Sengic family: Difference between revisions

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== 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

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: mapping generators: ~2, ~3, ~15/14

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 703.7873, ~15/14 = 129.6451

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.7873{{c}}, ~15/14 = 129.6451{{c}}

[[Badness]]: 0.320 × 10-3

[[Badness]] (Smith): 0.320 × 10-3

[[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 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 704.5918, ~14/13 = 129.7585

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 704.5918{{c}}, ~14/13 = 129.7585{{c}}

{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}

{{Optimal ET sequence|legend=0| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}

Badness: 0.320 × 10-3

Badness (Smith): 0.320 × 10-3

== Demeter ==

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[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~3/2 = 705.518, ~15/14 = 130.039

[[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 }}

[[Badness]]: 1.32 × 10-3

[[Badness]] (Smith): 1.32 × 10-3

[[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 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~3/2 = 705.113, ~14/13 = 129.673

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.113{{c}}, ~14/13 = 129.673{{c}}

{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}

{{Optimal ET sequence|legend=0| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}

Badness: 0.977 × 10-3

Badness (Smith): 0.977 × 10-3

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 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~3/2 = 705.147, ~14/13 = 129.700

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.147{{c}}, ~14/13 = 129.700{{c}}

{{Optimal ET sequence|legend=1| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}

{{Optimal ET sequence|legend=0| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}

Badness: 0.830 × 10-3

Badness (Smith): 0.830 × 10-3

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11

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[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~3/2 = 705.978, ~12/11 = 132.544

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 705.978{{c}}, ~12/11 = 132.544{{c}}

[[Badness]]: 0.856 × 10-3

[[Badness]] (Smith): 0.856 × 10-3

[[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 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~3/2 = 706.029, ~14/13 = 132.428

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 706.029{{c}}, ~14/13 = 132.428{{c}}

Badness: 0.727 × 10-3

Badness (Smith): 0.727 × 10-3

Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11

== ~~Notes~~ ==

== References ==

[[Category:Temperament families]]

@@ Line 5: / Line 5: @@
 == 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
@@ Line 12: / Line 12: @@
 {{Mapping|legend=1| 1 0 2 1 | 0 1 0 1 | 0 0 3 2 }}
 : mapping generators: ~2, ~3, ~15/14
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 703.7873, ~15/14 = 129.6451
+[[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 }}
-[[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
@@ Line 30: / Line 29: @@
 Sval mapping: {{mapping| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 704.5918, ~14/13 = 129.7585
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 704.5918{{c}}, ~14/13 = 129.7585{{c}}
-{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df }}
+{{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 ==
@@ Line 45: / Line 44: @@
 {{Mapping|legend=1| 1 0 2 1 -3 | 0 1 0 1 4 | 0 0 3 2 1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.518, ~15/14 = 130.039
+[[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 }}
-[[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
@@ Line 60: / Line 59: @@
 Mapping: {{mapping| 1 0 2 1 -3 2 | 0 1 0 1 4 1 | 0 0 3 2 1 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.113, ~14/13 = 129.673
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.113{{c}}, ~14/13 = 129.673{{c}}
-{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46, 102, 148f }}
+{{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
@@ Line 75: / Line 74: @@
 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 = 1\1, ~3/2 = 705.147, ~14/13 = 129.700
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 705.147{{c}}, ~14/13 = 129.700{{c}}
-{{Optimal ET sequence|legend=1| 10, 17cg, 19eg, 27eg, 29g, 46, 102, 148f }}
+{{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
@@ Line 90: / Line 89: @@
 {{Mapping|legend=1| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.978, ~12/11 = 132.544
+[[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 }}
-[[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
@@ Line 105: / Line 104: @@
 Mapping: {{mapping| 1 0 2 1 2 2 | 0 1 0 1 1 1 | 0 0 3 2 -1 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 706.029, ~14/13 = 132.428
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 706.029{{c}}, ~14/13 = 132.428{{c}}
-{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }}
+{{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
-== Notes ==
+== References ==
 [[Category:Temperament families]]