Lehmerismic temperaments: Difference between revisions

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== Lehmerismic ==
== Lehmerismic ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[3025/3024]]
[[Comma list]]: [[3025/3024]]
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{{See also| Landscape family }}
{{See also| Landscape family }}


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 102487/102400
[[Comma list]]: 3025/3024, 102487/102400
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Mapping: [{{val| 3 0 0 4 8 -3 }}, {{val| 0 2 0 -4 1 3 }}, {{val| 0 0 1 2 0 1 }}]
Mapping: [{{val| 3 0 0 4 8 -3 }}, {{val| 0 2 0 -4 1 3 }}, {{val| 0 0 1 2 0 1 }}]


Vals: {{Val list| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }}
Optimal GPV sequence: {{Val list| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }}


Badness: 0.549 × 10<sup>-3</sup>
Badness: 0.549 × 10<sup>-3</sup>
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== Skadi ==
== Skadi ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 703125/702464
[[Comma list]]: 3025/3024, 703125/702464
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== Ganesha ==
== Ganesha ==
=== 7-limit (quasiorwellismic) ===
=== 7-limit (quasiorwellismic) ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 29360128/29296875
[[Comma list]]: 29360128/29296875
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Mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5


[[POTE generator]]s: ~3/2 = 702.0901, ~5/4 = 386.6931
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.0901, ~5/4 = 386.6931


{{Val list|legend=1| 31, 87, 118, 152, 239, 270, 571, 723, 841, 993, 1263, 1564c, 1834c, 2104c }}
{{Val list|legend=1| 31, 87, 118, 152, 239, 270, 571, 723, 841, 993, 1263, 1564c, 1834c, 2104c }}
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=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 5632/5625
[[Comma list]]: 3025/3024, 5632/5625
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[[Mapping]]: [{{val| 1 0 0 -22 -9 }}, {{val| 0 1 0 1 2 }}, {{val| 0 0 1 10 4 }}]
[[Mapping]]: [{{val| 1 0 0 -22 -9 }}, {{val| 0 1 0 1 2 }}, {{val| 0 0 1 10 4 }}]


[[POTE generator]]s: ~3/2 = 702.2251, ~5/4 = 386.6908
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2251, ~5/4 = 386.6908


{{Val list|legend=1| 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502 }}
{{Val list|legend=1| 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502 }}
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== Hanuman ==
== Hanuman ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 4000/3993
[[Comma list]]: 3025/3024, 4000/3993
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Mapping: [{{val| 1 2 0 -8 1 -6 }}, {{val| 0 -3 0 11 1 3 }}, {{val| 0 0 1 4 1 4 }}]
Mapping: [{{val| 1 2 0 -8 1 -6 }}, {{val| 0 -3 0 11 1 3 }}, {{val| 0 0 1 4 1 4 }}]


Vals: {{Val list| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}
Optimal GPV sequence: {{Val list| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}


Badness: 0.653 × 10<sup>-3</sup>
Badness: 0.653 × 10<sup>-3</sup>
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== Lux ==
== Lux ==
=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 26843545600/26795786661
[[Comma list]]: 26843545600/26795786661
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{{Multival|legend=1|rank=3| 5 2 13 30 }}
{{Multival|legend=1|rank=3| 5 2 13 30 }}


[[POTE generator]]s: ~3/2 = 702.1152, ~40960/35721 = 235.5941
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1152, ~40960/35721 = 235.5941


{{Val list|legend=1| 41, 87, 128, 142, 178, 183, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}
{{Val list|legend=1| 41, 87, 128, 142, 178, 183, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}
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=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 131072/130977
[[Comma list]]: 3025/3024, 131072/130977
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{{Multival|legend=1|rank=3| 5 2 -4 13 -1 10 30 25 34 71 }}
{{Multival|legend=1|rank=3| 5 2 -4 13 -1 10 30 25 34 71 }}


[[POTE generator]]s: ~3/2 = 702.1117, ~55/48 = 235.5784
[[Optimal tuning]] ([[POTE]]): ~2 = 1/1, ~3/2 = 702.1117, ~55/48 = 235.5784


{{Val list|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}
{{Val list|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}
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Mapping: [{{val| 1 0 0 6 5 6 }}, {{val| 0 1 4 -1 -3 -5 }}, {{val| 0 0 -5 -2 4 7 }}]
Mapping: [{{val| 1 0 0 6 5 6 }}, {{val| 0 1 4 -1 -3 -5 }}, {{val| 0 0 -5 -2 4 7 }}]


POTE generators: ~3/2 = 702.0674, ~55/48 = 235.5870
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0674, ~55/48 = 235.5870


Vals: {{Val list| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
Optimal GPV sequence: {{Val list| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}


Badness: 0.360 × 10<sup>-3</sup>
Badness: 0.360 × 10<sup>-3</sup>


== Galaxy ==
== Galaxy ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 20614528/20588575
[[Comma list]]: 3025/3024, 20614528/20588575
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{{Multival|legend=1| 9 -14 -16 -14 -20 5 34 35 6 21 }}
{{Multival|legend=1| 9 -14 -16 -14 -20 5 34 35 6 21 }}


[[POTE generator]]s: ~3/2 = 701.9856, ~6125/3456 = 991.0688
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.9856, ~6125/3456 = 991.0688


