Septimal ennealimma: Difference between revisions

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]])9 / ([[2/1]])2 ~~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]])9 / ([[2/1]])2.

== 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]]~~: ~~[{{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 × 10-3

~~[[Badness]]~~: 0.~~191~~

== See also ==

@@ 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 ==