that are close to 34edf and 58edo.
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 572145834917888/571919811374025
Mapping: [⟨2 0 3 -10 -4], ⟨0 1 2 4 4], ⟨0 0 8 -5 3]]
Mapping generators: ~99/70, ~3, ~17364375/14172488
Badness: 2.10 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 371293/371250
Mapping: [⟨2 0 3 -10 -4 2], ⟨0 1 2 4 4 3], ⟨0 0 8 -5 3 7]]
Mapping generators: ~99/70, ~3, ~364/297
Badness: 0.505 × 10-3