Gamelismic family: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
"Gamelan" -> "gamelismic" (cuz "gamelan" needs to go)
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The '''gamelismic family''' of temperaments are rank-3 temperaments tempering out [[1029/1024]].  
The '''gamelismic family''' of temperaments are rank-3 temperaments tempering out [[1029/1024]].  


== Gamelan ==
== Gamelismic ==
{{Main| Portent }}
{{Main| Portent }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024
[[Comma list]]: 1029/1024
Line 16: Line 16:
: Angle (8/7, 5/4) = 90 degrees
: Angle (8/7, 5/4) = 90 degrees


[[POTE generator]]s: ~8/7 = 233.6875, ~5/4 = 385.1853
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 233.6875, ~5/4 = 385.1853


[[Minimax tuning]]: c = 1029/1024
[[Minimax tuning]]: c = 1029/1024
Line 41: Line 41:
Portent has a [[Normal lists #normal interval list|normal comma list]] [1029/1024, 385/384] and also tempers out 441/440.
Portent has a [[Normal lists #normal interval list|normal comma list]] [1029/1024, 385/384] and also tempers out 441/440.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 385/384, 441/440
[[Comma list]]: 385/384, 441/440
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: Angle (8/7, 12/11) = 86.657 degrees
: Angle (8/7, 12/11) = 86.657 degrees


[[POTE generator]]s: ~8/7 = 233.6884, ~5/4 = 385.1618
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 233.6884, ~5/4 = 385.1618


[[Minimax tuning]]: c = 1029/1024, e = 385/384
[[Minimax tuning]]: c = 1029/1024, e = 385/384
Line 77: Line 77:
Mapping: [{{val| 1 1 0 3 5 6 }}, {{val| 0 3 0 -1 4 12 }}, {{val| 0 0 1 0 -1 -2 }}]
Mapping: [{{val| 1 1 0 3 5 6 }}, {{val| 0 3 0 -1 4 12 }}, {{val| 0 0 1 0 -1 -2 }}]


POTE generators: ~8/7 = 233.9128, ~5/4 = 384.7278
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.9128, ~5/4 = 384.7278


{{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 159, 477cdf }}
{{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 159, 477cdf }}
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Mapping: [{{val| 1 1 0 3 5 -5 }}, {{val| 0 3 0 -1 4 -3 }}, {{val| 0 0 1 0 -1 4 }}]
Mapping: [{{val| 1 1 0 3 5 -5 }}, {{val| 0 3 0 -1 4 -3 }}, {{val| 0 0 1 0 -1 4 }}]


POTE generators: ~8/7 = 233.6843, ~5/4 = 384.9830
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.6843, ~5/4 = 384.9830


{{Optimal ET sequence|legend=1| 15, 31, 56, 72, 87, 103, 159, 190, 262df, 349f, 452cdef, 611cdef }}
{{Optimal ET sequence|legend=1| 15, 31, 56, 72, 87, 103, 159, 190, 262df, 349f, 452cdef, 611cdef }}
Line 103: Line 103:
Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 0 -1 4 -10 }}, {{val| 0 0 1 0 -1 2 }}]
Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 0 -1 4 -10 }}, {{val| 0 0 1 0 -1 2 }}]


POTE generators: ~8/7 = 233.4088, ~5/4 = 385.3825
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.4088, ~5/4 = 385.3825


{{Optimal ET sequence|legend=1| 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdef }}
{{Optimal ET sequence|legend=1| 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdef }}
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: Angle (8/7, 6/5) = 98.6015
: Angle (8/7, 6/5) = 98.6015


POTE generators: ~8/7 = 233.4313, ~5/4 = 385.2890
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.4313, ~5/4 = 385.2890


Minimax tuning:  
Minimax tuning:  
Line 142: Line 142:
Mapping: [{{val| 1 1 0 3 5 7 }}, {{val| 0 3 0 -4 1 -5 }}, {{val| 0 0 1 0 -1 -1 }}]
Mapping: [{{val| 1 1 0 3 5 7 }}, {{val| 0 3 0 -4 1 -5 }}, {{val| 0 0 1 0 -1 -1 }}]


POTE generators: ~8/7 = 233.4088, ~5/4 = 385.3825
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.4088, ~5/4 = 385.3825


