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 }}.
{{URWTC}}


See [[Ragismic microtemperaments #Ennealimmal]] for more technical data.
'''Ennealimmal temperament''' is a [[regular 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 [[Septiennealimmal clan #Ennealimmal]] for technical data.
 
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"
|- style="white-space: nowrap;"
! Period (steps) !! Generator (steps) !! Generator (cents)<br />(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
|}
 
== Interval chain ==
{| class="wikitable"
|+ style="font-size: 105%;" | Generator based off 11-limit ennealimmal
|- style="white-space: nowrap;"
! Generator
! Period 1
! Period 2
! Period 3
! Period 4
! Period 5
! 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
|-
! &minus;1
| 84.430
| 217.764
| 351.097
| 484.430
| 617.764
| 751.097
| 884.430
| 1017.764
| 1151.097
|-
! &minus;2
| 35.527
| 168.861
| 302.194
| 435.527
| 568.861
| 702.194
| 835.527
| 968.861
| 1102.194
|-
! &minus;3
| 1186.624
| 119.958
| 253.291
| 386.624
| 519.958
| 653.291
| 786.624
| 919.958
| 1053.291
|-
! &minus;4
| 1137.721
| 71.055
| 204.388
| 337.721
| 471.055
| 604.388
| 737.721
| 871.055
| 1004.388
|-
! &minus;5
| 1088.818
| 22.152
| 155.485
| 288.818
| 422.152
| 555.485
| 688.818
| 822.152
| 955.485
|-
! &minus;6
| 1039.915
| 1173.249
| 106.582
| 239.915
| 373.249
| 506.582
| 639.915
| 773.249
| 906.582
|-
! &minus;7
| 991.012
| 1124.346
| 57.679
| 191.012
| 324.346
| 457.679
| 591.012
| 724.346
| 857.679
|-
! &minus;8
| 942.109
| 1075.443
| 8.776
| 142.109
| 275.443
| 408.776
| 542.109
| 675.443
| 808.776
|-
! &minus;9
| 893.206
| 1026.540
| 1159.873
| 93.206
| 226.540
| 359.873
| 493.206
| 626.540
| 759.873
|-
! &minus;10
| 844.303
| 977.637
| 1110.970
| 44.303
| 177.637
| 310.970
| 444.303
| 577.637
| 710.970
|-
! &minus;11
| 795.400
| 928.734
| 1062.067
| 1195.400
| 128.734
| 262.067
| 395.400
| 528.734
| 662.067
|-
! &minus;12
| 746.497
| 879.831
| 1013.164
| 1146.497
| 79.831
| 213.164
| 346.497
| 479.831
| 613.164
|-
! &minus;13
| 697.594
| 830.928
| 964.261
| 1097.594
| 30.928
| 164.261
| 297.594
| 430.928
| 564.261
|-
! &minus;14
| 648.691
| 782.025
| 915.358
| 1048.691
| 1182.025
| 115.358
| 248.691
| 382.025
| 515.358
|-
! &minus;15
| 599.788
| 733.122
| 866.455
| 999.788
| 1133.122
| 66.455
| 199.788
| 333.122
| 466.455
|-
! &minus;16
| 550.885
| 684.219
| 817.552
| 950.885
| 1084.219
| 17.552
| 150.885
| 284.219
| 417.552
|-
! &minus;17
| 501.982
| 635.316
| 768.649
| 901.982
| 1035.316
| 1168.649
| 101.982
| 235.316
| 368.649
|}


== Ennealimmal extensions ==
== Ennealimmal extensions ==
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 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 &amp; 270) - tempering out 2401/2400, 4375/4374, 5632/5625
* Ennealimmal (99e &amp; 270) &ndash; tempering out 2401/2400, 4375/4374, 5632/5625
* Ennealimmia (171 &amp; 270) - tempering out 2401/2400, 4375/4374, 131072/130977
* Ennealimmia (171 &amp; 270) &ndash; tempering out 2401/2400, 4375/4374, 131072/130977
* Ennealimnic (72 &amp; 99e) - tempering out 243/242, 441/440, 4375/4356
* Ennealimnic (72 &amp; 99e) &ndash; tempering out 243/242, 441/440, 4375/4356
* Ennealiminal (72 &amp; 171e) - tempering out 385/384, 1375/1372, 4375/4374
* Ennealiminal (72 &amp; 171e) &ndash; tempering out 385/384, 1375/1372, 4375/4374


== Scales ==
== Scales ==
* [[Ennealimmal27]] - proper [[18L 9s]]. Ninth-octave analog of haplotonic scale
* [[Ennealimmal27]] &ndash; proper [[18L 9s]]. Ninth-octave analog of haplotonic scale
* [[Ennealimmal45]] - improper [[27L 18s]]. Ninth-octave analog of mega-haplotonic scale
* [[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]]. Ninth-octave analog of albitonic scale
* [[Ennealimmal72]] &ndash; proper [[27L 45s]]. Ninth-octave analog of albitonic scale
* [[Ennealimmal99]] - proper [[72L 27s]]. Ninth-octave analog of chromatic scale
* [[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:Breedsmic temperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Temperaments]]
[[Category:Landscape microtemperaments]]
[[Category:Ennealimmal| ]]

Latest revision as of 14:06, 28 April 2025

This page on a regular temperament, temperament collection, or aspect of regular temperament theory is being revised for clarity as part of WikiProject TempClean.

Ennealimmal temperament is a regular 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

Interval chain

Generator based off 11-limit ennealimmal
Generator Period 1 Period 2 Period 3 Period 4 Period 5 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 1088.818 22.152 155.485 288.818 422.152 555.485 688.818 822.152 955.485
−6 1039.915 1173.249 106.582 239.915 373.249 506.582 639.915 773.249 906.582
−7 991.012 1124.346 57.679 191.012 324.346 457.679 591.012 724.346 857.679
−8 942.109 1075.443 8.776 142.109 275.443 408.776 542.109 675.443 808.776
−9 893.206 1026.540 1159.873 93.206 226.540 359.873 493.206 626.540 759.873
−10 844.303 977.637 1110.970 44.303 177.637 310.970 444.303 577.637 710.970
−11 795.400 928.734 1062.067 1195.400 128.734 262.067 395.400 528.734 662.067
−12 746.497 879.831 1013.164 1146.497 79.831 213.164 346.497 479.831 613.164
−13 697.594 830.928 964.261 1097.594 30.928 164.261 297.594 430.928 564.261
−14 648.691 782.025 915.358 1048.691 1182.025 115.358 248.691 382.025 515.358
−15 599.788 733.122 866.455 999.788 1133.122 66.455 199.788 333.122 466.455
−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

Ennealimmal extensions

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
  • Ennealimmia (171 & 270) – tempering out 2401/2400, 4375/4374, 131072/130977
  • Ennealimnic (72 & 99e) – tempering out 243/242, 441/440, 4375/4356
  • Ennealiminal (72 & 171e) – tempering out 385/384, 1375/1372, 4375/4374

Scales

Music

Gene Ward Smith
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