Ennealimmal: Difference between revisions

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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 {{EDOs| 27, 45, 72, 99, 171, 270, 441, and 612 }}.
{{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 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:
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:


{| class="wikitable
{| class="wikitable"
! Period (steps) !! Generator (steps) !! Generator (cents) <br/> (pure octave) !! EDO
|- style="white-space: nowrap;"
! Period (steps) !! Generator (steps) !! Generator (cents)<br />(pure octave) !! Edo
|-
|-
| 3 || 1 || 44.444 || 27
| 3 || 1 || 44.444 || 27
Line 20: Line 33:
| 5 || 2 || 53.333 || 45
| 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 ==
== Interval chain ==
{| class="wikitable"
In the following table, odd harmonics 1–9 are labeled in '''bold'''.
|+Generator based off 11-limit ennealimmal
 
!Generator
{| class="wikitable center-1 right-2 right-4 right-6 right-8"
!Period 1
! rowspan="2" | Period
!Period 2
! colspan="2" | Generator 0
!Period 3
! colspan="2" | Generator 1
!Period 4
! colspan="2" | Generator 2
!Period 5
! colspan="2" | Generator 3
!Period 6
!Period 7
!Period 8
!Period 9 (0)
|-
!0
|133.333
|266.666
|400.000
|533.333
|666.666
|800.000
|933.333
|1066.666
|1200.000
|-
!-1
|84.430
|217.764
|351.097
|484.430
|617.764
|751.097
|884.430
|1017.764
|1151.097
|-
!-2
|35.527
|168.861
|302.194
|435.527
|568.861
|702.194
|835.527
|968.861
|1102.194
|-
!-3
|1186.624
|119.958
|253.291
|386.624
|519.958
|653.291
|786.624
|919.958
|1053.291
|-
!-4
|1137.721
|71.055
|204.388
|337.721
|471.055
|604.388
|737.721
|871.055
|1004.388
|-
|-
!-5
! Cents*
|1088.818
! Approx. ratios
|22.152
! Cents*
|155.485
! Approx. ratios
|288.818
! Cents*
|422.152
! Approx. ratios
|555.485
! Cents*
|688.818
! Approx. ratios
|822.152
|955.485
|-
|-
!-6
| 0
|1039.915
| 0.000
|1173.249
| '''1/1'''
|106.582
|  
|239.915
|  
|373.249
|  
|506.582
|  
|639.915
|  
|773.249
|  
|906.582
|-
|-
!-7
| 1
|991.012
| 133.333
|1124.346
| 27/25
|57.679
| 84.322
|191.012
| 21/20
|324.346
| 35.310
|457.679
| 49/48, 50/49
|591.012
| 1186.298
|724.346
| 125/63
|857.679
|-
|-
!-8
| 2
|942.109
| 266.667
|1075.443
| 7/6
|8.776
| 217.655
|142.109
| 245/216
|275.443
| 168.643
|408.776
| 54/49
|542.109
| 119.631
|675.443
| 15/14
|808.776
|-
|-
!-9
| 3
|893.206
| 400.000
|1026.540
| 63/50
|1159.873
| 350.988
|93.206
| 49/40, 60/49
|226.540
| 301.976
|359.873
| 25/21
|493.206
| 252.965
|626.540
| 81/70, 125/108
|759.873
|-
|-
!-10
| 4
|844.303
| 533.333
|977.637
| 49/36
|1110.970
| 484.322
|44.303
| 250/189
|177.637
| 435.310
|310.970
| 9/7
|444.303
| 386.298
|577.637
| '''5/4'''
|710.970
|-
|-
!-11
| 5
|795.400
| 666.667
|928.734
| 72/49
|1062.067
| 617.655
|1195.400
| 10/7
|128.734
| 568.643
|262.067
| 25/18
|395.400
| 519.631
|528.734
| 27/20
|662.067
|-
|-
!-12
| 6
|746.497
| 800.000
|879.831
| 100/63
|1013.164
| 750.988
|1146.497
| 54/35
|79.831
| 701.976
|213.164
| '''3/2'''
|346.497
| 652.965
|479.831
| 35/24
|613.164
|-
|-
!-13
| 7
|697.594
| 933.333
|830.928
| 12/7
|964.261
| 884.322
|1097.594
| 5/3
|30.928
| 835.310
|164.261
| 81/50
|297.594
| 786.298
|430.928
| 63/40
|564.261
|-
|-
!-14
| 8
|648.691
| 1066.667
|782.025
| 50/27
|915.358
| 1017.655
|1048.691
| 9/5
|1182.025
| 968.643
|115.358
| '''7/4'''
|248.691
| 919.631
|382.025
| 245/144
|515.358
|-
|-
!-15
| 9
|599.788
| 1200.000
|733.122
| 2/1
|866.455
| 1150.988
|999.788
| 35/18
|1133.122
| 1101.976
|66.455
| 189/100
|199.788
| 1052.965
|333.122
| 147/80
|466.455
|-
!-16
|550.885
|684.219
|817.552
|950.885
|1084.219
|17.552
|150.885
|284.219
|417.552
|-
!-17
|501.982
|635.316
|768.649
|901.982
|1035.316
|1168.649
|101.982
|235.316
|368.649
|}
|}
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced


== Ennealimmal extensions ==
== Scales ==
Ennealimmal temperament has various extensions to the 11-limit. These are all members of the ennealimmal family, but in addition they are linear temperaments:
* [[Ennealimmal27]] &ndash; proper [[18L 9s]]. Ninth-octave analog of haplotonic scale
* Ennealimmal (99e &amp; 270) - tempering out 2401/2400, 4375/4374, 5632/5625
* [[Ennealimmal45]] &ndash; improper [[27L 18s]]. Ninth-octave analog of mega-haplotonic scale
* Ennealimmia (171 &amp; 270) - tempering out 2401/2400, 4375/4374, 131072/130977
** [[Ennealimmal45trans]] &ndash; symmetric 5-limit transversal version
* Ennealimnic (72 &amp; 99e) - tempering out 243/242, 441/440, 4375/4356
* [[Ennealimmal72]] &ndash; proper [[27L 45s]]. Ninth-octave analog of albitonic scale
* Ennealiminal (72 &amp; 171e) - tempering out 385/384, 1375/1372, 4375/4374
* [[Ennealimmal99]] &ndash; proper [[72L 27s]]. Ninth-octave analog of chromatic scale
* [[Ennealimmal171]] &ndash; [[99L 72s]] scale. The boundary of propriety is [[270edo]].


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


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