Ennealimmal: Difference between revisions

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== Interval chain ==
== Interval chain ==
{| class="wikitable"
In the following table, odd harmonics 1–9 are labeled in '''bold'''.
|+ style="font-size: 105%;" | Generator based off 11-limit ennealimmal
 
|- style="white-space: nowrap;"
{| class="wikitable center-1 right-2 right-4 right-6 right-8"
! Generator
! rowspan="2" | Period
! Period 1
! colspan="2" | Generator 0
! Period 2
! colspan="2" | Generator 1
! Period 3
! colspan="2" | Generator 2
! Period 4
! colspan="2" | Generator 3
! Period 5
! Period 6
! Period 7
! Period 8
! Period 9 (0)
|-
|-
! 0
! Cents*
| 133.333
! Approx. ratios
| 266.666
! Cents*
| 400.000
! Approx. ratios
| 533.333
! Cents*
| 666.666
! Approx. ratios
| 800.000
! Cents*
| 933.333
! Approx. ratios
| 1066.666
| 1200.000
|-
|-
! −1
| 0
| 84.430
| 0.000
| 217.764
| '''1/1'''
| 351.097
|  
| 484.430
|  
| 617.764
|  
| 751.097
|  
| 884.430
|  
| 1017.764
|  
| 1151.097
|-
|-
! −2
| 1
| 35.527
| 133.333
| 168.861
| 27/25
| 302.194
| 84.322
| 435.527
| 21/20
| 568.861
| 35.310
| 702.194
| 49/48, 50/49
| 835.527
| 1186.298
| 968.861
| 125/63
| 1102.194
|-
|-
! −3
| 2
| 1186.624
| 266.667
| 119.958
| 7/6
| 253.291
| 217.655
| 386.624
| 245/216
| 519.958
| 168.643
| 653.291
| 54/49
| 786.624
| 119.631
| 919.958
| 15/14
| 1053.291
|-
|-
! −4
| 3
| 1137.721
| 400.000
| 71.055
| 63/50
| 204.388
| 350.988
| 337.721
| 49/40, 60/49
| 471.055
| 301.976
| 604.388
| 25/21
| 737.721
| 252.965
| 871.055
| 81/70, 125/108
| 1004.388
|-
|-
! −5
| 4
| 1088.818
| 533.333
| 22.152
| 49/36
| 155.485
| 484.322
| 288.818
| 250/189
| 422.152
| 435.310
| 555.485
| 9/7
| 688.818
| 386.298
| 822.152
| '''5/4'''
| 955.485
|-
|-
! −6
| 5
| 1039.915
| 666.667
| 1173.249
| 72/49
| 106.582
| 617.655
| 239.915
| 10/7
| 373.249
| 568.643
| 506.582
| 25/18
| 639.915
| 519.631
| 773.249
| 27/20
| 906.582
|-
|-
! −7
| 6
| 991.012
| 800.000
| 1124.346
| 100/63
| 57.679
| 750.988
| 191.012
| 54/35
| 324.346
| 701.976
| 457.679
| '''3/2'''
| 591.012
| 652.965
| 724.346
| 35/24
| 857.679
|-
|-
! −8
| 7
| 942.109
| 933.333
| 1075.443
| 12/7
| 8.776
| 884.322
| 142.109
| 5/3
| 275.443
| 835.310
| 408.776
| 81/50
| 542.109
| 786.298
| 675.443
| 63/40
| 808.776
|-
|-
! −9
| 8
| 893.206
| 1066.667
| 1026.540
| 50/27
| 1159.873
| 1017.655
| 93.206
| 9/5
| 226.540
| 968.643
| 359.873
| '''7/4'''
| 493.206
| 919.631
| 626.540
| 245/144
| 759.873
|-
|-
! −10
| 9
| 844.303
| 1200.000
| 977.637
| 2/1
| 1110.970
| 1150.988
| 44.303
| 35/18
| 177.637
| 1101.976
| 310.970
| 189/100
| 444.303
| 1052.965
| 577.637
| 147/80
| 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
|}
|}
* In 7-limit CWE tuning, octave reduced


== Ennealimmal extensions ==
== Ennealimmal extensions ==

Revision as of 12:50, 29 November 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

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

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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