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 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
-| Name = septimal ennealimma, no-five ennealimma
+| Name = septimal ennealimma, septiennealimma, no-five ennealimma
 | Color name = tritrizo comma
 | Comma = yes
 }}
-The '''septimal ennealimma''' or '''no-five ennealimma''' is an unnoticeable 7-limit comma defined as ([[7/6]])<sup>9</sup> / ([[2/1]])<sup>2</sup> and measuring 1.838 cents.
+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]].
+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.
-It can only possibly be tempered out in an EDO if that EDO is a multiple of 9. This temperament is documented 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 ===
-The rank 2 temperament septiennealimmal is of interest to anyone who wants a different generator for the "ennealimmal-like structure" by detempering [[2401/2400|S49]] and/or because it represents the part of ennealimmal supported by non-ennealimmal EDOs of interest that do well in the 2.3.7 subgroup, such as [[36edo]], which adds [[1029/1024|S7/S8]], or [[63edo]], which in the 7-limit can be used for [[septimal magic]] and in higher limits for [[parapyth]] (among other things).
+[[Subgroup]]: 2.3.5.7
-[[Subgroup]]: 2.3.7
+{{Mapping|legend=1| 9 0 0 11 | 0 1 0 1 | 0 0 1 0 }}
-[[Mapping]]: [{{val| 9 0 11 }}, {{val| 0 1 1 }}]
+: mapping generators: ~2592/2401, ~3, ~5
-Generators ([[CTE]]): ~2592/2401 = (27/25)/[[2401/2400|S49]] = 1\9, ~3 = 1902.004{{cent}}
+{{Optimal ET sequence|legend=1| 27, 45, 63, 72, 99, 171, 441, 612 }}
-[[Patent val]] non-ennealimmal EDO tunings < 311 with the [[7-limit]]ed [[9-odd-limit]] (or here equivalently [[21-odd-limit]]) [[consistent]]: 36, 63, 108, 135, 162, 207, 234, 279, 306
+Badness (Smith): 0.502 × 10<sup>-3</sup>
-[[Badness]]: 0.191
 == See also ==

Septimal ennealimma: Difference between revisions

Revision as of 10:09, 6 March 2025

Temperaments

Septiennealimmal

See also

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