Kalismic temperaments: Difference between revisions

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Badness: 0.416 × 10<sup>-3</sup>
Badness: 0.416 × 10<sup>-3</sup>
== Loki ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 5632/5625, 9801/9800
[[Mapping]]: [{{val| 2 0 0 -21 -18 }}, {{val| 0 1 0 4 2 }}, {{val| 0 0 1 3 4 }}]
Mapping generators: ~99/70, ~3, ~5
{{Val list|legend=1| 12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}
[[Badness]]: 0.493 × 10<sup>-3</sup>


== Van Gogh ==
== Van Gogh ==
Line 98: Line 85:


[[Badness]]: 0.297 × 10<sup>-3</sup>
[[Badness]]: 0.297 × 10<sup>-3</sup>
== Pessoal ==
{{See also| Pessoalisma }}
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 9801/9800, 131072/130977
[[Mapping]]: [{{val| 2 0 1 10 14 }}, {{val| 0 1 0 -1 -3 }}, {{val| 0 0 3 -1 2 }}]
Mapping generators: ~99/70, ~3, ~32/21
[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 702.0759, ~32/21 = 728.7910
{{Val list|legend=1| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde }}
[[Badness]]: 0.499 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 4096/4095
Mapping: [{{val| 2 0 1 10 14 13 }}, {{val| 0 1 0 -1 -3 -1 }}, {{val| 0 0 3 -1 2 -2 }}]
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.0637, ~32/21 = 728.7786
Optimal GPV sequence: {{Val list| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d }}
Badness: 0.391 × 10<sup>-3</sup>
== Rishi ==
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.
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]]: [{{val| 2 0 3 -10 -4 }}, {{val| 0 1 2 4 4 }}, {{val| 0 0 8 -5 3 }}]
Mapping generators: ~99/70, ~3, ~17364375/14172488
{{Val list|legend=1| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 7414, 9874, 12828e }}
[[Badness]]: 2.10 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 371293/371250
Mapping: [{{val| 2 0 3 -10 -4 2 }}, {{val| 0 1 2 4 4 3 }}, {{val| 0 0 8 -5 3 7 }}]
Mapping generators: ~99/70, ~3, ~364/297
Optimal GPV sequence: {{Val list| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e }}
[[Badness]]: 0.505 × 10<sup>-3</sup>


== Hnoss ==
== Hnoss ==
Line 267: Line 194:


Badness: 1.23 × 10<sup>-3</sup>
Badness: 1.23 × 10<sup>-3</sup>
== Loki ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 5632/5625, 9801/9800
[[Mapping]]: [{{val| 2 0 0 -21 -18 }}, {{val| 0 1 0 4 2 }}, {{val| 0 0 1 3 4 }}]
Mapping generators: ~99/70, ~3, ~5
{{Val list|legend=1| 12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}
[[Badness]]: 0.493 × 10<sup>-3</sup>
== Pessoal ==
{{See also| Pessoalisma }}
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 9801/9800, 131072/130977
[[Mapping]]: [{{val| 2 0 1 10 14 }}, {{val| 0 1 0 -1 -3 }}, {{val| 0 0 3 -1 2 }}]
Mapping generators: ~99/70, ~3, ~32/21
[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 702.0759, ~32/21 = 728.7910
{{Val list|legend=1| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde }}
[[Badness]]: 0.499 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 4096/4095
Mapping: [{{val| 2 0 1 10 14 13 }}, {{val| 0 1 0 -1 -3 -1 }}, {{val| 0 0 3 -1 2 -2 }}]
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.0637, ~32/21 = 728.7786
Optimal GPV sequence: {{Val list| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d }}
Badness: 0.391 × 10<sup>-3</sup>
== Rishi ==
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.
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]]: [{{val| 2 0 3 -10 -4 }}, {{val| 0 1 2 4 4 }}, {{val| 0 0 8 -5 3 }}]
Mapping generators: ~99/70, ~3, ~17364375/14172488
{{Val list|legend=1| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 7414, 9874, 12828e }}
[[Badness]]: 2.10 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 371293/371250
Mapping: [{{val| 2 0 3 -10 -4 2 }}, {{val| 0 1 2 4 4 3 }}, {{val| 0 0 8 -5 3 7 }}]
Mapping generators: ~99/70, ~3, ~364/297
Optimal GPV sequence: {{Val list| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e }}
[[Badness]]: 0.505 × 10<sup>-3</sup>


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Kalismic temperaments| ]] <!-- main article -->
[[Category:Kalismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 14:51, 31 January 2023

These are rank-3 temperaments tempering out 9801/9800. Temperaments discussed elsewhere are:

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

Template:Val list

Odin

See also: Landscape family

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

Template:Val list

Badness: 0.116 × 10-3

Lycoris

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 1771561/1771470

Mapping: [⟨2 0 4 -6 1], ⟨0 1 1 3 2], ⟨0 0 -6 5 -1]]

Mapping generators: ~99/70, ~3, ~81/70

Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.9439, ~81/70 = 252.6028

Template:Val list

Badness: 0.249 × 10-3

Higanbana

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 10648/10647, 1771561/1771470

Mapping: [⟨2 0 4 -6 1 8], ⟨0 2 2 6 4 1], ⟨0 0 -6 5 -1 4]]

Mapping generators: ~99/70, ~1458/1001, ~81/70

Optimal tuning (CTE): ~99/70 = 1\2, ~1458/1001 = 650.9715, ~81/70 = 252.6037

Optimal GPV sequence: Template:Val list

Badness: 0.416 × 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

Template:Val list

Badness: 0.297 × 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

Template:Val list

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]]

Optimal GPV sequence: Template:Val list

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]]

Optimal GPV sequence: Template:Val list

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]]

Optimal GPV sequence: Template:Val list

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]]

Optimal GPV sequence: Template:Val list

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

Optimal GPV sequence: Template:Val list

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]]

Optimal GPV sequence: Template:Val list

Badness: 1.46 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Optimal GPV sequence: Template:Val list

Badness: 1.11 × 10-3

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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]]

Optimal GPV sequence: Template:Val list

Badness: 1.23 × 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

Template:Val list

Badness: 0.493 × 10-3

Pessoal

See also: Pessoalisma

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977

Mapping: [⟨2 0 1 10 14], ⟨0 1 0 -1 -3], ⟨0 0 3 -1 2]]

Mapping generators: ~99/70, ~3, ~32/21

Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.0759, ~32/21 = 728.7910

Template:Val list

Badness: 0.499 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095

Mapping: [⟨2 0 1 10 14 13], ⟨0 1 0 -1 -3 -1], ⟨0 0 3 -1 2 -2]]

Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.0637, ~32/21 = 728.7786

Optimal GPV sequence: Template:Val list

Badness: 0.391 × 10-3

Rishi

The 7-limit comma [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)-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

Template:Val list

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

Optimal GPV sequence: Template:Val list

Badness: 0.505 × 10-3

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