Lehmerismic temperaments: Difference between revisions
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{{Mapping|legend=1| 3 0 0 4 8 | 0 2 0 -4 1 | 0 0 1 2 0 }} | {{Mapping|legend=1| 3 0 0 4 8 | 0 2 0 -4 1 | 0 0 1 2 0 }} | ||
: mapping generators: ~63/50, ~400/231, ~5 | : mapping generators: ~63/50, ~400/231, ~5 | ||
{{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 }} | {{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]] (Smith): 0.259 × 10<sup>-3</sup> | ||
=== Borneo === | === Borneo === | ||
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Mapping: {{mapping| 3 0 0 4 8 -3 | 0 2 0 -4 1 3 | 0 0 1 2 0 1 }} | Mapping: {{mapping| 3 0 0 4 8 -3 | 0 2 0 -4 1 3 | 0 0 1 2 0 1 }} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }} | ||
Badness: 0.549 × 10<sup>-3</sup> | Badness (Smith): 0.549 × 10<sup>-3</sup> | ||
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 | 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 | ||
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{{Mapping|legend=1| 1 0 5 8 1 | 0 1 1 3 2 | 0 0 -6 -14 -1 }} | {{Mapping|legend=1| 1 0 5 8 1 | 0 1 1 3 2 | 0 0 -6 -14 -1 }} | ||
: mapping generators: ~2, ~3, ~18/11 | : mapping generators: ~2, ~3, ~18/11 | ||
{{Optimal ET sequence|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]] (Smith): 0.349 × 10<sup>-3</sup> | ||
== Ganesha == | == Ganesha == | ||
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{{Mapping|legend=1| 1 0 0 -22 -9 | 0 1 0 1 2 | 0 0 1 10 4 }} | {{Mapping|legend=1| 1 0 0 -22 -9 | 0 1 0 1 2 | 0 0 1 10 4 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.2251{{c}}, ~5/4 = 386.6908{{c}} | ||
{{Optimal ET sequence|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]] (Smith): 0.390 × 10<sup>-3</sup> | ||
== Hanuman == | == Hanuman == | ||
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{{Mapping|legend=1| 1 2 0 -8 1 | 0 -3 0 11 1 | 0 0 1 4 1 }} | {{Mapping|legend=1| 1 2 0 -8 1 | 0 -3 0 11 1 | 0 0 1 4 1 }} | ||
: mapping generators: ~2, ~11/10, ~5 | : mapping generators: ~2, ~11/10, ~5 | ||
{{Optimal ET sequence|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]] (Smith): 0.500 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }} | Mapping: {{mapping| 1 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }} | ||
Badness: 0.653 × 10<sup>-3</sup> | Badness (Smith): 0.653 × 10<sup>-3</sup> | ||
== Lux == | == Lux == | ||
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{{Mapping|legend=1| 1 0 0 6 5 | 0 1 4 -1 -3 | 0 0 -5 -2 4 }} | {{Mapping|legend=1| 1 0 0 6 5 | 0 1 4 -1 -3 | 0 0 -5 -2 4 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.1117{{c}}, ~55/48 = 235.5784{{c}} | ||
{{Optimal ET sequence|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]] (Smith): 0.508 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 0 0 6 5 6 | 0 1 4 -1 -3 -5 | 0 0 -5 -2 4 7 }} | Mapping: {{mapping| 1 0 0 6 5 6 | 0 1 4 -1 -3 -5 | 0 0 -5 -2 4 7 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.0674{{c}}, ~55/48 = 235.5870{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }} | ||
Badness: 0.360 × 10<sup>-3</sup> | Badness (Smith): 0.360 × 10<sup>-3</sup> | ||
== Galaxy == | == Galaxy == | ||
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{{Mapping|legend=1| 1 0 5 -4 -5 | 0 1 3 -3 -3 | 0 0 -9 14 16 }} | {{Mapping|legend=1| 1 0 5 -4 -5 | 0 1 3 -3 -3 | 0 0 -9 14 16 }} | ||
: mapping generators: ~2, ~3, ~6125/3456 | : mapping generators: ~2, ~3, ~6125/3456 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9856{{c}}, ~6125/3456 = 991.0688{{c}} | ||
{{Optimal ET sequence|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]] (Smith): 0.841 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 0 5 -4 -5 -4 | 0 1 3 -3 -3 -4 | 0 0 -9 14 16 17 }} | Mapping: {{mapping| 1 0 5 -4 -5 -4 | 0 1 3 -3 -3 -4 | 0 0 -9 14 16 17 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.9368{{c}}, ~484/273 = 991.0597{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }} | ||
Badness: 0.463 × 10<sup>-3</sup> | Badness (Smith): 0.463 × 10<sup>-3</sup> | ||
=== 17-limit === | === 17-limit === | ||
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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 ET sequence|legend= | {{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }} | ||
Badness: 0.933 × 10<sup>-3</sup> | Badness (Smith): 0.933 × 10<sup>-3</sup> | ||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Lehmerismic temperaments| ]] <!-- main article --> | [[Category:Lehmerismic temperaments| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
Revision as of 15:06, 19 April 2026
- 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.
These are rank-3 temperaments tempering out 3025/3024. Temperaments discussed elsewhere are:
- Oxpecker (+121/120 or 126/125) → Biyatismic clan
- Manwe (+176/175) → Valinorsmic clan
- Spectacle (+225/224 or 243/242) → Rastmic rank-3 clan
- Portent (+385/384 or 441/440) → Gamelismic family
- Indra (+540/539) → Mirkwai family
- Tolerant (+896/891) → Pentacircle clan
- Freya (+2401/2400) → Breed family
- Triglav (+3136/3125) → Hemimean family
- Thor (+4375/4374) → Ragismic family
- Kapo (+5120/5103) → Hemifamity family
Considered below are tyr, skadi, ganesha, hanuman, lux, and galaxy. For the rank-4 temperament, see Rank-4 temperament #Lehmerismic (3025/3024).
Tyr
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
Optimal ET sequence: 15, 24d, 33cd, 39d, 54c, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee
Badness (Smith): 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 ET sequence: 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270
Badness (Smith): 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
Optimal ET sequence: 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce
Badness (Smith): 0.349 × 10-3
Ganesha
- For the 7-limit version, see Miscellaneous 7-limit temperaments #Quasiorwellismic.
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 = 1200.0000 ¢, ~3/2 = 702.2251 ¢, ~5/4 = 386.6908 ¢
Optimal ET sequence: 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502
Badness (Smith): 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
Optimal ET sequence: 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998
Badness (Smith): 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 ET sequence: 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c
Badness (Smith): 0.653 × 10-3
Lux
The last generator of lux, represented by 55/48, exceeds 8/7 by 385/384, which is equated with a number of important superparticular ratios in the 13-limit: 325/324, 352/351, 364/363, and 441/440.
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]]
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 702.1117 ¢, ~55/48 = 235.5784 ¢
Optimal ET sequence: 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d
Badness (Smith): 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 = 1200.0000 ¢, ~3/2 = 702.0674 ¢, ~55/48 = 235.5870 ¢
Optimal ET sequence: 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d
Badness (Smith): 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
Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 701.9856 ¢, ~6125/3456 = 991.0688 ¢
Optimal ET sequence: 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665
Badness (Smith): 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 = 1200.0000 ¢, ~3/2 = 701.9368 ¢, ~484/273 = 991.0597 ¢
Optimal ET sequence: 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125
Badness (Smith): 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 ET sequence: 46, 75e, 103, 121, 149, 167, 224, 270, 494g
Badness (Smith): 0.933 × 10-3