Septimal ennealimma: Difference between revisions

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

Temperaments

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.

Septiennealimmal

Subgroup: 2.3.5.7

Mapping: [⟨9 0 0 11], ⟨0 1 0 1], ⟨0 0 1 0]]

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

Optimal tunings:

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

error map: ⟨0.0000 +0.0494 0.0000 -0.1549]

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

error map: ⟨0.0000 +0.0283 -0.0260 -0.1759]

Optimal ET sequence: 27, 45, 63, 72, 99, 171, 441, 612

Badness (Smith): 0.502 × 10^-3

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Ratio = 40353607/40310784
-| Monzo = -11 -9 0 9
+| Name = septimal ennealimma, septiennealimma, no-five ennealimma
-| Cents = 1.83815
-| Name = septimal ennealimma, <br>septiennealimma, <br>no-five ennealimma
 | Color name = tritrizo comma
+| Comma = yes
 }}
-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>.
-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 if and only if the edo is divisible by 9.
+== Temperaments ==
+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.
+=== Septiennealimmal ===
+[[Subgroup]]: 2.3.5.7
+{{Mapping|legend=1| 9 0 0 11 | 0 1 0 1 | 0 0 1 0 }}
+: 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]] (Smith): 0.502 × 10<sup>-3</sup>
 == See also ==
-* [[Unnoticeable comma]]
+* [[Ennealimma]]
-[[Category:7-limit]]
-[[Category:Unnoticeable commas]]
 [[Category:Septiennealimmal]]
+[[Category:Commas named for their periods per equave]]

Septimal ennealimma: Difference between revisions

Latest revision as of 10:14, 6 March 2025

Temperaments

Septiennealimmal

See also

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