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'''Ennealimmal temperament''' has a period of 1/9 octave and tempers out [[2401/2400]] and [[4375/4374]]. EDOs that support ennealimmal include {{EDOs| 27, 45, 72, 99, 171, 270, 441, and 612 }} EDOs.
{{Infobox regtemp
| Title = Ennealimmal
| Subgroups = 2.3.5.7
| Comma basis = [[2401/2400]], [[4375/4374]]
| Edo join 1 = 27 | Edo join 2 = 45
| Mapping = 9; 2 3 2
| Generators = 5/3 | Generators tuning = 884.322 | Optimization method = CWE
| MOS scales = [[18L 9s]], [[27L 18s]], [[27L 45s]]
| Pergen = (P8/9, P5/2)
| Odd limit 1 = 9 | Mistuning 1 = 0.204 | Complexity 1 = 45
| Odd limit 2 = 7-limit 81 | Mistuning 2 = 0.408 | Complexity 2 = 99
}}
'''Ennealimmal''' is a [[regular temperament|temperament]] with a period of {{frac|1|9}} octave and tempers out [[2401/2400]] and [[4375/4374]]. Edos that support ennealimmal include {{EDOs| 27, 45, 72, 99, 171, 270, 441, and 612 }}.


See [[Ragismic microtemperaments #Ennealimmal]] for more technical data.
See [[Septiennealimmal clan #Ennealimmal]] for technical data.


== Ennealimmal extensions ==
Ennealimmal scales are built from a ''period'' (which is exactly {{frac|1|9}} of an octave), and a ''generator'' (which is approximately 49 cents and represents several small intervals including 36/35). Depending on the size of the generator and the period in steps, the above listed edos make sense:
Ennealimmal temperament has various extensions to the 11-limit. These are all members of the ennealimmal family, but in addition they are linear temperaments:
 
* Ennealimmal (99e & 270) - tempering out 2401/2400, 4375/4374, 5632/5625
{| class="wikitable"
* Ennealimnic (72 & 99e) - tempering out 243/242, 441/440, 4375/4356
|- style="white-space: nowrap;"
* Ennealiminal (72 & 171e) - tempering out 385/384, 1375/1372, 4375/4374
! Period (steps) !! Generator (steps) !! Generator (cents)<br />(pure octave) !! Edo
* 171 &amp; 270 - tempering out 2401/2400, 4375/4374, 131072/130977
|-
| 3 || 1 || 44.444 || 27
|-
| 11 || 4 || 48.485 || 99
|-
| 30 || 11 || 48.889 || 270
|-
| 19 || 7 || 49.123 || 171
|-
| 8 || 3 || 50.000 || 72
|-
| 5 || 2 || 53.333 || 45
|}
 
Ennealimmal extends less well to the [[11-limit]]. Extensions include enneabiotic (99e & 270), ennealympic (171 & 270), ennealimnic (72 & 99e), and ennealiminal (72 & 171e).
 
See [[Ennealimmal extensions]] for a discussion on 11-limit extensions.
 
== Interval chain ==
In the following table, odd harmonics 1–9 are labeled in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4 right-6 right-8"
! rowspan="2" | Period
! colspan="2" | Generator 0
! colspan="2" | Generator 1
! colspan="2" | Generator 2
! colspan="2" | Generator 3
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| 0
| 0.000
| '''1/1'''
|
|
|
|
|
|
|-
| 1
| 133.333
| 27/25
| 84.322
| 21/20
| 35.310
| 49/48, 50/49
| 1186.298
| 125/63
|-
| 2
| 266.667
| 7/6
| 217.655
| 245/216
| 168.643
| 54/49
| 119.631
| 15/14
|-
| 3
| 400.000
| 63/50
| 350.988
| 49/40, 60/49
| 301.976
| 25/21
| 252.965
| 81/70, 125/108
|-
| 4
| 533.333
| 49/36
| 484.322
| 250/189
| 435.310
| 9/7
| 386.298
| '''5/4'''
|-
| 5
| 666.667
| 72/49
| 617.655
| 10/7
| 568.643
| 25/18
| 519.631
| 27/20
|-
| 6
| 800.000
| 100/63
| 750.988
| 54/35
| 701.976
| '''3/2'''
| 652.965
| 35/24
|-
| 7
| 933.333
| 12/7
| 884.322
| 5/3
| 835.310
| 81/50
| 786.298
| 63/40
|-
| 8
| 1066.667
| 50/27
| 1017.655
| 9/5
| 968.643
| '''7/4'''
| 919.631
| 245/144
|-
| 9
| 1200.000
| 2/1
| 1150.988
| 35/18
| 1101.976
| 189/100
| 1052.965
| 147/80
|}
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced


