Septimal ennealimma: Difference between revisions
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The '''septimal ennealimma''', '''septiennealimma''' or '''no-five ennealimma''' is an unnoticeable 7-limit comma | 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 == | == Temperaments == | ||
In the 2.3.7 subgroup, tempering it out leads to the 2.3.7 | 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 === | === Septiennealimmal === | ||
[[Subgroup]]: 2.3.5.7 | [[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 }} | {{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 == | == See also == | ||
* [[ | * [[Ennealimma]] | ||
[[Category:Septiennealimmal]] | [[Category:Septiennealimmal]] | ||
[[Category:Commas named for their periods per equave]] | |||
Latest revision as of 10:14, 6 March 2025
| Interval information |
septiennealimma,
no-five ennealimma
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
- 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