5th-octave temperaments: Difference between revisions

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Thunder is a weak extension of slendrismic (above), [[rainy]] and [[cata]], with a generator of a slightly sharp ~6/5 (befitting of any [[kleismic]] temperament), three of which making [[26/15]]~[[19/11]]. More interesting though is that the period is [[5edo|1\5]]; slendrismic gives this a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])<sup>2</sup> = [[8/7]] * [[1029/1024|S7/S8]] which is a significant interval as it is the "harmonic 5edostep" (in that it's a [[rooted]] (/2^n) interval in the 2.3.7 subgroup that approximates 1\5 very well). This temperament gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of this temperament. In fact, this temperament combines many convergents and semiconvergents to intervals of [[5edo]] into a single, high-limit temperament:

1\5 = [[23/20]] = [[31/27]] = [[54/47]] = [[85/74]] and 2\5 = [[29/22]] = [[33/25]] = [[62/47]] = [[95/72]] = [[128/97]].

1\5 = [[23/20]] = [[31/27]] = [[85/74]] = [[54/47]] (which Thunder also equates with [[63/50]]), and 2\5 = [[33/25]] = [[95/72]] = [[29/22]] = [[62/47]] = [[128/97]] (which Thunder also equates with [[37/28]]).

Thunder can be thought of as the [[125edo|125f]] & [[140edo|140]] temperament in the [[37-limit]] add-47 add-97, with both tunings notable in all corresponding limits.

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[[Badness]] (Dirichlet): 1.715

== Pentonismic (rank-5) ==

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 Thunder is a weak extension of slendrismic (above), [[rainy]] and [[cata]], with a generator of a slightly sharp ~6/5 (befitting of any [[kleismic]] temperament), three of which making [[26/15]]~[[19/11]]. More interesting though is that the period is [[5edo|1\5]]; slendrismic gives this a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])<sup>2</sup> = [[8/7]] * [[1029/1024|S7/S8]] which is a significant interval as it is the "harmonic 5edostep" (in that it's a [[rooted]] (/2^n) interval in the 2.3.7 subgroup that approximates 1\5 very well). This temperament gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of this temperament. In fact, this temperament combines many convergents and semiconvergents to intervals of [[5edo]] into a single, high-limit temperament:
-\5 = [[23/20]] = [[31/27]] = [[54/47]] = [[85/74]] and 2\5 = [[29/22]] = [[33/25]] = [[62/47]] = [[95/72]] = [[128/97]].
+\5 = [[23/20]] = [[31/27]] = [[85/74]] = [[54/47]] (which Thunder also equates with [[63/50]]), and 2\5 = [[33/25]] = [[95/72]] = [[29/22]] = [[62/47]] = [[128/97]] (which Thunder also equates with [[37/28]]).
 Thunder can be thought of as the [[125edo|125f]] & [[140edo|140]] temperament in the [[37-limit]] add-47 add-97, with both tunings notable in all corresponding limits.
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 [[Badness]] (Dirichlet): 1.715
 == Pentonismic (rank-5) ==