Septimal ennealimma: Difference between revisions
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m redefined septiennealimmal to be more practically useful as there's no real reason to use the rank 3; it's 7/6 = 2\9 that is most useful and this allows future extension to prime 5 in non-ennealimmal ways so that the gen can be something different |
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{{Infobox Interval | {{Infobox Interval | ||
| Ratio = 40353607/40310784 | | Ratio = 40353607/40310784 | ||
| Name = septimal ennealimma | | Name = septimal ennealimma, no-five ennealimma | ||
| Color name = tritrizo comma | | Color name = tritrizo comma | ||
| Comma = yes | | Comma = yes | ||
}} | }} | ||
The '''septimal ennealimma' | 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. | ||
== Temperaments == | == 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 version of the [[ennealimmal temperament]], which is a member of the [[tritrizo clan]]. | ||
It can only possibly be tempered out in an EDO if that EDO is a multiple of 9. This temperament is documented below: | |||
=== Septiennealimmal === | === 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]]: 2.3.7 | ||
Mapping | [[Mapping]]: [{{val| 9 0 11 }}, {{val| 0 1 1 }}] | ||
Generators ([[CTE]]): ~2592/2401 = (27/25)/[[2401/2400|S49]] = 1\9, ~3 = 1902.004{{cent}} | |||
[[Badness]]: 0. | [[Patent val]] non-ennealimmal EDOs with increasingly good approximations of 3: 9, 18, 36, 63, 90, 117, 135, 333, 360, 378, 747, 945, 990, 1044, 1089 | ||
[[Badness]]: 0.191 | |||
== See also == | == See also == | ||
Revision as of 20:11, 8 April 2024
| Interval information |
no-five ennealimma
The septimal ennealimma' or no-five ennealimma is an unnoticeable 7-limit comma defined as (7/6)9 / (2/1)2 and measuring 1.838 cents.
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.
It can only possibly be tempered out in an EDO if that EDO is a multiple of 9. This temperament is documented below:
Septiennealimmal
The rank 2 temperament septiennealimmal is of interest to anyone who wants a different generator for the "ennealimmal-like structure" by detempering 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 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: 2.3.7
Mapping: [⟨9 0 11], ⟨0 1 1]]
Generators (CTE): ~2592/2401 = (27/25)/S49 = 1\9, ~3 = 1902.004 ¢
Patent val non-ennealimmal EDOs with increasingly good approximations of 3: 9, 18, 36, 63, 90, 117, 135, 333, 360, 378, 747, 945, 990, 1044, 1089
Badness: 0.191