1224edo: Difference between revisions
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1224edo is [[ | 1224edo is [[enfactoring|enfactored]] in the [[11-limit]], with the same tuning as [[612edo]], but it corrects the [[harmonic]]s [[13/1|13]] and [[17/1|17]] to work better with the flat tendency of the lower harmonics. It [[tempering out|tempers out]] [[4225/4224]], [[10648/10647]] in the 13-limit; [[2431/2430]], [[4914/4913]] in the 17-limit; [[1729/1728]], [[2926/2925]] among others in the 19-limit. It provides the [[optimal patent val]] for the 19-limit [[semihemiennealimmal]] temperament with fine tunes of [[23/1|23]], [[29/1|29]] and [[31/1|31]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 1224 factors into | Since 1224 factors into {{factorization|1224}}, 1224edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 17, 18, 24, 34, 36, 51, 68, 72, 102, 136, 153, 204, 306, 408, and 612 }}. | ||
Latest revision as of 05:46, 21 February 2025
| ← 1223edo | 1224edo | 1225edo → |
1224 equal divisions of the octave (abbreviated 1224edo or 1224ed2), also called 1224-tone equal temperament (1224tet) or 1224 equal temperament (1224et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1224 equal parts of about 0.98 ¢ each. Each step represents a frequency ratio of 21/1224, or the 1224th root of 2.
1224edo is enfactored in the 11-limit, with the same tuning as 612edo, but it corrects the harmonics 13 and 17 to work better with the flat tendency of the lower harmonics. It tempers out 4225/4224, 10648/10647 in the 13-limit; 2431/2430, 4914/4913 in the 17-limit; 1729/1728, 2926/2925 among others in the 19-limit. It provides the optimal patent val for the 19-limit semihemiennealimmal temperament with fine tunes of 23, 29 and 31.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.006 | -0.039 | -0.198 | -0.338 | -0.332 | -0.053 | -0.454 | +0.157 | -0.165 | +0.062 |
| Relative (%) | +0.0 | +0.6 | -4.0 | -20.2 | -34.4 | -33.8 | -5.5 | -46.3 | +16.0 | -16.9 | +6.4 | |
| Steps (reduced) |
1224 (0) |
1940 (716) |
2842 (394) |
3436 (988) |
4234 (562) |
4529 (857) |
5003 (107) |
5199 (303) |
5537 (641) |
5946 (1050) |
6064 (1168) | |
Subsets and supersets
Since 1224 factors into 23 × 32 × 17, 1224edo has subset edos 2, 3, 4, 6, 8, 9, 12, 17, 18, 24, 34, 36, 51, 68, 72, 102, 136, 153, 204, 306, 408, and 612.