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Lehmerismic temperaments: Difference between revisions

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→Lux: -7-limit (not a reasonable comma itself, replaced with a brief intro in the 13-limit)
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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 22: Line 22:
Mapping generators: ~2, ~3, ~5, ~55/36
Mapping generators: ~2, ~3, ~5, ~55/36


{{Val list|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
{{Optimal ET sequence|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}


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


{{Val list|legend=1| 15, 24d, 33cd, 39d, 54c, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee }}
{{Optimal ET sequence|legend=1| 15, 24d, 33cd, 39d, 54c, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee }}


[[Badness]]: 0.259 × 10<sup>-3</sup>
[[Badness]]: 0.259 × 10<sup>-3</sup>
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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 }}]


Optimal GPV sequence: {{Val list| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }}
{{Optimal ET sequence|legend=1| 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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Mapping generators: ~2, ~3, ~18/11
Mapping generators: ~2, ~3, ~18/11


{{Val list|legend=1| 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce }}
{{Optimal ET sequence|legend=1| 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce }}


[[Badness]]: 0.349 × 10<sup>-3</sup>
[[Badness]]: 0.349 × 10<sup>-3</sup>
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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}
{{Optimal ET sequence|legend=1| 31, 87, 118, 152, 239, 270, 571, 723, 841, 993, 1263, 1564c, 1834c, 2104c }}


[[Badness]]: 1.13 × 10<sup>-3</sup>
[[Badness]]: 1.13 × 10<sup>-3</sup>
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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}
{{Optimal ET sequence|legend=1| 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502 }}


[[Badness]]: 0.390 × 10<sup>-3</sup>
[[Badness]]: 0.390 × 10<sup>-3</sup>
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Mapping generators: ~2, ~11/10, ~5
Mapping generators: ~2, ~11/10, ~5


{{Val list|legend=1| 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998 }}
{{Optimal ET sequence|legend=1| 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998 }}


[[Badness]]: 0.500 × 10<sup>-3</sup>
[[Badness]]: 0.500 × 10<sup>-3</sup>
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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 }}]


Optimal GPV sequence: {{Val list| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}
{{Optimal ET sequence|legend=1| 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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[[Optimal tuning]] ([[POTE]]): ~2 = 1/1, ~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 }}
{{Optimal ET sequence|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}


[[Badness]]: 0.508 × 10<sup>-3</sup>
[[Badness]]: 0.508 × 10<sup>-3</sup>
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Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0674, ~55/48 = 235.5870
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0674, ~55/48 = 235.5870


Optimal GPV sequence: {{Val list| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
{{Optimal ET sequence|legend=1| 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>
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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}
{{Optimal ET sequence|legend=1| 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665 }}


[[Badness]]: 0.841 × 10<sup>-3</sup>
[[Badness]]: 0.841 × 10<sup>-3</sup>
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Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9368, ~484/273 = 991.0597
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9368, ~484/273 = 991.0597


Optimal GPV sequence: {{val list| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }}
{{Optimal ET sequence|legend=1| 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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Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1144, ~85/48 = 991.1035
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1144, ~85/48 = 991.1035


Optimal GPV sequence: {{val list| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}
{{Optimal ET sequence|legend=1| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}


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