Sengic family: Difference between revisions
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== Sengic == | == Sengic == | ||
Sengic is naturally a 2.3.5.7.13 subgroup temperament due to the identity 686/675 = (91/90)(196/195) and 91/90 = (169/168)(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group | ''It's the "thirds", stupid!'']</ref>. | Sengic is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 = (91/90)(196/195) and 91/90 = (169/168)(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by [[Keenan Pepper]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19390.html Yahoo! Tuning Group | ''It's the "thirds", stupid!'']</ref>. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 0 2 1 | 0 1 0 1 | 0 0 3 2 }} | {{Mapping|legend=1| 1 0 2 1 | 0 1 0 1 | 0 0 3 2 }} | ||
: mapping generators: ~2, ~3, ~15/14 | : mapping generators: ~2, ~3, ~15/14 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.7873{{c}}, ~15/14 = 129.6451{{c}} | ||
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46 }} | {{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46 }} | ||
[[Badness]]: 0.320 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.320 × 10<sup>-3</sup> | ||
[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5 | [[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5 | ||
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Sval mapping: {{mapping| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 1 }} | Sval mapping: {{mapping| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 1 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 704.5918{{c}}, ~14/13 = 129.7585{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }} | ||
Badness: 0.320 × 10<sup>-3</sup> | Badness (Smith): 0.320 × 10<sup>-3</sup> | ||
== Demeter == | == Demeter == | ||
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{{Mapping|legend=1| 1 0 2 1 -3 | 0 1 0 1 4 | 0 0 3 2 1 }} | {{Mapping|legend=1| 1 0 2 1 -3 | 0 1 0 1 4 | 0 0 3 2 1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 705.518{{c}}, ~15/14 = 130.039{{c}} | ||
{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 46, 102, 148 }} | {{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 46, 102, 148 }} | ||
[[Badness]]: 1.32 × 10<sup>-3</sup> | [[Badness]] (Smith): 1.32 × 10<sup>-3</sup> | ||
[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11 | [[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11 | ||
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Mapping: {{mapping| 1 0 2 1 -3 2 | 0 1 0 1 4 1 | 0 0 3 2 1 1 }} | Mapping: {{mapping| 1 0 2 1 -3 2 | 0 1 0 1 4 1 | 0 0 3 2 1 1 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.113{{c}}, ~14/13 = 129.673{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 10, 17c, 19e, 27e, 29, 46, 102, 148f }} | ||
Badness: 0.977 × 10<sup>-3</sup> | Badness (Smith): 0.977 × 10<sup>-3</sup> | ||
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11 | Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11 | ||
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Mapping: {{mapping| 1 0 2 1 -3 2 -1 | 0 1 0 1 4 1 3 | 0 0 3 2 1 1 3 }} | Mapping: {{mapping| 1 0 2 1 -3 2 -1 | 0 1 0 1 4 1 3 | 0 0 3 2 1 1 3 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.147{{c}}, ~14/13 = 129.700{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }} | ||
Badness: 0.830 × 10<sup>-3</sup> | Badness (Smith): 0.830 × 10<sup>-3</sup> | ||
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11 | Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11 | ||
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{{Mapping|legend=1| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 -1 }} | {{Mapping|legend=1| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 -1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 705.978{{c}}, ~12/11 = 132.544{{c}} | ||
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }} | {{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }} | ||
[[Badness]]: 0.856 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.856 × 10<sup>-3</sup> | ||
[[Projection pair]]s: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11 | [[Projection pair]]s: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11 | ||
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Mapping: {{mapping| 1 0 2 1 2 2 | 0 1 0 1 1 1 | 0 0 3 2 -1 1 }} | Mapping: {{mapping| 1 0 2 1 2 2 | 0 1 0 1 1 1 | 0 0 3 2 -1 1 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 706.029{{c}}, ~14/13 = 132.428{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 8d, 9, 10, 17c, 19, 27e, 36 }} | ||
Badness: 0.727 × 10<sup>-3</sup> | Badness (Smith): 0.727 × 10<sup>-3</sup> | ||
Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11 | Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11 | ||
== | == References == | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
Revision as of 12:10, 10 October 2025
- 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.
The sengic family of rank-3 temperaments tempers out the senga a.k.a. sengic comma, 686/675.
Temperament discussed elsewhere include sensigh (→ Sensamagic family). Considered below are demeter and krypton.
Sengic
Sengic is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 = (91/90)(196/195) and 91/90 = (169/168)(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by Keenan Pepper in 2011[1].
Subgroup: 2.3.5.7
Mapping: [⟨1 0 2 1], ⟨0 1 0 1], ⟨0 0 3 2]]
- mapping generators: ~2, ~3, ~15/14
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 703.7873 ¢, ~15/14 = 129.6451 ¢
Optimal ET sequence: 8d, 9, 10, 17c, 19, 27, 46
Badness (Smith): 0.320 × 10-3
Projection pairs: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5
2.3.5.7.13 subgroup
Subgroup: 2.3.5.7.13
Comma list: 91/90, 169/168
Sval mapping: [⟨1 0 2 1 2], ⟨0 1 0 1 1], ⟨0 0 3 2 1]]
Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 704.5918 ¢, ~14/13 = 129.7585 ¢
Optimal ET sequence: 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df
Badness (Smith): 0.320 × 10-3
Demeter
Named by Graham Breed in 2011, demeter was found to be locally efficient in the 17-limit among all rank-3 temperaments[2].
Subgroup: 2.3.5.7.11
Comma list: 441/440, 686/675
Mapping: [⟨1 0 2 1 -3], ⟨0 1 0 1 4], ⟨0 0 3 2 1]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 705.518 ¢, ~15/14 = 130.039 ¢
Optimal ET sequence: 10, 17c, 19e, 27e, 46, 102, 148
Badness (Smith): 1.32 × 10-3
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 169/168, 352/351
Mapping: [⟨1 0 2 1 -3 2], ⟨0 1 0 1 4 1], ⟨0 0 3 2 1 1]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 705.113 ¢, ~14/13 = 129.673 ¢
Optimal ET sequence: 10, 17c, 19e, 27e, 29, 46, 102, 148f
Badness (Smith): 0.977 × 10-3
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 136/135, 154/153, 169/168
Mapping: [⟨1 0 2 1 -3 2 -1], ⟨0 1 0 1 4 1 3], ⟨0 0 3 2 1 1 3]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 705.147 ¢, ~14/13 = 129.700 ¢
Optimal ET sequence: 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f
Badness (Smith): 0.830 × 10-3
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11
Krypton
Subgroup: 2.3.5.7.11
Comma list: 56/55, 540/539
Mapping: [⟨1 0 2 1 2], ⟨0 1 0 1 1], ⟨0 0 3 2 -1]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 705.978 ¢, ~12/11 = 132.544 ¢
Optimal ET sequence: 8d, 9, 10, 17c, 19, 27e, 36
Badness (Smith): 0.856 × 10-3
Projection pairs: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 78/77, 91/90
Mapping: [⟨1 0 2 1 2 2], ⟨0 1 0 1 1 1], ⟨0 0 3 2 -1 1]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 706.029 ¢, ~14/13 = 132.428 ¢
Optimal ET sequence: 8d, 9, 10, 17c, 19, 27e, 36
Badness (Smith): 0.727 × 10-3
Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11