Porwell family: Difference between revisions

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== Hewuermity ==
== Hewuermity ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[6144/6125]]
[[Comma list]]: [[6144/6125]]


[[Mapping]]: [{{val| 1 0 1 4 }}, {{val| 0 1 1 -1 }}, {{val| 0 0 -2 3 }}]
{{Mapping|legend=1| 1 0 1 4 | 0 1 1 -1 | 0 0 -2 3 }}


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


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


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 11/5 1/5 2/5 -2/5 }}, {{monzo| 11/5 1/5 -3/5 3/5 }}]
: {{monzo list| 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]]s (unchanged-intervals): 4/3, 7/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3.7/5


{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 84, 99, 229, 251, 282, 381 }}
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 84, 99, 229, 251, 282, 381 }}
Line 24: Line 24:


== Zeus ==
== Zeus ==
Subgroup: 2.3.5.7.11
{{See also| Biyatismic clan #Zeus }}
 
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 121/120, 176/175
[[Comma list]]: 121/120, 176/175


[[Mapping]]: [{{val| 1 0 1 4 2 }}, {{val| 0 1 1 -1 1 }}, {{val| 0 0 -2 3 1 }}]
{{Mapping|legend=1| 1 0 1 4 2 | 0 1 1 -1 1 | 0 0 -2 3 1 }}
 
Mapping generators: ~2, ~3, ~11/10


[[Mapping to lattice]]: [{{val| 0 1 -1 2 0 }}, {{val| 0 1 1 -1 1 }}]
[[Mapping to lattice]]: [{{val| 0 1 -1 2 0 }}, {{val| 0 1 1 -1 1 }}]
Line 38: Line 38:
: Angle (11/10, 11/8) = 87.464 degrees
: Angle (11/10, 11/8) = 87.464 degrees


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


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 9/7, 7/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.9/7


{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 68, 77, 99, 130e }}
Line 69: Line 69:
Comma list: 121/120, 176/175, 351/350
Comma list: 121/120, 176/175, 351/350


Mapping: [{{val| 1 0 1 4 2 7 }}, {{val| 0 1 1 -1 1 -2 }}, {{val| 0 0 -2 3 -1 -1 }}]
Mapping: {{mapping| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}


Mapping to lattice: [{{val| 0 1 -1 2 0 -3 }}, {{val| 0 1 1 -1 1 -2 }}]
Mapping to lattice: [{{val| 0 1 -1 2 0 -3 }}, {{val| 0 1 1 -1 1 -2 }}]
Line 77: Line 77:
: Angle (11/10, 11/8) = 106.7439 degrees
: Angle (11/10, 11/8) = 106.7439 degrees


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


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit  
* 13-odd-limit  
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 0 }}, {{monzo| 14/3 -8/3 1 1/3 0 0 }}]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 11/9 10/9 -1/3 -2/9 0 0 }}, {{monzo| 22/9 2/9 1/3 -4/9 0 0 }}, {{monzo| 22/9 2/9 -2/3 5/9 0 0 }}, {{monzo| 10/3 2/3 0 -1/3 0 0 }}, {{monzo| 14/3 -8/3 1 1/3 0 0 }}]
: Eigenmonzos (unchanged-intervals): 2, 9/7, 7/5
: eigenmonzo (unchanged-interval) basis: 2.9/5.9/7
* 15-odd-limit
* 15-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 11/5 1/5 2/5 -2/5 0 0 }}, {{monzo| 11/5 1/5 -3/5 3/5 0 0 }}, {{monzo| 13/5 3/5 1/5 -1/5 0 0 }}, {{monzo| 38/5 -12/5 1/5 -1/5 0 0 }}]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 11/5 1/5 2/5 -2/5 0 0 }}, {{monzo| 11/5 1/5 -3/5 3/5 0 0 }}, {{monzo| 13/5 3/5 1/5 -1/5 0 0 }}, {{monzo| 38/5 -12/5 1/5 -1/5 0 0 }}]
: Eigenmonzos (unchanged-intervals): 2, 4/3, 7/5
: eigenmonzo (unchanged-interval) basis: 2.3.7/5


{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 77, 99, 130e }}
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 77, 99, 130e }}
Line 101: Line 101:
Comma list: 66/65, 121/120, 176/175
Comma list: 66/65, 121/120, 176/175


Mapping: [{{val| 1 0 1 4 2 2 }}, {{val| 0 1 1 -1 1 1 }}, {{val| 0 0 -2 3 -1 -1 }}]
Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}


