1984edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 1983edo1984edo1985edo →
Prime factorization 26 × 31
Step size 0.604839¢ 
Fifth 1161\1984 (702.218¢)
Semitones (A1:m2) 191:147 (115.5¢ : 88.91¢)
Dual sharp fifth 1161\1984 (702.218¢)
Dual flat fifth 1160\1984 (701.613¢) (→145\248)
Dual major 2nd 337\1984 (203.831¢)
Consistency limit 7
Distinct consistency limit 7

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 3, 5, 7, 11, and 17 all tuned sharp. Harmonics 9 and 15, though, are tuned flat, 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

Approximation of odd harmonics in 1984edo
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.