Sengic temperaments: Difference between revisions

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* ''[[Niner]]'' (+128/125) → [[Augmented family #Niner|Augmented family]]

* ''[[Ammonite]]'' (+250/243) → [[Porcupine family #Ammonite|Porcupine family]]

* ''[[Sengasec]]'' (+16807/16384) → [[Quintosec family #Sengasec|Quintosec family]]

* ''[[Sycamore]]'' (+875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]

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Comma list: 121/120, 441/440, 686/675

~~Mapping:~~ {{~~mapping~~| 1 0 -31 -21 -14 | 0 1 21 15 11 }}

Optimal tunings:

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Comma list: 91/90, 121/120, 169/168, 352/351

~~Mapping:~~ {{~~mapping~~| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}

Optimal tunings:

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Comma list: 91/90, 121/120, 136/135, 154/153, 169/168

~~Mapping:~~ {{~~mapping~~| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}

{{Mapping|legend=0| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}

Optimal tunings:

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Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168

~~Mapping:~~ {{~~mapping~~| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}

{{Mapping|legend=0| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}

Optimal tunings:

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== Twothirdtonic ==

Twothirdtonic tempers out 6144/6125, the [[porwell comma]], in addition to the senga, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.

Twothirdtonic tempers out 6144/6125, the [[porwell comma]], in addition to the senga, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot (or triseph). [[46edo]] may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

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[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}

* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}} (~15/14 = 130.3255{{c}})

: [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}} (~15/14 = 130.4254{{c}})

: error map: {{val| 0.000 +2.515 +4.962 -3.505 }}

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Comma list: 121/120, 176/175, 686/675

~~Mapping:~~ {{~~mapping~~| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}

Optimal tunings:

* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}

* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}} (~15/14 = 130.3983{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}} (~15/14 = 130.4400{{c}})

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Comma list: 91/90, 121/120, 169/168, 176/175

~~Mapping:~~ {{~~mapping~~| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}

Optimal tunings:

* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}

* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}} (~14/13 = 130.4039{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}} (~14/13 = 130.4107{{c}})

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[[Category:Sengic temperaments| ]]

[[Category:Temperament collections]]

[[Category:Catalogs of rank-2 temperaments]]

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 * ''[[Niner]]'' (+128/125) → [[Augmented family #Niner|Augmented family]]
 * ''[[Ammonite]]'' (+250/243) → [[Porcupine family #Ammonite|Porcupine family]]
+* ''[[Sengasec]]'' (+16807/16384) → [[Quintosec family #Sengasec|Quintosec family]]
 * ''[[Sycamore]]'' (+875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
@@ Line 50: / Line 51: @@
 Comma list: 121/120, 441/440, 686/675
-Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
+{{Mapping|legend=0| 1 0 -31 -21 -14 | 0 1 21 15 11 }}
 Optimal tunings:
@@ Line 65: / Line 66: @@
 Comma list: 91/90, 121/120, 169/168, 352/351
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
+{{Mapping|legend=0| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}
 Optimal tunings:
@@ Line 80: / Line 81: @@
 Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
+{{Mapping|legend=0| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
 Optimal tunings:
@@ Line 95: / Line 96: @@
 Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168
-Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}
+{{Mapping|legend=0| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}
 Optimal tunings:
@@ Line 106: / Line 107: @@
 == Twothirdtonic ==
-Twothirdtonic tempers out 6144/6125, the [[porwell comma]], in addition to the senga, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.
+Twothirdtonic tempers out 6144/6125, the [[porwell comma]], in addition to the senga, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot (or triseph). [[46edo]] may be recommended as a tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 116: / Line 117: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}
+* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}} (~15/14 = 130.3255{{c}})
 : [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}} (~15/14 = 130.4254{{c}})
 : error map: {{val| 0.000 +2.515 +4.962 -3.505 }}
@@ Line 130: / Line 131: @@
 Comma list: 121/120, 176/175, 686/675
-Mapping: {{mapping| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}
+{{Mapping|legend=0| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}
 Optimal tunings:
-* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}
+* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}} (~15/14 = 130.3983{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}} (~15/14 = 130.4400{{c}})
 {{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
@@ Line 145: / Line 146: @@
 Comma list: 91/90, 121/120, 169/168, 176/175
-Mapping: {{mapping| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}
+{{Mapping|legend=0| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}
 Optimal tunings:
-* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}
+* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}} (~14/13 = 130.4039{{c}})
-* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}} (~14/13 = 130.4107{{c}})
 {{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
@@ Line 156: / Line 157: @@
 [[Category:Sengic temperaments| ]] <!-- main article -->
+[[Category:Temperament collections]]
+[[Category:Catalogs of rank-2 temperaments]]