Porwell family

The porwell family of rank-3 temperaments tempers out the porwell comma, [11 1 -3 -2⟩ = 6144/6125.

Porwell aka hewuermity

Subgroup: 2.3.5.7

Comma list: 6144/6125

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

mapping generators: ~2, ~3, ~35/32

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.3482, ~35/32 = 157.4677

Minimax tuning:

• 7- and 9-odd-limit

[[1 0 0 0⟩, [0 1 0 0⟩, [11/5 1/5 2/5 -2/5⟩, [11/5 1/5 -3/5 3/5⟩]

eigenmonzo (unchanged-interval) basis: 2.3.7/5

Optimal ET sequence: 15, 22, 31, 46, 53, 84, 99, 229, 251, 282, 381

Badness: 0.142 × 10^-3

Projection pairs: 3 6125/2048 to 2.5.7

Zeus

Main article: Zeus

See also: Biyatismic clan § Zeus

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175

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

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

Lattice basis:

11/10, 11/8

Angle (11/10, 11/8) = 87.464 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0⟩, [22/9 2/9 1/3 -4/9 0⟩, [22/9 2/9 -2/3 5/9 0⟩, [10/3 2/3 0 -1/3 0⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5.9/7

Optimal ET sequence: 15, 22, 31, 46, 53, 68, 77, 99, 130e

Badness: 0.400 × 10^-3

Projection pairs: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5

Zeus11[22] hobbit transversal

33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,

11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,

11/6, 15/8, 64/33, 2

Zeus11[24] hobbit transversal

33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,

11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,

11/6, 15/8, 64/33, 2

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350

Mapping: [⟨1 0 1 4 2 7], ⟨0 1 1 -1 1 -2], ⟨0 0 -2 3 -1 -1]]

Mapping to lattice: [⟨0 1 -1 2 0 -3], ⟨0 1 1 -1 1 -2]]

Lattice basis:

11/10 length = 0.7898, 11/8 length = 1.002

Angle (11/10, 11/8) = 106.7439 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582

Minimax tuning:

• 13-odd-limit

[[1 0 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0 0⟩, [22/9 2/9 1/3 -4/9 0 0⟩, [22/9 2/9 -2/3 5/9 0 0⟩, [10/3 2/3 0 -1/3 0 0⟩, [14/3 -8/3 1 1/3 0 0⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5.9/7

• 15-odd-limit

[[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [11/5 1/5 2/5 -2/5 0 0⟩, [11/5 1/5 -3/5 3/5 0 0⟩, [13/5 3/5 1/5 -1/5 0 0⟩, [38/5 -12/5 1/5 -1/5 0 0⟩]

eigenmonzo (unchanged-interval) basis: 2.3.7/5

Optimal ET sequence: 15, 22, 31, 46, 53, 77, 99, 130e

Badness: 0.934 × 10^-3

Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5

Zeus13[22] hobbit transversal

260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2

Tinia

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 176/175

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.3420, ~11/10 = 155.3666

Optimal ET sequence: 7, 9, 15, 22f, 24, 31

Badness: 0.808 × 10^-3

Jupiter

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625

Mapping: [⟨1 0 1 4 -5], ⟨0 1 1 -1 6], ⟨0 0 -2 3 -8]]

Optimal ET sequence: 22, 31, 53, 84e, 99e, 121, 130, 152, 282, 434de, 465d, 617de, 747cde, 899cdde

Badness: 0.562 × 10^-3

Hera

Subgroup: 2.3.5.7.11

Comma list: 6144/6125, 8019/8000

Mapping: [⟨1 0 1 4 9], ⟨0 1 1 -1 -3], ⟨0 0 -2 3 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8035, ~35/32 = 157.1649

Optimal ET sequence: 46, 53, 84, 99, 130, 183, 313, 359, 542c, 672c

Badness: 2.23 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 2080/2079, 4096/4095

Mapping: [⟨1 0 1 4 9 7], ⟨0 1 1 -1 -3 -2], ⟨0 0 -2 3 -6 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.6641, ~35/32 = 157.1899

Optimal ET sequence: 46, 53, 84, 99, 130, 183, 313

Badness: 1.30 × 10^-3

Semiporwell

Subgroup: 2.3.5.7.11

Comma list: 6144/6125, 9801/9800

Mapping: [⟨2 0 0 11 14], ⟨0 1 1 -1 -2], ⟨0 0 2 -3 -1]]

mapping generators: ~99/70, ~3, ~128/99

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 702.4613, ~128/99 = 442.4903

Optimal ET sequence: 22, 46, 68, 84, 106, 130, 152, 282, 328, 480, 1112bccddee, 1242ccddee, 1722bcccdddeee

Badness: 1.25 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3584/3575

Mapping: [⟨2 0 0 11 14 15], ⟨0 1 1 -1 -2 -1], ⟨0 0 2 -3 -1 -6]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 702.3964, ~84/65 = 442.5902

Optimal ET sequence: 22f, 38e, 46, 68, 84, 130, 198, 282, 328, 480f, 610cdef, 808cdeff, 1090ccdeefff, 1418bcccddeeefff

Badness: 1.22 × 10^-3