== Scales ==
== Scales ==
* [[Ennealimmal27]] - proper [[18L 9s]]
* [[Ennealimmal27]] &ndash; proper [[18L 9s]]. Ninth-octave analog of haplotonic scale
* [[Ennealimmal45]] - improper [[27L 18s]]
* [[Ennealimmal45]] &ndash; improper [[27L 18s]]. Ninth-octave analog of mega-haplotonic scale
** [[Ennealimmal45trans]] - symmetric 5-limit transversal version
** [[Ennealimmal45trans]] &ndash; symmetric 5-limit transversal version
* [[Ennealimmal72]] - proper [[27L 45s]]
* [[Ennealimmal72]] &ndash; proper [[27L 45s]]. Ninth-octave analog of albitonic scale
* [[Ennealimmal99]] - proper [[72L 27s]]
* [[Ennealimmal99]] &ndash; proper [[72L 27s]]. Ninth-octave analog of chromatic scale
* [[Ennealimmal171]] - [[99L 72s]] scale. The boundary of propriety is [[270edo|270EDO]].
* [[Ennealimmal171]] &ndash; [[99L 72s]] scale. The boundary of propriety is [[270edo]].
 
== Music ==
; [[Gene Ward Smith]]
* [https://archive.org/details/fingers_201403 ''The 45000 fingers of Dr. S''] (2003) – Ennealimmal[54] in TOP tuning


[[Category:Ennealimmal| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Microtemperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Temperament]]
[[Category:Landscape microtemperaments]]
[[Category:Ennealimmal| ]]

Latest revision as of 13:52, 28 May 2026

Ennealimmal
Subgroups 2.3.5.7
Comma basis 2401/2400, 4375/4374
Reduced mapping ⟨9; 2 3 2]
ET join 27 & 45
Generators (CWE) ~5/3 = 884.322 ¢
MOS scales 18L 9s, 27L 18s, 27L 45s
Ploidacot enneaploid dicot
Pergen (P8/9, P5/2)
Minimax error 9-odd-limit: 0.204 ¢;
7-limit 81-odd-limit: 0.408 ¢
Target scale size 9-odd-limit: 45 notes;
7-limit 81-odd-limit: 99 notes

Ennealimmal is a temperament with a period of 1⁄9 octave and tempers out 2401/2400 and 4375/4374. Edos that support ennealimmal include 27, 45, 72, 99, 171, 270, 441, and 612.

See Septiennealimmal clan #Ennealimmal for technical data.

Ennealimmal scales are built from a period (which is exactly 1⁄9 of an octave), and a generator (which is approximately 49 cents and represents several small intervals including 36/35). Depending on the size of the generator and the period in steps, the above listed edos make sense:

Period (steps) Generator (steps) Generator (cents)
(pure octave)		 Edo
3 1 44.444 27
11 4 48.485 99
30 11 48.889 270
19 7 49.123 171
8 3 50.000 72
5 2 53.333 45

Ennealimmal extends less well to the 11-limit. Extensions include enneabiotic (99e & 270), ennealympic (171 & 270), ennealimnic (72 & 99e), and ennealiminal (72 & 171e).

See Ennealimmal extensions for a discussion on 11-limit extensions.

Interval chain

In the following table, odd harmonics 1–9 are labeled in bold.

Period Generator 0 Generator 1 Generator 2 Generator 3
Cents* Approx. ratios Cents* Approx. ratios Cents* Approx. ratios Cents* Approx. ratios
0 0.000 1/1
1 133.333 27/25 84.322 21/20 35.310 49/48, 50/49 1186.298 125/63
2 266.667 7/6 217.655 245/216 168.643 54/49 119.631 15/14
3 400.000 63/50 350.988 49/40, 60/49 301.976 25/21 252.965 81/70, 125/108
4 533.333 49/36 484.322 250/189 435.310 9/7 386.298 5/4
5 666.667 72/49 617.655 10/7 568.643 25/18 519.631 27/20
6 800.000 100/63 750.988 54/35 701.976 3/2 652.965 35/24
7 933.333 12/7 884.322 5/3 835.310 81/50 786.298 63/40
8 1066.667 50/27 1017.655 9/5 968.643 7/4 919.631 245/144
9 1200.000 2/1 1150.988 35/18 1101.976 189/100 1052.965 147/80

* In 7-limit CWE tuning, octave reduced

Scales

Music

Gene Ward Smith
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