218edo: Difference between revisions
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128 and 90 are the patent val. Patent val works by combining nearest match for primes, and neither 127 or 91 are nearest. 37 is not patent val of 9 because 128+128-218=38. |
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218edo is in[[consistent]] to the [[5-odd-limit]], with [[harmonic]] [[3/1|3]] falling about halfway between its steps. However, it contains very accurate ratios, such as [[7/4]], [[9/7]], [[9/8]], [[10/9]], [[11/10]], [[17/16]], and [[19/16]], which are approximated within 0.55-cent deviation (10% the step size). The suggested [[subgroup]]s are therefore 2.9.7.17.19 and 2.9.5.7.11.17.19.23. | |||
Commas using the [[13-limit]] patent val: | |||
; [[5-limit]]: 20000/19683, 1220703125/1207959552 | |||
; [[7-limit]]: 4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640 | |||
; [[11-limit]]: 4000/3993, 12005/11979, 16384/16335, 4375/4356, 78125/77616, 896/891, 67228/66825, 1375/1372, 6875/6804, 5632/5625, 385/384, 94325/93312, 15488/15435, 75625/75264, 15488/15309, 3388/3375, 1331/1323, 6655/6561, 65219/64800, 43923/43904, 73205/72576, | |||
; [[13-limit]]: 28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512 | |||
=== Odd harmonics === | |||
{{Harmonics in equal|218}} | |||
=== Subsets and supersets === | |||
Since 218 factors into {{factorization|218}}, 218edo contains [[2edo]] and [[109edo]] as its subsets. [[436edo]], which doubles it, is worth exploring. | |||
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Latest revision as of 14:17, 20 February 2025
| ← 217edo | 218edo | 219edo → |
218 equal divisions of the octave (abbreviated 218edo or 218ed2), also called 218-tone equal temperament (218tet) or 218 equal temperament (218et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 218 equal parts of about 5.5 ¢ each. Each step represents a frequency ratio of 21/218, or the 218th root of 2.
218edo is inconsistent to the 5-odd-limit, with harmonic 3 falling about halfway between its steps. However, it contains very accurate ratios, such as 7/4, 9/7, 9/8, 10/9, 11/10, 17/16, and 19/16, which are approximated within 0.55-cent deviation (10% the step size). The suggested subgroups are therefore 2.9.7.17.19 and 2.9.5.7.11.17.19.23.
Commas using the 13-limit patent val:
- 5-limit
- 20000/19683, 1220703125/1207959552
- 7-limit
- 4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640
- 11-limit
- 4000/3993, 12005/11979, 16384/16335, 4375/4356, 78125/77616, 896/891, 67228/66825, 1375/1372, 6875/6804, 5632/5625, 385/384, 94325/93312, 15488/15435, 75625/75264, 15488/15309, 3388/3375, 1331/1323, 6655/6561, 65219/64800, 43923/43904, 73205/72576,
- 13-limit
- 28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.63 | -0.99 | -0.02 | -0.24 | -0.86 | +1.67 | +1.64 | -0.37 | -0.27 | +2.61 | -0.75 |
| Relative (%) | +47.8 | -18.0 | -0.3 | -4.4 | -15.6 | +30.4 | +29.8 | -6.7 | -4.8 | +47.5 | -13.7 | |
| Steps (reduced) |
346 (128) |
506 (70) |
612 (176) |
691 (37) |
754 (100) |
807 (153) |
852 (198) |
891 (19) |
926 (54) |
958 (86) |
986 (114) | |
Subsets and supersets
Since 218 factors into 2 × 109, 218edo contains 2edo and 109edo as its subsets. 436edo, which doubles it, is worth exploring.