Sengic family: Difference between revisions

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The '''sengic family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the senga a.k.a. sengic comma, [[686/675]].

~~Temperament discussed elsewhere include [[sensigh]] (→ [[Sensamagic family #Sensigh|Sensamagic family]]). Considered~~ below are demeter and ~~krypton~~.

Temperaments considered below are demeter, krypton, and sensigh.

== Sengic ==

Sengic is generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 ({{S|13⋅S14<sup>2</sup>}}) = ([[91/90]])⋅([[196/195]]) and 91/90 (S13⋅S14) = ([[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 generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It is naturally a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament due to the identity 686/675 ({{S|13⋅S14<sup>2</sup>}}) = ([[91/90]])⋅([[196/195]]) and 91/90 (S13⋅S14) = ([[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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Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11

== Sensigh ==

Sensigh uses the same mapping as 7-limit [[sensi]] with an independent generator for prime 11.

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 126/125, 245/243

: mapping generators: ~2, ~9/7, ~11

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}, ~11/8 = 552.1736{{c}}

: [[error map]]: {{val| -0.2919 +1.2608 +3.4518 -5.6691 -0.0202 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}, ~11/8 = 551.8069{{c}}

: error map: {{val| 0.0000 +1.4899 +3.8297 -5.2854 +0.4890 }}

[[Badness]] (Sintel): 1.48

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168

Mapping: {{mapping| 1 -1 -1 -2 3 0 | 0 7 9 13 0 10 | 0 0 0 0 1 0 }}

Optimal tunings:

* WE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4379{{c}}, ~11/8 = 550.3462{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}, ~11/8 = 550.7092{{c}}

Badness (Sintel): 0.878

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168

Mapping: {{mapping| 1 -1 -1 -2 3 0 4 | 0 7 9 13 0 10 -1 | 0 0 0 0 1 0 1 }}

Optimal tunings:

* WE: ~2 = 1200.2286{{c}}, ~9/7 = 443.4291{{c}}, ~11/8 = 549.2790{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3707{{c}}, ~11/8 = 549.5775{{c}}

Badness (Sintel): 0.917

== References ==

@@ Line 2: / Line 2: @@
 The '''sengic family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the senga a.k.a. sengic comma, [[686/675]].
-Temperament discussed elsewhere include [[sensigh]] (→ [[Sensamagic family #Sensigh|Sensamagic family]]). Considered below are demeter and krypton.
+Temperaments considered below are demeter, krypton, and sensigh.
 == Sengic ==
-Sengic is generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It is naturally a 2.3.5.7.13-subgroup temperament due to the identity 686/675 ({{S|13⋅S14<sup>2</sup>}}) = ([[91/90]])⋅([[196/195]]) and 91/90 (S13⋅S14) = ([[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 generated by a perfect fifth and a wide semitone of ~[[15/14]], two of which make ~[[7/6]] and three make ~[[5/4]]. It is naturally a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament due to the identity 686/675 ({{S|13⋅S14<sup>2</sup>}}) = ([[91/90]])⋅([[196/195]]) and 91/90 (S13⋅S14) = ([[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 135: / Line 135: @@
 Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11
+== Sensigh ==
+Sensigh uses the same mapping as 7-limit [[sensi]] with an independent generator for prime 11.
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 126/125, 245/243
+{{Mapping|legend=1| 1 -1 -1 -2 3 | 0 7 9 13 0 | 0 0 0 0 1 }}
+: mapping generators: ~2, ~9/7, ~11
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}, ~11/8 = 552.1736{{c}}
+: [[error map]]: {{val| -0.2919 +1.2608 +3.4518 -5.6691 -0.0202 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}, ~11/8 = 551.8069{{c}}
+: error map: {{val| 0.0000 +1.4899 +3.8297 -5.2854 +0.4890 }}
+{{Optimal ET sequence|legend=1| 27e, 38df, 46, 111d }}
+[[Badness]] (Sintel): 1.48
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 91/90, 126/125, 169/168
+Mapping: {{mapping| 1 -1 -1 -2 3 0 | 0 7 9 13 0 10 | 0 0 0 0 1 0 }}
+Optimal tunings:
+* WE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4379{{c}}, ~11/8 = 550.3462{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}, ~11/8 = 550.7092{{c}}
+{{Optimal ET sequence|legend=0| 27e, 38df, 46, 111df }}
+Badness (Sintel): 0.878
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 91/90, 126/125, 154/153, 169/168
+Mapping: {{mapping| 1 -1 -1 -2 3 0 4 | 0 7 9 13 0 10 -1 | 0 0 0 0 1 0 1 }}
+Optimal tunings:
+* WE: ~2 = 1200.2286{{c}}, ~9/7 = 443.4291{{c}}, ~11/8 = 549.2790{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3707{{c}}, ~11/8 = 549.5775{{c}}
+{{Optimal ET sequence|legend=0| 27eg, 38df, 46 }}
+Badness (Sintel): 0.917
 == References ==