{{Val list|legend=1| 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665 }}
{{Val list|legend=1| 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665 }}
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Mapping: [{{val| 1 0 5 -4 -5 -4 }}, {{val| 0 1 3 -3 -3 -4 }}, {{val| 0 0 -9 14 16 17 }}]
Mapping: [{{val| 1 0 5 -4 -5 -4 }}, {{val| 0 1 3 -3 -3 -4 }}, {{val| 0 0 -9 14 16 17 }}]


POTE generators: ~3/2 = 701.9368, ~484/273 = 991.0597
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9368, ~484/273 = 991.0597


Vals: {{val list| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }}
Optimal GPV sequence: {{val list| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }}


Badness: 0.463 × 10<sup>-3</sup>
Badness: 0.463 × 10<sup>-3</sup>
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Mapping: [{{val| 1 0 5 -4 -5 -4 -1 }}, {{val| 0 1 3 -3 -3 -4 -2 }}, {{val| 0 0 -9 14 16 17 10 }}]
Mapping: [{{val| 1 0 5 -4 -5 -4 -1 }}, {{val| 0 1 3 -3 -3 -4 -2 }}, {{val| 0 0 -9 14 16 17 10 }}]


POTE generators: ~3/2 = 702.1144, ~85/48 = 991.1035
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1144, ~85/48 = 991.1035


Vals: {{val list| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}
Optimal GPV sequence: {{val list| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}


Badness: 0.933 × 10<sup>-3</sup>
Badness: 0.933 × 10<sup>-3</sup>

Revision as of 18:43, 11 December 2022

These are rank-3 temperaments tempering out 3025/3024. Temperaments discussed elsewhere are:

Considered below are tyr, skadi, ganesha, hanuman, lux, and galaxy, but we can begin by looking at the rank-4 temperament.

Lehmerismic

Subgroup: 2.3.5.7.11

Comma list: 3025/3024

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

Mapping generators: ~2, ~3, ~5, ~55/36

Template:Val list

Tyr

See also: Landscape family

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 102487/102400

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

Mapping generators: ~63/50, ~400/231, ~5

Template:Val list

Badness: 0.259 × 10-3

Borneo

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024

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

Optimal GPV sequence: Template:Val list

Badness: 0.549 × 10-3

Complexity spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7

Skadi

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 703125/702464

Mapping: [⟨1 0 5 8 1], ⟨0 1 1 3 2], ⟨0 0 -6 -14 -1]]

Mapping generators: ~2, ~3, ~18/11

Template:Val list

Badness: 0.349 × 10-3

Ganesha

7-limit (quasiorwellismic)

Subgroup: 2.3.5.7

Comma list: 29360128/29296875

Mapping: [⟨1 0 0 -22], ⟨0 1 0 1], ⟨0 0 1 10]]

Mapping generators: ~2, ~3, ~5

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0901, ~5/4 = 386.6931

Template:Val list

Badness: 1.13 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625

Mapping: [⟨1 0 0 -22 -9], ⟨0 1 0 1 2], ⟨0 0 1 10 4]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2251, ~5/4 = 386.6908

Template:Val list

Badness: 0.390 × 10-3

Hanuman

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993

Mapping: [⟨1 2 0 -8 1], ⟨0 -3 0 11 1], ⟨0 0 1 4 1]]

Mapping generators: ~2, ~11/10, ~5

Template:Val list

Badness: 0.500 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079

Mapping: [⟨1 2 0 -8 1 -6], ⟨0 -3 0 11 1 3], ⟨0 0 1 4 1 4]]

Optimal GPV sequence: Template:Val list

Badness: 0.653 × 10-3

Lux

7-limit

Subgroup: 2.3.5.7

Comma list: 26843545600/26795786661

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

Mapping generators: ~2, ~3, ~40960/35721

Wedgie: ⟨⟨⟨ 5 2 13 30 ]]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1152, ~40960/35721 = 235.5941

Template:Val list

Badness: 1.71 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 131072/130977

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

Wedgie: ⟨⟨⟨ 5 2 -4 13 -1 10 30 25 34 71 ]]]

Optimal tuning (POTE): ~2 = 1/1, ~3/2 = 702.1117, ~55/48 = 235.5784

Template:Val list

Badness: 0.508 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 4096/4095

Mapping: [⟨1 0 0 6 5 6], ⟨0 1 4 -1 -3 -5], ⟨0 0 -5 -2 4 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0674, ~55/48 = 235.5870

Optimal GPV sequence: Template:Val list

Badness: 0.360 × 10-3

Galaxy

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 20614528/20588575

Mapping: [⟨1 0 5 -4 -5], ⟨0 1 3 -3 -3], ⟨0 0 -9 14 16]]

Mapping generators: ~2, ~3, ~6125/3456

Wedgie: ⟨⟨ 9 -14 -16 -14 -20 5 34 35 6 21 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9856, ~6125/3456 = 991.0688

Template:Val list

Badness: 0.841 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 3025/3024, 4225/4224

Mapping: [⟨1 0 5 -4 -5 -4], ⟨0 1 3 -3 -3 -4], ⟨0 0 -9 14 16 17]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9368, ~484/273 = 991.0597

Optimal GPV sequence: Template:Val list

Badness: 0.463 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1225/1224, 4225/4224

Mapping: [⟨1 0 5 -4 -5 -4 -1], ⟨0 1 3 -3 -3 -4 -2], ⟨0 0 -9 14 16 17 10]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1144, ~85/48 = 991.1035

Optimal GPV sequence: Template:Val list

Badness: 0.933 × 10-3

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