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


{{Optimal ET sequence|legend=1| 7, 9, 15, 22f, 24, 31 }}
{{Optimal ET sequence|legend=1| 7, 9, 15, 22f, 24, 31 }}
Line 110: Line 110:


== Jupiter ==
== Jupiter ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 5632/5625
[[Comma list]]: 540/539, 5632/5625


[[Mapping]]: [{{val| 1 0 1 4 -5 }}, {{val| 0 1 1 -1 6 }}, {{val| 0 0 -2 3 -8 }}]
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 1 -1 6 | 0 0 -2 3 -8 }}
 
Mapping generators: ~2, ~3, ~35/32


{{Optimal ET sequence|legend=1| 22, 31, 53, 84e, 99e, 121, 130, 152, 282, 434de, 465d, 617de, 747cde, 899cdde }}
{{Optimal ET sequence|legend=1| 22, 31, 53, 84e, 99e, 121, 130, 152, 282, 434de, 465d, 617de, 747cde, 899cdde }}
Line 123: Line 121:


== Hera ==
== Hera ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 6144/6125, 8019/8000
[[Comma list]]: 6144/6125, 8019/8000


[[Mapping]]: [{{val| 1 0 1 4 9 }}, {{val| 0 1 1 -1 -3 }}, {{val| 0 0 -2 3 -6 }}]
{{Mapping|legend=1| 1 0 1 4 9 | 0 1 1 -1 -3 | 0 0 -2 3 -6 }}
 
Mapping generators: ~2, ~3, ~35/32


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


{{Optimal ET sequence|legend=1| 46, 53, 84, 99, 130, 183, 313, 359, 542c, 672c }}
{{Optimal ET sequence|legend=1| 46, 53, 84, 99, 130, 183, 313, 359, 542c, 672c }}
Line 142: Line 138:
Comma list: 351/350, 2080/2079, 4096/4095
Comma list: 351/350, 2080/2079, 4096/4095


Mapping: [{{val| 1 0 1 4 9 7 }}, {{val| 0 1 1 -1 -3 -2 }}, {{val| 0 0 -2 3 -6 -1 }}]
Mapping: {{mapping| 1 0 1 4 9 7 | 0 1 1 -1 -3 -2 | 0 0 -2 3 -6 -1 }}


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


{{Optimal ET sequence|legend=1| 46, 53, 84, 99, 130, 183, 313 }}
{{Optimal ET sequence|legend=1| 46, 53, 84, 99, 130, 183, 313 }}
Line 151: Line 147:


== Semiporwell ==
== Semiporwell ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 6144/6125, 9801/9800
[[Comma list]]: 6144/6125, 9801/9800


[[Mapping]]: [{{val| 2 0 0 11 14 }}, {{val| 0 1 1 -1 -2 }}, {{val| 0 0 2 -3 -1 }}]
{{Mapping|legend=1| 2 0 0 11 14 | 0 1 1 -1 -2 | 0 0 2 -3 -1 }}


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


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


{{Optimal ET sequence|legend=1| 22, 46, 68, 84, 106, 130, 152, 282, 328, 480, 1112bccddee, 1242ccddee, 1722bcccdddeee }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 84, 106, 130, 152, 282, 328, 480, 1112bccddee, 1242ccddee, 1722bcccdddeee }}
Line 170: Line 166:
Comma list: 1716/1715, 2080/2079, 3584/3575
Comma list: 1716/1715, 2080/2079, 3584/3575


Mapping: [{{val| 2 0 0 11 14 15 }}, {{val| 0 1 1 -1 -2 -1 }}, {{val| 0 0 2 -3 -1 -6 }}]
Mapping: {{mapping| 2 0 0 11 14 15 | 0 1 1 -1 -2 -1 | 0 0 2 -3 -1 -6 }}


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


{{Optimal ET sequence|legend=1| 22f, 38e, 46, 68, 84, 130, 198, 282, 328, 480f, 610cdef, 808cdeff, 1090ccdeefff, 1418bcccddeeefff }}
{{Optimal ET sequence|legend=1| 22f, 38e, 46, 68, 84, 130, 198, 282, 328, 480f, 610cdef, 808cdeff, 1090ccdeefff, 1418bcccddeeefff }}
Line 180: Line 176:
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Porwell family| ]] <!-- main article -->
[[Category:Porwell family| ]] <!-- main article -->
[[Category:Porwell| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 03:32, 8 September 2023

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

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:

[[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

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:

[[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

Scales:

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

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