Septimal ennealimma: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Ratio = 40353607/40310784
| Ratio = 40353607/40310784
| Name = septimal ennealimma, no-five ennealimma
| Name = septimal ennealimma, septiennealimma, no-five ennealimma
| Color name = tritrizo comma
| Color name = tritrizo comma
| Comma = yes
| 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 ==
== 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 ===
=== 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 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 }}


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


[[Badness]]: 0.191
[[Badness]] (Smith): 0.502 × 10<sup>-3</sup>


== See also ==
== See also ==
* [[Unnoticeable comma]]
* [[Ennealimma]]
* [[Ennealimma]]


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

Latest revision as of 10:14, 6 March 2025

Interval information
Ratio 40353607/40310784
Factorization 2-11 × 3-9 × 79
Monzo [-11 -9 0 9⟩
Size in cents 1.83815¢
Names septimal ennealimma,
septiennealimma,
no-five ennealimma
Color name tritrizo comma
FJS name [math]\displaystyle{ \text{dddd5}^{7,7,7,7,7,7,7,7,7} }[/math]
Special properties reduced
Tenney height (log2 nd) 50.5309
Weil height (log2 max(n, d)) 50.5324
Wilson height (sopfr(nd)) 112
Comma size unnoticeable
Open this interval in xen-calc

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

Optimal tunings:

  • 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

See also

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