User:VIxen/Sandbox: Difference between revisions

(7 intermediate revisions by one other user not shown)

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Mapping generators: ~2, ~3, ~648/245

{{~~Val list~~|legend=1| 5, 17, 22, 27, 45, 49, 50, 72, 99, 171, 441, 612, 2812, 3082, 3253, 3424, 3694, 3865, 4306, 4477d, 4918d, 5089d, 5701d }}

{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 45, 49, 50, 72, 99, 171, 441, 612, 2812, 3082, 3253, 3424, 3694, 3865, 4306, 4477d, 4918d, 5089d, 5701d }}

[[Badness]]: 0.0864 × 10-3

== Gersemi ==

~~{{see also~~| ~~Kalismic temperaments }}~~

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)3 ~ 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

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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 }}

{{Optimal ET sequence|legend=1| 10c, 18bcd, 22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}

[[Badness]]: 0.368 × 10-3

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Subgroup: 2.3.5.7.11.13

[[Comma list]]: ~~4096~~/~~4095~~, 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 generators: ~99/70, ~3, ~154/65

{{Optimal ET sequence|legend=1| 44, 50, 94, 144, 176, 220, 270, 590, 684, 954 }}

[[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]]: [{{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

{{Optimal ET sequence|legend=1| 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684 }}

[[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]]: [{{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

{{Optimal ET sequence|legend=1| 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h }}

[[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]]: [{{val| 2 0 1 2 6 21 }}, {{val| 0 1 ~~4 0~~ 2 -6 }}, {{val| 0 0 -~~5 2~~ -3 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 }}]

Mapping generators: ~99/70, ~3, ~~~144~~/77

Mapping generators: ~99/70, ~3, ~45/19

{{~~Val list~~|legend=1| 22, ~~72f~~, 94, ~~176~~, ~~248~~, 270, ~~612, 882~~, ~~1152~~, ~~2034~~ }}

{{Optimal ET sequence|legend=1| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }}

[[Badness]]: 0.~~867~~ × 10-3

[[Badness]]: 1.23 × 10-3

=== 13-limit (ibnsinmic) ===

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Mapping generators: ~99/70, ~3, ~144/77

{{~~Val list~~|legend=1| 10c, 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }}

{{Optimal ET sequence|legend=1| 10c, 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }}

[[Badness]]: 0.867 × 10-3

=== 13-limit (schisminic) ===

Subgroup: 2.3.5.7.11.13

[[Comma list]]: 4096/4095, 9801/9800, 41503/41472

[[Mapping]]: [{{val| 2 0 1 2 6 21 }}, {{val| 0 1 4 0 2 -6 }}, {{val| 0 0 -5 2 -3 3 }}]

Mapping generators: ~99/70, ~3, ~144/77

{{Optimal ET sequence|legend=1| 22, 72f, 94, 176, 248, 270, 612, 882, 1152, 2034 }}

[[Badness]]: 1.56 × 10-3

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Mapping generators: ~2, ~3, ~5

{{~~Val list~~|legend=1| 5, 7d, 12d, 39ddd, 44cdd, 46d, 48, 53, 111, 118, 171, 742, 795, 913, 966, 1137, 1308, 2050, 2221, 2392, 3358, 3529, 7229b, 10758bbd }}

{{Optimal ET sequence|legend=1| 5, 7d, 12d, 39ddd, 44cdd, 46d, 48, 53, 111, 118, 171, 742, 795, 913, 966, 1137, 1308, 2050, 2221, 2392, 3358, 3529, 7229b, 10758bbd }}

[[Badness]]: 0.333 × 10-3

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Mapping generators: ~99/70, ~3, ~5

{{~~Val list~~|legend=1| 10c, 12de, 44cddeee, 46de, 48, 58, 106, 118, 224, 342, 742, 966, 1084, 1308, 1650, 2392, 2734, 5126bdb, 6776bd }}

{{Optimal ET sequence|legend=1| 10c, 12de, 44cddeee, 46de, 48, 58, 106, 118, 224, 342, 742, 966, 1084, 1308, 1650, 2392, 2734, 5126bdb, 6776bd }}

[[Badness]]: 3.20 × 10-3

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Mapping generators: ~99/70, ~3, ~220/117

{{~~Val list~~|legend=1| 12de, 46def, 58, 166, 224, 460, 684, 1426, 2334, 3076 }}

{{Optimal ET sequence|legend=1| 12de, 46def, 58, 166, 224, 460, 684, 1426, 2334, 3076 }}

[[Badness]]: 1.31 × 10-3

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== Rishi ==

The 7-limit comma [65 -84 10 16⟩ ~ 0.13c has the ratio of the exponents of 3 and 2 that is ~~almost~~ the ~~same -4:3 as~~ in 81/8. The square root of the latter is close to 35/11. This suggests tempering out (81/8)(35/11)-2 ~~in 11-limit~~, 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)-2, which is the kalisma.