{{Optimal ET sequence|legend=1| 10, 31, 41, 46, 77, 87, 118, 164, 205d, 574def }}
{{Optimal ET sequence|legend=1| 10, 31, 41, 46, 77, 87, 118, 164, 205d, 574def }}
Line 155: Line 155:
Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 0 -1 4 2 }}, {{val| 0 0 1 0 -1 1 }}]
Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 0 -1 4 2 }}, {{val| 0 0 1 0 -1 1 }}]


POTE generators: ~8/7 = 233.3664, ~5/4 = 382.3425
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.3664, ~5/4 = 382.3425


{{Optimal ET sequence|legend=1| 5, 10, 15, 26, 31, 41, 72f, 185cf }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 26, 31, 41, 72f, 185cf }}
Line 168: Line 168:
Mapping: [{{val| 1 1 2 3 3 4 }}, {{val| 0 3 0 -1 4 -1 }}, {{val| 0 0 3 0 -3 -1 }}]
Mapping: [{{val| 1 1 2 3 3 4 }}, {{val| 0 3 0 -1 4 -1 }}, {{val| 0 0 3 0 -3 -1 }}]


POTE generators: ~8/7 = 233.6434, ~14/13 = 128.3440
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.6434, ~14/13 = 128.3440


{{Optimal ET sequence|legend=1| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff }}
{{Optimal ET sequence|legend=1| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff }}
Line 181: Line 181:
Mapping: [{{val| 1 1 2 3 3 4 4 }}, {{val| 0 3 0 -1 4 -1 1 }}, {{val| 0 0 3 0 -3 -1 -1 }}]
Mapping: [{{val| 1 1 2 3 3 4 4 }}, {{val| 0 3 0 -1 4 -1 1 }}, {{val| 0 0 3 0 -3 -1 -1 }}]


POTE generators: ~8/7 = 233.6803, ~14/13 = 128.4149
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.6803, ~14/13 = 128.4149


{{Optimal ET sequence|legend=1| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def }}
{{Optimal ET sequence|legend=1| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def }}
Line 188: Line 188:


== Gamel ==
== Gamel ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 1029/1024
[[Comma list]]: 540/539, 1029/1024
Line 194: Line 194:
[[Mapping]]: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 0 -1 11 }}, {{val| 0 0 1 0 1 }}]
[[Mapping]]: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 0 -1 11 }}, {{val| 0 0 1 0 1 }}]


[[POTE generator]]s: ~8/7 = 233.2909, ~5/4 = 384.2120
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 233.2909, ~5/4 = 384.2120


{{Optimal ET sequence|legend=1| 10, 31, 41, 72 }}
{{Optimal ET sequence|legend=1| 10, 31, 41, 72 }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Gamelismic family| ]] <!-- main article -->
[[Category:Gamelismic family| ]] <!-- main article -->
[[Category:Gamelismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 03:16, 22 May 2023

The gamelismic family of temperaments are rank-3 temperaments tempering out 1029/1024.

Gamelismic

Main article: Portent

Subgroup: 2.3.5.7

Comma list: 1029/1024

Mapping: [⟨1 1 0 3], ⟨0 3 0 -1],⟨0 0 1 0]]

Mapping to lattice: [⟨0 3 0 -1], ⟨0 0 1 0]]

Minkowski lattice basis:

8/7 length = 0.5192, 5/4 length = log25
Angle (8/7, 5/4) = 90 degrees

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.6875, ~5/4 = 385.1853

Minimax tuning: c = 1029/1024

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [5/2 -1/4 1 -3/4⟩, [5/2 -1/4 0 1/4⟩]
Eigenmonzos (unchanged-intervals): 2, 7/6, 6/5
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [10/7 -1/7 1 -3/7⟩, [20/7 -2/7 0 1/7⟩]
Eigenmonzos (unchanged-intervals): 2, 6/5, 9/7

TE tuning map: ⟨1200.486 1901.833 2786.314 3367.676]

Optimal ET sequence: 5, 10, 15, 26, 31, 41, 72, 118, 159, 190

Badness: 0.176 × 10-3

Projection pair: 3 1024/343 to 2.5.7

Scales: portent26

Portent

Main article: Portent

Portent has a normal comma list [1029/1024, 385/384] and also tempers out 441/440.

