2048edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 2047edo2048edo2049edo →
Prime factorization 211
Step size 0.585938¢
Fifth 1198\2048 (701.953¢) (→599\1024)
Semitones (A1:m2) 194:154 (113.7¢ : 90.23¢)
Consistency limit 9
Distinct consistency limit 9

2048 equal divisions of the octave (2048edo), or 2048-tone equal temperament (2048tet), 2048 equal temperament (2048et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 2048 equal parts of about 0.586 ¢ each.

Theory

Approximation of prime harmonics in 2048edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error absolute (¢) +0.000 -0.002 -0.181 -0.271 +0.049 +0.293 -0.073 +0.143 -0.149 -0.085
relative (%) +0 -0 -31 -46 +8 +50 -12 +24 -25 -15
Steps
(reduced)
2048
(0)
3246
(1198)
4755
(659)
5749
(1653)
7085
(941)
7579
(1435)
8371
(179)
8700
(508)
9264
(1072)
9949
(1757)

2048edo has an excellent perfect fifth, which derives from 1024edo. It tempers out the breedsma (2401/2400) in the 7-limit, and supports the rank 3 breed temperament.

2048edo is usable as high as 43-limit, with the 2048f val regular temperament having an error of 0.023 cents (0.04 edosteps) per octave, and the no-13s limit patent val is unambiguous and has an error of only 0.019 cents/octave.

2048edo is the 11th power of two EDO.

In the 5-limit, it supports monzismic temperament.

In the 11-limit, 2048edo tempers out the quartisma.

Regular temperament properties

Subgroup Comma list Mapping Optimal

8ve stretch (¢)

Tuning error
Absolute (¢) Relative (%)
2.3.5 [54, -37, 2,[-111, -12, 56 [2048 3246 4755]] 0.0264 0.0364 6.22
2.3.5.7 2401/2400, [-18, -13, 13, 3, [49, -38, 0, 4 [2048 3246 4755 5749]] 0.0439 0.0438 7.48
2.3.5.7.11 2401/2400, 1890625/1889568, 1953125/1951488, 117440512/117406179 [2048 3246 4755 5749 7085]] 0.0323 0.0456 7.78