Ennealimmal: Difference between revisions
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| Comma basis = [[2401/2400]], [[4375/4374]] | | Comma basis = [[2401/2400]], [[4375/4374]] | ||
| Edo join 1 = 27 | Edo join 2 = 45 | | 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]] | | MOS scales = [[18L 9s]], [[27L 18s]], [[27L 45s]] | ||
| Pergen = (P8/9, P5/2) | | Pergen = (P8/9, P5/2) | ||
| Odd limit 1 = 9 | Mistuning 1 = 0.204 | Complexity 1 = 45 | | Odd limit 1 = 9 | Mistuning 1 = 0.204 | Complexity 1 = 45 | ||
| Odd limit 2 = 7-limit 81 | Mistuning 2 = 0.408 | Complexity 2 = 99 | | 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 }}. | |||
'''Ennealimmal | |||
See [[Septiennealimmal clan #Ennealimmal]] for technical data. | See [[Septiennealimmal clan #Ennealimmal]] for technical data. | ||
Revision as of 09:10, 9 February 2026
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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 |
7-limit 81-odd-limit: 0.408 ¢
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 |
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
- 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.
Music
- The 45000 fingers of Dr. S (2003) – Ennealimmal[54] in TOP tuning