1984edo: Difference between revisions
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1984edo is [[consistent]] in the [[7-odd-limit]] and is a mostly sharp system, with [[ | 1984edo is [[consistent]] in the [[7-odd-limit]] and is a mostly sharp system, with all primes up to [[23/1|23]] tuned sharp. However, harmonics 9, 15, and 21 are more than 50% of a step sharp by patent val, and flat by direct approximation, which results in [[consistency|inconsistencies]]; that is, their [[direct approximation]]s are not the same as the sum of their constituent odd harmonics' direct approximations: 9/1 is 6289 steps while 3/1 is 3145 steps (and {{nowrap|3145 + 3145 {{=}} 6290}}, ≠ 6289), and 15/1 is 7751 steps while 5/1 is 4607 steps (and {{nowrap|3145 + 4607 {{=}} 7752}}, ≠ 7751). 1984edo does, however, approximate the 2.9.19.31.33 [[subgroup]] well. | ||
In the 7-limit the equal temperament [[tempering out|tempers out]] the [[wizma]] (420175/419904), the [[garischisma]] (33554432/33480783), and the [[pessoalisma]] (2147483648/2144153025). | In the 7-limit the equal temperament [[tempering out|tempers out]] the [[wizma]] (420175/419904), the [[garischisma]] (33554432/33480783), and the [[pessoalisma]] (2147483648/2144153025). | ||
Latest revision as of 01:14, 19 December 2025
| ← 1983edo | 1984edo | 1985edo → |
1984 equal divisions of the octave (abbreviated 1984edo or 1984ed2), also called 1984-tone equal temperament (1984tet) or 1984 equal temperament (1984et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1984 equal parts of about 0.605 ¢ each. Each step represents a frequency ratio of 21/1984, or the 1984th root of 2.
1984edo is consistent in the 7-odd-limit and is a mostly sharp system, with all primes up to 23 tuned sharp. However, harmonics 9, 15, and 21 are more than 50% of a step sharp by patent val, and flat by direct approximation, which results in inconsistencies; that is, their direct approximations are not the same as the sum of their constituent odd harmonics' direct approximations: 9/1 is 6289 steps while 3/1 is 3145 steps (and 3145 + 3145 = 6290, ≠ 6289), and 15/1 is 7751 steps while 5/1 is 4607 steps (and 3145 + 4607 = 7752, ≠ 7751). 1984edo does, however, approximate the 2.9.19.31.33 subgroup well.
In the 7-limit the equal temperament tempers out the wizma (420175/419904), the garischisma (33554432/33480783), and the pessoalisma (2147483648/2144153025).
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.263 | +0.178 | +0.126 | -0.079 | +0.295 | +0.198 | -0.164 | +0.287 | +0.068 | -0.216 | +0.153 |
| Relative (%) | +43.4 | +29.5 | +20.8 | -13.1 | +48.8 | +32.8 | -27.1 | +47.4 | +11.2 | -35.8 | +25.3 | |
| Steps (reduced) |
3145 (1161) |
4607 (639) |
5570 (1602) |
6289 (337) |
6864 (912) |
7342 (1390) |
7751 (1799) |
8110 (174) |
8428 (492) |
8714 (778) |
8975 (1039) | |
Subsets and supersets
Since 1984 factors into 26 × 31, 1984edo has subset edos 2, 4, 8, 16, 31, 32, 62, 64, 124, 248, 496, and 992.