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440

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

Mapping generators: ~2, ~8/7, ~5

Mapping to lattice: [⟨0 3 1 -1 3], ⟨0 0 1 0 -1]]

Minkowski lattice basis:

8/7 length = 0.46467, 12/11 length = 1.931
Angle (8/7, 12/11) = 86.657 degrees

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.6884, ~5/4 = 385.1618

Minimax tuning: c = 1029/1024, e = 385/384

  • 11-odd-limit: 3 1/7c flat, 5 and 7 2/7c flat, 11 e-3/7c flat
[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [39/14 4/7 1/2 -2/7 -1/2⟩, [20/7 -2/7 0 1/7 0⟩, [39/14 4/7 -1/2 -2/7 1/2⟩]
Eigenmonzos (unchanged-intervals): 2, 11/10, 9/7

Optimal ET sequence: 10, 15, 26, 31, 41, 72, 118, 159, 190

Badness: 0.234 × 10-3

Projection pairs: 3 1024/343 11 131072/12005 to 2.5.7

Scales: portent26

Portending

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384

Mapping: [⟨1 1 0 3 5 6], ⟨0 3 0 -1 4 12], ⟨0 0 1 0 -1 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.9128, ~5/4 = 384.7278

Optimal ET sequence: 15, 26, 41, 46, 72, 87, 159, 477cdf

Badness: 0.627 × 10-3

Portentous

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624

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

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.6843, ~5/4 = 384.9830

Optimal ET sequence: 15, 31, 56, 72, 87, 103, 159, 190, 262df, 349f, 452cdef, 611cdef

Badness: 0.662 × 10-3

Ominous

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384, 441/440

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 0 -1 4 -10], ⟨0 0 1 0 -1 2]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.4088, ~5/4 = 385.3825

Optimal ET sequence: 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdef

Badness: 0.751 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440

Mapping: [⟨1 1 0 3 5 1 1], ⟨0 3 0 -1 4 -10 -8], ⟨0 0 1 0 -1 2 2]]

Mapping generators: 2, 8/7, 5

Mapping to lattice: [⟨0 1 1 0 0 -1 0], ⟨0 -1 -1 0 -1 2 1]]

Lattice basis:

8/7 length = 0.3859, 6/5 length = 1.1303
Angle (8/7, 6/5) = 98.6015

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.4313, ~5/4 = 385.2890

Minimax tuning:

  • 17-odd-limit
[[1 0 0 0 0 0 0⟩, [7/4 9/10 0 0 -3/10 -3/20 0⟩, [5/2 7/5 0 0 -4/5 1/10 0⟩, [11/4 -3/10 0 0 1/10 1/20 0⟩, [7/2 -1/5 0 0 2/5 -3/10 0⟩, [7/2 -1/5 0 0 -3/5 7/10 0⟩, [4 2/5 0 0 -4/5 3/5 0⟩]
Eigenmonzos (unchanged-intervals): 2, 13/11, 11/9

Optimal ET sequence: 26, 31, 46, 72, 103, 149, 175f, 221ef

Badness: 0.612 × 10-3

Momentous

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384

Mapping: [⟨1 1 0 3 5 7], ⟨0 3 0 -4 1 -5], ⟨0 0 1 0 -1 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.4088, ~5/4 = 385.3825

Optimal ET sequence: 10, 31, 41, 46, 77, 87, 118, 164, 205d, 574def

Badness: 0.832 × 10-3

Foreboding

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 275/273

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

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.3664, ~5/4 = 382.3425

Optimal ET sequence: 5, 10, 15, 26, 31, 41, 72f, 185cf

Badness: 0.873 × 10-3

Portannic

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 10985/10976

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

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.6434, ~14/13 = 128.3440

Optimal ET sequence: 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff

Badness: 1.783 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 385/384, 441/440, 8624/8619

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

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.6803, ~14/13 = 128.4149

Optimal ET sequence: 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def

Badness: 1.303 × 10-3

Gamel

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1029/1024

Mapping: [⟨1 1 0 3 -1], ⟨0 3 0 -1 11], ⟨0 0 1 0 1]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 233.2909, ~5/4 = 384.2120

Optimal ET sequence: 10, 31, 41, 72

Badness: 0.853 × 10-3

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