Septimal ennealimma: Difference between revisions

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The '''septimal ennealimma''', '''septiennealimma''' or '''no-five ennealimma''' is an unnoticeable 7-limit comma ~~defined as 40353607/40310784 and~~ measuring 1.838 ~~cents~~. It ~~arises as~~ a stack ~~between~~ nine [[7/6]]'s and two octaves.

The '''septimal ennealimma''', '''septiennealimma''' or '''no-five ennealimma''' is an [[unnoticeable comma|unnoticeable]] [[7-limit]] [[comma]] measuring 1.838 [[cent]]s. It is the difference between a stack of nine [[7/6|septimal minor thirds (7/6)]] and two [[2/1|octaves]], or ([[7/6]])9 / ([[2/1]])2.

== Temperaments ==

In the 2.3.7 subgroup, tempering it out leads to the 2.3.7 ~~version~~ of the [[ennealimmal ~~temperament~~]], which is ~~a member~~ of the [[~~tritrizo~~ clan]]. ~~For edos up to circa 3000, it is~~ tempered out if ~~and only if the~~ edo is ~~divisible by~~ 9.

In the 2.3.7 subgroup, tempering it out leads to the 2.3.7-subgroup [[restriction]] of the [[ennealimmal]] temperament, which is the head of the [[septiennealimmal clan]]. It can only possibly be tempered out in an edo if that edo is a multiple of 9.

In the full 7-limit, tempering it out leads to the '''septiennealimmal temperament''' (cf. septimal ennealimmal temperament). You may find a list of good equal temperaments that support this temperament below.

In the full 7-limit, tempering it out leads to the '''septiennealimmal temperament''' (''cf''. septimal ennealimmal temperament). You may find a list of good equal temperaments that support this temperament below.

=== Septiennealimmal ===

[[Subgroup]]: 2.3.5.7

~~[[Mapping]]: [~~{{~~val~~| 9 0 0 11 ~~}}, {{val~~| 0 1 0 1 ~~}}, {{val~~| 0 0 1 0 }}]

~~Mapping~~ generators: ~2592/2401, ~3, ~5

: mapping generators: ~2592/2401, ~3, ~5

[[Optimal tuning]]s:

* [[CTE]]: ~2592/2401 = 133.3333, ~3/2 = 702.0044, ~5/4 = 386.3137

: [[error map]]: {{val| 0.0000 +0.0494 0.0000 -0.1549 }}

* [[CWE]]: ~2592/2401 = 133.3333, ~3/2 = 701.9833, ~5/4 = 386.2877

: error map: {{val| 0.0000 +0.0283 -0.0260 -0.1759 }}

{{Optimal ET sequence|legend=1| 27, 45, 63, 72, 99, 171, 441, 612 }}

[[Badness]]: 0.502 × 10-3

[[Badness]] (Smith): 0.502 × 10-3

== See also ==

* [[Unnoticeable comma]]

* [[Ennealimma]]

[[Category:Septiennealimmal]]

[[Category:Commas named for their periods per equave]]

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-The '''septimal ennealimma''', '''septiennealimma''' or '''no-five ennealimma''' is an unnoticeable 7-limit comma defined as 40353607/40310784 and measuring 1.838 cents. It arises as a stack between nine [[7/6]]'s and two octaves.
+The '''septimal ennealimma''', '''septiennealimma''' or '''no-five ennealimma''' is an [[unnoticeable comma|unnoticeable]] [[7-limit]] [[comma]] measuring 1.838 [[cent]]s. It is the difference between a stack of nine [[7/6|septimal minor thirds (7/6)]] and two [[2/1|octaves]], or ([[7/6]])<sup>9</sup> / ([[2/1]])<sup>2</sup>.
 == Temperaments ==
-In the 2.3.7 subgroup, tempering it out leads to the 2.3.7 version of the [[ennealimmal temperament]], which is a member of the [[tritrizo clan]]. For edos up to circa 3000, it is tempered out if and only if the edo is divisible by 9.
+In the 2.3.7 subgroup, tempering it out leads to the 2.3.7-subgroup [[restriction]] of the [[ennealimmal]] temperament, which is the head of the [[septiennealimmal clan]]. It can only possibly be tempered out in an edo if that edo is a multiple of 9.
-In the full 7-limit, tempering it out leads to the '''septiennealimmal temperament''' (cf. septimal ennealimmal temperament). You may find a list of good equal temperaments that support this temperament below.
+In the full 7-limit, tempering it out leads to the '''septiennealimmal temperament''' (''cf''. septimal ennealimmal temperament). You may find a list of good equal temperaments that support this temperament below.
 === Septiennealimmal ===
 [[Subgroup]]: 2.3.5.7
-[[Mapping]]: [{{val| 9 0 0 11 }}, {{val| 0 1 0 1 }}, {{val| 0 0 1 0 }}]
+{{Mapping|legend=1| 9 0 0 11 | 0 1 0 1 | 0 0 1 0 }}
-Mapping generators: ~2592/2401, ~3, ~5
+: mapping generators: ~2592/2401, ~3, ~5
+[[Optimal tuning]]s:
+* [[CTE]]: ~2592/2401 = 133.3333, ~3/2 = 702.0044, ~5/4 = 386.3137
+: [[error map]]: {{val| 0.0000 +0.0494 0.0000 -0.1549 }}
+* [[CWE]]: ~2592/2401 = 133.3333, ~3/2 = 701.9833, ~5/4 = 386.2877
+: error map: {{val| 0.0000 +0.0283 -0.0260 -0.1759 }}
 {{Optimal ET sequence|legend=1| 27, 45, 63, 72, 99, 171, 441, 612 }}
-[[Badness]]: 0.502 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.502 × 10<sup>-3</sup>
 == See also ==
-* [[Unnoticeable comma]]
 * [[Ennealimma]]
 [[Category:Septiennealimmal]]
+[[Category:Commas named for their periods per equave]]