618edo: Difference between revisions
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== | 618edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. Nonetheless, as every other step of [[1236edo]], 618edo is excellent in approximating harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], [[13/1|13]], and [[17/1|17]], making it suitable for a 2.9.5.7.11.13.17 [[subgroup]] interpretation, where the equal temperament notably [[tempering out|tempers out]] [[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], and [[10648/10647]]. With a reasonable approximation of 19, it further tempers out 2926/2925, 5985/5984, and 6175/6174. | ||
=== Odd harmonics === | |||
{{Harmonics in equal|618}} | {{Harmonics in equal|618}} | ||
=== Subsets and supersets === | |||
Since 618 factors into {{factorization|618}}, 618edo has subset edos {{EDOs| 2, 3, 6, 103, 206, and 309 }}. 1236edo, which doubles it, provides a good correction for harmonic 3. | |||
Latest revision as of 23:03, 20 February 2025
| ← 617edo | 618edo | 619edo → |
618 equal divisions of the octave (abbreviated 618edo or 618ed2), also called 618-tone equal temperament (618tet) or 618 equal temperament (618et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 618 equal parts of about 1.94 ¢ each. Each step represents a frequency ratio of 21/618, or the 618th root of 2.
618edo is consistent to the 7-odd-limit, but harmonic 3 is about halfway between its steps. Nonetheless, as every other step of 1236edo, 618edo is excellent in approximating harmonics 5, 7, 9, 11, 13, and 17, making it suitable for a 2.9.5.7.11.13.17 subgroup interpretation, where the equal temperament notably tempers out 2601/2600, 4096/4095, 5832/5831, 6656/6655, 9801/9800, and 10648/10647. With a reasonable approximation of 19, it further tempers out 2926/2925, 5985/5984, and 6175/6174.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.958 | +0.094 | +0.106 | -0.027 | +0.138 | +0.249 | -0.890 | -0.101 | -0.426 | -0.878 | +0.852 |
| Relative (%) | +49.3 | +4.8 | +5.5 | -1.4 | +7.1 | +12.8 | -45.8 | -5.2 | -21.9 | -45.2 | +43.9 | |
| Steps (reduced) |
980 (362) |
1435 (199) |
1735 (499) |
1959 (105) |
2138 (284) |
2287 (433) |
2414 (560) |
2526 (54) |
2625 (153) |
2714 (242) |
2796 (324) | |
Subsets and supersets
Since 618 factors into 2 × 3 × 103, 618edo has subset edos 2, 3, 6, 103, 206, and 309. 1236edo, which doubles it, provides a good correction for harmonic 3.