Apart from 35/11, 35/33, and the equivalents of their squares, 81/8 and 9/8, another ~~possible~~ [[equave]] is 3/2, ~~considering that adding~~ the [[chalmersia]] ~~- the ratio of the kalisma to the [[10648/10647|harmonisma]] - is an option in 13-limit~~. When 3/2 is ~~used~~ as the equave, [[Fokker block]]s of 34 pitches per equave can be used that are close to [[34edf]] and [[58edo]].

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 block]]s of 34 pitches per equave can be used that are close to [[34edf]] and [[58edo]].

Subgroup: 2.3.5.7.11

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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 }}

{{Optimal ET sequence|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 }}

[[Badness]]: 2.10 × 10-3

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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 ET sequence|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 }}

[[Badness]]: 0.505 × 10-3

== Odin (harmonismic) ==

(Equave 3/2: q63ef & q70p & q95p)

Subgroup: 2.3.5.7.11.13

[[Comma list]]: 9801/9800, 10648/10647, 105644/105625

[[Mapping]]: [{{val| 6 0 1 10 20 34 }}, {{val| 0 1 0 -2 -4 -6 }}, {{val| 0 0 2 4 6 7 }}]

Mapping generators: ~55/49, ~3, ~325/154

{{Optimal ET sequence|legend=1| 12f, 108cddeeefff, 120cdeeff, 126, 132de, 144, 258ef, 270, 552, 684, 954, 1236, 1506, 2190, 3696, 5886, 6156, 8346, 12042 }}

