Septimal ennealimma: Difference between revisions

Line 1:

{{Infobox Interval

| Ratio = 40353607/40310784

| Name = septimal ennealimma~~, septiennealimma~~, no-five ennealimma

| Name = septimal ennealimma, 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'''' 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]]~~. 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 version of the [[ennealimmal temperament]], which is a member of the [[tritrizo clan]].

~~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.

It can only possibly be tempered out in an EDO if that EDO is a multiple of 9. This temperament is documented below:

=== Septiennealimmal ===

[[~~Subgroup~~]]: 2.3.5.7

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).

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

[[Subgroup]]: 2.3.7

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

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

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

Generators ([[CTE]]): ~2592/2401 = (27/25)/[[2401/2400|S49]] = 1\9, ~3 = 1902.004{{cent}}

[[Badness]]: 0.~~502 × 10-3~~

[[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 ==

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Ratio = 40353607/40310784
-| Name = septimal ennealimma, septiennealimma, no-five ennealimma
+| Name = septimal ennealimma, 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'''' 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 ==
-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 version of the [[ennealimmal temperament]], which is a member of the [[tritrizo clan]].
-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.
+It can only possibly be tempered out in an EDO if that EDO is a multiple of 9. This temperament is documented below:
 === Septiennealimmal ===
-[[Subgroup]]: 2.3.5.7
+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).
-[[Mapping]]: [{{val| 9 0 0 11 }}, {{val| 0 1 0 1 }}, {{val| 0 0 1 0 }}]
+[[Subgroup]]: 2.3.7
-Mapping generators: ~2592/2401, ~3, ~5
+[[Mapping]]: [{{val| 9 0 11 }}, {{val| 0 1 1 }}]
-{{Optimal ET sequence|legend=1| 27, 45, 63, 72, 99, 171, 441, 612 }}
+Generators ([[CTE]]): ~2592/2401 = (27/25)/[[2401/2400|S49]] = 1\9, ~3 = 1902.004{{cent}}
-[[Badness]]: 0.502 × 10<sup>-3</sup>
+[[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 ==