Sengic temperaments: Difference between revisions

Line 106:

== 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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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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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}})

@@ Line 106: / Line 106: @@
 == 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 116: @@
 [[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 133: / Line 133: @@
 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 148: / Line 148: @@
 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 }}