Septimal ennealimma: Difference between revisions

(14 intermediate revisions by 6 users not shown)

Line 1:

{{Infobox Interval

| Ratio = 40353607/40310784

~~| Monzo = -11 -9 0 9~~

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

~~| Cents = 1.83815~~

| Name = septimal ennealimma, no-five ennealimma

| Color name = tritrizo comma

| Comma = yes

}}

'''~~Septimal~~ ennealimma''', '''~~no-five ennealimma~~''', or '''~~tritrizo comma~~''' is an unnoticeable 7-limit comma ~~defined as 40353607/40310784 and~~ measuring 1.838 ~~cents~~. It ~~arises as~~ a stack ~~between 9~~ [[7/6]]<~~nowiki~~/>~~s and~~ 2 ~~octaves~~.

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.

~~Tempering~~ it out leads to the 7-~~limit version~~ of the [[ennealimmal ~~temperament~~]], which is ~~a subset~~ of the [[~~tritrizo~~ clan]]. ~~For EDOs up to circa 3000, it is~~ tempered if ~~and only if the EDO~~ is ~~divisible by~~ 9.

== 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 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 }}

[[Badness]] (Smith): 0.502 × 10-3

== See also ==

* [[Ennealimma]]

[[Category:Septiennealimmal]]

[[Category:Commas named for their periods per equave]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Ratio = 40353607/40310784
-| Monzo = -11 -9 0 9
+| Name = septimal ennealimma, septiennealimma, no-five ennealimma
-| Cents = 1.83815
-| Name = septimal ennealimma, no-five ennealimma
 | Color name = tritrizo comma
+| Comma = yes
 }}
-'''Septimal ennealimma''', '''no-five ennealimma''', or '''tritrizo comma''' is an unnoticeable 7-limit comma defined as 40353607/40310784 and measuring 1.838 cents. It arises as a stack between 9 [[7/6]]<nowiki/>s and 2 octaves.
+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>.
-Tempering it out leads to the 7-limit version of the [[ennealimmal temperament]], which is a subset of the [[tritrizo clan]]. For EDOs up to circa 3000, it is tempered if and only if the EDO is divisible by 9.
+== 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|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 }}
+[[Badness]] (Smith): 0.502 × 10<sup>-3</sup>
+== See also ==
+* [[Ennealimma]]
+[[Category:Septiennealimmal]]
+[[Category:Commas named for their periods per equave]]