Kalismic temperaments: Difference between revisions

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

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

@@ Line 18: / Line 18: @@
 == Kalismic ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 9801/9800
@@ Line 31: / Line 31: @@
 {{see also| Landscape family }}
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 9801/9800, 151263/151250
@@ Line 44: / Line 44: @@
 == Loki ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 5632/5625, 9801/9800
@@ Line 57: / Line 57: @@
 == Van Gogh ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 9801/9800, 199297406/199290375
@@ Line 70: / Line 70: @@
 == 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.
-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)<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
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 9801/9800, 572145834917888/571919811374025
@@ Line 83: / Line 82: @@
 Mapping generators: ~99/70, ~3, ~17364375/14172488
-{{Val list|legend=1| 24, 34d, 58, 150cdee, 208ccddee, 252ccddeee, 262ccdee, 286ccdee, 310cddee, 320ccee, 344cdee, 378ce, 402de, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 7414, 9874, 12828e }}
+{{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
+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: [{{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
-{{Val list|legend=1| 24, 34d, 58, 150cdeef, 208ccddeeff, 252ccddeeefff, 262ccdeefff, 286ccdeeff, 310cddeeff, 320cceeff, 344cdeef, 378cef, 402def, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e }}
+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 ==
+To the wizma {{monzo| -6 -8 2 5 }} = 420175/419904, the kalisma is a natural complement, as their product is the [[tinge]].
-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
-Subgroup: 2.3.5.7.11
 [[Comma list]]: 9801/9800, 41503/41472
@@ Line 113: / Line 110: @@
 Mapping generators: ~99/70, ~3, ~144/77
-{{Val list|legend=1| 10c, 18bcd, 22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}
+{{Val list|legend=1| 22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}
 [[Badness]]: 0.368 × 10<sup>-3</sup>
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 1716/1715, 2080/2079, 17303/17280
+Comma list: 1716/1715, 2080/2079, 17303/17280
-[[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 }}
+Optimal GPV sequence: {{Val list| 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 ===
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 715/714, 1089/1088, 1225/1224, 2025/2023
+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 }}]
+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 }}
+Optimal GPV sequence: {{Val list| 22f, 54cffgg, 72, 166g, 198g, 270, 364, 436, 634g, 706f }}
-[[Badness]]: 0.862 × 10<sup>-3</sup>
+Badness: 0.862 × 10<sup>-3</sup>
-=== 19-limit ===
+==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
-[[Comma list]]: 715/714, 1225/1224, 1540/1539, 2080/2079, 4200/4199
+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 }}]
+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 }}
+Optimal GPV sequence: {{Val list| 72, 94, 166g, 198g, 270, 436, 634g, 706f }}
-[[Badness]]: 0.901 × 10<sup>-3</sup>
+Badness: 0.901 × 10<sup>-3</sup>
-=== 23-limit ===
+==== 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
+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 }}]
+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 }}
+Optimal GPV sequence: {{Val list| 72, 94, 166g, 270, 342f, 436, 706fi }}
-[[Badness]]: 1.14 × 10<sup>-3</sup>
+Badness: 1.14 × 10<sup>-3</sup>
 === 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)<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
-[[Comma list]]: 4225/4224, 9801/9800, 41503/41472
+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: [{{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 }}
+Optimal GPV sequence: {{Val list| 44, 50, 94, 144, 176, 220, 270, 590, 684, 954 }}
-[[Badness]]: 1.06 × 10<sup>-3</sup>
+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
+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: [{{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 }}
+Optimal GPV sequence: {{Val list| 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684 }}
-[[Badness]]: 1.46 × 10<sup>-3</sup>
+Badness: 1.46 × 10<sup>-3</sup>
 ==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
-[[Comma list]]: 1089/1088, 1225/1224, 1729/1728, 2926/2925, 3762/3757
+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: [{{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 }}
+Optimal GPV sequence: {{Val list| 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h }}
-[[Badness]]: 1.11 × 10<sup>-3</sup>
+Badness: 1.11 × 10<sup>-3</sup>
 ==== 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
+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: [{{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 }}
+Optimal GPV sequence: {{Val list| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }}
-[[Badness]]: 1.23 × 10<sup>-3</sup>
+Badness: 1.23 × 10<sup>-3</sup>
 [[Category:Temperament collections]]
 [[Category:Kalismic temperaments| ]] <!-- main article -->
 [[Category:Rank 3]]

Kalismic temperaments: Difference between revisions

Revision as of 18:33, 11 December 2022

Contents

Kalismic

Odin

Loki

Van Gogh

Rishi

13-limit

Hnoss

13-limit

17-limit

19-limit

23-limit

Gersemi

17-limit

19-limit

23-limit

Navigation menu

Kalismic temperaments: Difference between revisions

Revision as of 18:33, 11 December 2022

Kalismic

Odin

Loki

Van Gogh

Rishi

13-limit

Hnoss

13-limit

17-limit

19-limit

23-limit

Gersemi

17-limit

19-limit

23-limit

Navigation menu

Search