Gamelismic family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The gamelismic family of rank-3 temperaments tempers out the gamelisma, 1029/1024. The head of this family, gamelismic, tempers out 1029/1024 alone in the full 7-limit, so it has the same 2.3.7-subgroup structure as slendric but giving prime 5 an independent generator.

See Gamelismic clan for the rank-2 temperament without the last generator of gamelismic, and its various extensions.

Gamelismic

Main article: Gamelismic and portent

Subgroup: 2.3.5.7

Comma list: 1029/1024

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

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

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 tunings:

• WE: ~2 = 1200.4859 ¢, ~8/7 = 233.7822 ¢, ~5/4 = 385.3412 ¢

error map: ⟨+0.486 -0.123 -0.001 -1.151]

• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7474 ¢, ~5/4 = 385.5205 ¢

error map: ⟨0.000 -0.713 -0.793 -2.573]

Minimax tuning: c = 1029/1024

• 7-odd-limit: 3, 5, and 7 (1/4)c flat

[[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⟩]

unchanged-interval (eigenmonzo) basis: 2.7/3.5/3

• 9-odd-limit: 3 1/7-comma flat, 5 and 7 2/7-comma flat

[[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⟩]

unchanged-interval (eigenmonzo) basis: 2.5/3.9/7

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

Badness (Sintel): 0.777

Projection pair: 3 1024/343 to 2.5.7

Scales: portent26

Portent

Main article: Gamelismic and portent

Portent tempers out 385/384 and 441/440 and is the main extension of gamelismic. Notice the identity 1029/1024 = (385/384)⋅(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 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 tunings:

• WE: ~2 = 1200.4902 ¢, ~8/7 = 233.7839 ¢, ~5/4 = 385.3191 ¢

error map: ⟨+0.490 -0.113 -0.014 -1.139 -0.031]

• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7616 ¢, ~5/4 = 385.3149 ¢

error map: ⟨0.000 -0.670 -0.999 -2.587 -1.586]

Minimax tuning: c₁ = 1029/1024, c₂ = 385/384

• 11-odd-limit: 3 (1/7)c₁ flat, 5 and 7 (2/7)c₁ flat, 11 (c₂ - (3/7)c₁) 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⟩]

unchanged-interval (eigenmonzo) basis: 2.9/7.11/5

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

Badness (Sintel): 0.281

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 tunings:

• WE: ~2 = 1200.4540 ¢, ~8/7 = 234.0013 ¢, ~5/4 = 384.8733 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.9748 ¢, ~5/4 = 384.8812 ¢

Optimal ET sequence: 15, 26, 31f, 41, 46, 72, 87, 159

Badness (Sintel): 0.587

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 tunings:

• WE: ~2 = 1200.4888 ¢, ~8/7 = 233.7795 ¢, ~5/4 = 385.1398 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7575 ¢, ~5/4 = 385.1447 ¢

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

Badness (Sintel): 0.618

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 tunings:

• WE: ~2 = 1200.7019 ¢, ~8/7 = 233.5453 ¢, ~5/4 = 385.6079 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.4510 ¢, ~5/4 = 385.6739 ¢

Optimal ET sequence: 15f, 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff

Badness (Sintel): 0.702

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 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 tunings:

• WE: ~2 = 1200.6745 ¢, ~8/7 = 233.5625 ¢, ~5/4 = 385.5056 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.4679 ¢, ~5/4 = 385.5892 ¢

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⟩]

unchanged-interval (eigenmonzo) basis: 2.11/9.13/9

Optimal ET sequence: 15f, 20c, 26, 31, 46, 72, 103, 149, 221ef

Badness (Sintel): 0.582

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 tunings:

• WE: ~2 = 1200.0652 ¢, ~8/7 = 234.1856 ¢, ~5/4 = 386.6199 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.1748 ¢, ~5/4 = 386.5951 ¢

Optimal ET sequence: 15f, 21e, 31, 41, 46, 72f, 77, 87, 118, 164, 205d

Badness (Sintel): 0.778

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 tunings:

• WE: ~2 = 1200.2251 ¢, ~8/7 = 233.4102 ¢, ~5/4 = 382.4142 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.4017 ¢, ~5/4 = 382.4261 ¢

Optimal ET sequence: 5, 10, 15, 25e, 26, 31, 41, 72f

Badness (Sintel): 0.816

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]]

mapping generators: ~2, ~8/7, ~14/13

Optimal tunings:

• WE: ~2 = 1200.5451 ¢, ~8/7 = 233.7495 ¢, ~14/13 = 128.4023 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6930 ¢, ~14/13 = 128.3684 ¢

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

Badness (Sintel): 1.67

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 tunings:

• WE: ~2 = 1200.4416 ¢, ~8/7 = 233.7663 ¢, ~14/13 = 128.4622 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7173 ¢, ~14/13 = 128.4269 ¢

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

Badness (Sintel): 1.24

Gamel

This esoteric alternative extension tempers out 540/539, and sometimes comes up in temperament searches. In practice however, it is almost always desirable to further temper it to miracle, in which case is is identical to portent anyway. This enables many more essentially tempered chords while introducing virtually no additional error.

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 tunings:

• WE: ~2 = 1200.6462 ¢, ~8/7 = 233.4166 ¢, ~5/4 = 384.4189 ¢

error map: ⟨+0.646 -1.059 -0.602 -0.304 +1.329]

• CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.3289 ¢, ~5/4 = 384.5627 ¢

error map: ⟨0.000 -1.968 -1.751 -2.155 -0.137]

Optimal ET sequence: 5e, 10, 21e, 26e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde

Badness (Sintel): 1.02