[[Badness]]: 0.418 × 10-3

@@ Line 13: / Line 13: @@
 Mapping generators: ~2, ~3, ~648/245
-{{Val list|legend=1| 5, 17, 22, 27, 45, 49, 50, 72, 99, 171, 441, 612, 2812, 3082, 3253, 3424, 3694, 3865, 4306, 4477d, 4918d, 5089d, 5701d }}
+{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 45, 49, 50, 72, 99, 171, 441, 612, 2812, 3082, 3253, 3424, 3694, 3865, 4306, 4477d, 4918d, 5089d, 5701d }}
 [[Badness]]: 0.0864 × 10<sup>-3</sup>
 == Gersemi ==
-{{see also| Kalismic temperaments }}
+To the wizma [-6 -8 2 5⟩ = 420175/419904, the kalisma is a natural complement, as their product is the [[tinge]].
+/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
@@ Line 28: / Line 31: @@
 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 }}
+{{Optimal ET sequence|legend=1| 10c, 18bcd, 22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}
 [[Badness]]: 0.368 × 10<sup>-3</sup>
@@ Line 36: / Line 39: @@
 Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 4096/4095, 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 generators: ~99/70, ~3, ~154/65
+{{Optimal ET sequence|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
+{{Optimal ET sequence|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
+{{Optimal ET sequence|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 21 }}, {{val| 0 1 4 0 2 -6 }}, {{val| 0 0 -5 2 -3 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 }}]
-Mapping generators: ~99/70, ~3, ~144/77
+Mapping generators: ~99/70, ~3, ~45/19
-{{Val list|legend=1| 22, 72f, 94, 176, 248, 270, 612, 882, 1152, 2034 }}
+{{Optimal ET sequence|legend=1| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }}
-[[Badness]]: 0.867 × 10<sup>-3</sup>
+[[Badness]]: 1.23 × 10<sup>-3</sup>
 === 13-limit (ibnsinmic) ===
@@ Line 56: / Line 101: @@
 Mapping generators: ~99/70, ~3, ~144/77
-{{Val list|legend=1| 10c, 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }}
+{{Optimal ET sequence|legend=1| 10c, 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }}
+[[Badness]]: 0.867 × 10<sup>-3</sup>
+=== 13-limit (schisminic) ===
+Subgroup: 2.3.5.7.11.13
+[[Comma list]]: 4096/4095, 9801/9800, 41503/41472
+[[Mapping]]: [{{val| 2 0 1 2 6 21 }}, {{val| 0 1 4 0 2 -6 }}, {{val| 0 0 -5 2 -3 3 }}]
+Mapping generators: ~99/70, ~3, ~144/77
+{{Optimal ET sequence|legend=1| 22, 72f, 94, 176, 248, 270, 612, 882, 1152, 2034 }}
 [[Badness]]: 1.56 × 10<sup>-3</sup>
@@ Line 74: / Line 133: @@
 Mapping generators: ~2, ~3, ~5
-{{Val list|legend=1| 5, 7d, 12d, 39ddd, 44cdd, 46d, 48, 53, 111, 118, 171, 742, 795, 913, 966, 1137, 1308, 2050, 2221, 2392, 3358, 3529, 7229b, 10758bbd }}
+{{Optimal ET sequence|legend=1| 5, 7d, 12d, 39ddd, 44cdd, 46d, 48, 53, 111, 118, 171, 742, 795, 913, 966, 1137, 1308, 2050, 2221, 2392, 3358, 3529, 7229b, 10758bbd }}
 [[Badness]]: 0.333 × 10<sup>-3</sup>
@@ Line 89: / Line 148: @@
 Mapping generators: ~99/70, ~3, ~5
-{{Val list|legend=1| 10c, 12de, 44cddeee, 46de, 48, 58, 106, 118, 224, 342, 742, 966, 1084, 1308, 1650, 2392, 2734, 5126bdb, 6776bd }}
+{{Optimal ET sequence|legend=1| 10c, 12de, 44cddeee, 46de, 48, 58, 106, 118, 224, 342, 742, 966, 1084, 1308, 1650, 2392, 2734, 5126bdb, 6776bd }}
 [[Badness]]: 3.20 × 10<sup>-3</sup>
@@ Line 103: / Line 162: @@
 Mapping generators: ~99/70, ~3, ~220/117
-{{Val list|legend=1| 12de, 46def, 58, 166, 224, 460, 684, 1426, 2334, 3076 }}
+{{Optimal ET sequence|legend=1| 12de, 46def, 58, 166, 224, 460, 684, 1426, 2334, 3076 }}
 [[Badness]]: 1.31 × 10<sup>-3</sup>
@@ Line 111: / Line 170: @@
 == Rishi ==
-The 7-limit comma [65 -84 10 16⟩ ~ 0.13c has the ratio of the exponents of 3 and 2 that is almost the same -4:3 as 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> in 11-limit, 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 possible [[equave]] is 3/2, considering that adding the [[chalmersia]] - the ratio of the kalisma to the [[10648/10647|harmonisma]] - is an option in 13-limit. When 3/2 is used as the equave, [[Fokker block]]s of 34 pitches per equave can be used that are close to [[34edf]] and [[58edo]].
+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 block]]s of 34 pitches per equave can be used that are close to [[34edf]] and [[58edo]].
 Subgroup: 2.3.5.7.11
@@ Line 123: / Line 182: @@
 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 }}
+{{Optimal ET sequence|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 }}
 [[Badness]]: 2.10 × 10<sup>-3</sup>
@@ Line 137: / Line 196: @@
 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 ET sequence|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 }}
 [[Badness]]: 0.505 × 10<sup>-3</sup>
+== Odin (harmonismic) ==
+(Equave 3/2: q63ef & q70p & q95p)
+Subgroup: 2.3.5.7.11.13
+[[Comma list]]: 9801/9800, 10648/10647, 105644/105625
+[[Mapping]]: [{{val| 6 0 1 10 20 34 }}, {{val| 0 1 0 -2 -4 -6 }}, {{val| 0 0 2 4 6 7 }}]
+Mapping generators: ~55/49, ~3, ~325/154
+{{Optimal ET sequence|legend=1| 12f, 108cddeeefff, 120cdeeff, 126, 132de, 144, 258ef, 270, 552, 684, 954, 1236, 1506, 2190, 3696, 5886, 6156, 8346, 12042 }}
+[[Badness]]: 0.418 × 10<sup>-3</sup>