Kalismic temperaments: Difference between revisions
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[[Mapping]]: [{{val| 2 0 1 2 6 -3 }}, {{val| 0 1 4 0 2 1 }}, {{val| 0 0 -5 2 -3 4 }}] | [[Mapping]]: [{{val| 2 0 1 2 6 -3 }}, {{val| 0 1 4 0 2 1 }}, {{val| 0 0 -5 2 -3 4 }}] | ||
{{Val list|legend=1| 10c, 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }} | {{Val list|legend=1| 10c, 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }} | ||
[[Badness]]: 0.867 × 10<sup>-3</sup> | [[Badness]]: 0.867 × 10<sup>-3</sup> | ||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
[[Comma list]]: 715/714, 1089/1088, 1225/1224, 2025/2023 | |||
[[Mapping]]: [{{val| 2 0 1 2 6 -3 0 }}, {{val| 0 1 4 0 2 1 6 }}, {{val| 0 0 -5 2 -3 4 -6 }}] | |||
{{Val list|legend=1| 22f, 54cffgg, 72, 166g, 198g, 270, 364, 436, 634g, 706f }} | |||
[[Badness]]: 0.862 × 10<sup>-3</sup> | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
[[Comma list]]: 715/714, 1225/1224, 1540/1539, 2080/2079, 4200/4199 | |||
[[Mapping]]: [{{val| 2 0 1 2 6 -3 0 13 }}, {{val| 0 1 4 0 2 1 6 2 }}, {{val| 0 0 -5 2 -3 4 -6 -6 }}] | |||
{{Val list|legend=1| 72, 94, 166g, 198g, 270, 436, 634g, 706f }} | |||
[[Badness]]: 0.901 × 10<sup>-3</sup> | |||
=== 23-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19.23 | |||
[[Comma list]]: 715/714, 1225/1224, 1540/1539, 2080/2079, 2530/2527, 2737/2736 | |||
[[Mapping]]: [{{val| 2 0 1 2 6 -3 0 13 19 }}, {{val| 0 1 4 0 2 1 6 2 -2 }}, {{val| 0 0 -5 2 -3 4 -6 -6 -2 }}] | |||
{{Val list|legend=1| 72, 94, 166g, 270, 342f, 436, 706fi }} | |||
[[Badness]]: 1.14 × 10<sup>-3</sup> | |||
=== Gersemi === | === Gersemi === | ||
The extension to 13-limit with [[4225/4224]] is | The extension to 13-limit with [[4225/4224]] is weak but facilitates the use of 18/7 as the equave. Fokker blocks 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. | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 154: | Line 188: | ||
[[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]]: [{{val| 2 0 1 2 6 9 0 }}, {{val| 0 1 9 -2 5 -6 12 }}, {{val| 0 0 -10 4 -6 7 -12 }}] | ||
{{Val list|legend=1| 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684 }} | {{Val list|legend=1| 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684 }} | ||
| Line 182: | Line 214: | ||
[[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]]: [{{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 }}] | ||
{{Val list|legend=1| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }} | {{Val list|legend=1| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }} | ||
Revision as of 21:43, 24 June 2022
These are rank-3 temperaments tempering out 9801/9800. Temperaments discussed elsewhere are:
- Jubilee, {50/49, 99/98} → Jubilismic family
- Fantastic, {225/224, 4375/4356} → Marvel family
- Bisector, {121/120, 245/243} → Sensamagic family
- Julius or varda, {176/175, 896/891} → Diaschismic rank three family
- Hagrid, {243/242, 9801/9800} → Cataharry family
- Uniwiz, {385/384, 9801/9800} → Keenanismic temperaments
- Varuna, {441/440, 8019/8000} → Werckismic temperaments
- Hades, {540/539, 4000/3993} → Swetismic temperaments
- Dimcomp, {1375/1372, 6250/6237} → Dimcomp family
- Baldur, {2401/2400, 9801/9800} → Breed family
- Thor, {3025/3024, 4375/4374} → Ragismic family
- Semiporwell, {6144/6125, 9801/9800} → Porwell family
- Semicanou, {9801/9800, 14641/14580} → Canou family
Considered below are odin, loki, van gogh, rishi, hnoss, and gersemi, but we can begin by looking at the rank-4 temperament.
Kalismic
Subgroup: 2.3.5.7.11
Comma list: 9801/9800
Mapping: [⟨2 0 0 0 3], ⟨0 1 0 0 -2], ⟨0 0 1 0 1], ⟨0 0 0 1 1]]
Mapping generators: ~99/70, ~3, ~5, ~7
Odin
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 151263/151250
Mapping: [⟨6 0 0 8 17], ⟨0 1 0 -2 -4], ⟨0 0 1 2 3]]
Mapping generators: ~55/49, ~3, ~5
Badness: 0.116 × 10-3
Loki
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 9801/9800
Mapping: [⟨2 0 0 -21 -18], ⟨0 1 0 4 2], ⟨0 0 1 3 4]]
Mapping generators: ~99/70, ~3, ~5
Badness: 0.493 × 10-3
Van Gogh
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 199297406/199290375
Mapping: [⟨2 0 8 0 11], ⟨0 1 1 2 1], ⟨0 0 -9 -1 -10]]
Mapping generators: ~99/70, ~3, ~9/7
Badness: 0.297 × 10-3
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 notes 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
Hnoss
To the wizma [-6 -8 2 5⟩ = 420175/419904, the kalisma is a natural complement, as their product is the tinge.
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: 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]]
Badness: 0.867 × 10-3
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 1089/1088, 1225/1224, 2025/2023
Mapping: [⟨2 0 1 2 6 -3 0], ⟨0 1 4 0 2 1 6], ⟨0 0 -5 2 -3 4 -6]]
Badness: 0.862 × 10-3
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 4200/4199
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]]
Badness: 0.901 × 10-3
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 2530/2527, 2737/2736
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]]
Badness: 1.14 × 10-3
Gersemi
The extension to 13-limit with 4225/4224 is weak but facilitates the use of 18/7 as the equave. Fokker blocks 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)3 ~ 17/1 via the chlorisma. However, the tones 9/8 and (19/15)/(9/8) = 152/135 have distinct mappings.
Subgroup: 2.3.5.7.11.13
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]]
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]]
Badness: 1.23 × 10-3