Ennealimmal
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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.
See Ragismic microtemperaments #Ennealimmal for more 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 | 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
- Ennealimmal27 - proper 18L 9s. Ninth-octave analog of haplotonic scale
- Ennealimmal45 - improper 27L 18s. Ninth-octave analog of mega-haplotonic scale
- 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.