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]].

~~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|2.3.5.7.13-subgroup]] temperament due to the identity 686/675 ({{S|13⋅S142}}) = ([[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 was discovered and named in 2005.

[[Subgroup]]: 2.3.5.7

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[[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5

=== Overview to extensions ===

First 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|2.3.5.7.13-subgroup]] temperament due to the identity 686/675 = ([[169/168]])⋅([[196/195]])2, as we can see from its [[S-expression]], S13⋅S142. This identifies the last generator as [[13/12]]~[[14/13]]~15/14. This extension is considered immediately below.

11-limit temperaments considered below are demeter, krypton, and sensigh.

=== 2.3.5.7.13 subgroup ===

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 {{Technical data page}}
 The '''sengic family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the senga a.k.a. sengic comma, [[686/675]].
-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|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 was discovered and named in 2005.
 [[Subgroup]]: 2.3.5.7
@@ Line 25: / Line 23: @@
 [[Projection pair]]s: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5
+=== Overview to extensions ===
+First 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|2.3.5.7.13-subgroup]] temperament due to the identity 686/675 = ([[169/168]])⋅([[196/195]])<sup>2</sup>, as we can see from its [[S-expression]], S13⋅S14<sup>2</sup>. This identifies the last generator as [[13/12]]~[[14/13]]~15/14. This extension is considered immediately below.
+-limit temperaments considered below are demeter, krypton, and sensigh.
 === 2.3.5.7.13 subgroup ===