2048edo

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← 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 (abbreviated 2048edo or 2048ed2), also called 2048-tone equal temperament (2048tet) or 2048 equal temperament (2048et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2048 equal parts of about 0.586 ¢ each. Each step represents a frequency ratio of 21/2048, or the 2048th root of 2.

Theory

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-13's limit patent val is unambiguous and has an error of only 0.019 cents/octave.

In the 5-limit, it supports monzismic temperament. In the 11-limit, 2048edo tempers out the quartisma.

Prime harmonics

Approximation of prime harmonics in 2048edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.002 -0.181 -0.271 +0.049 +0.293 -0.073 +0.143 -0.149 -0.085 -0.114
Relative (%) +0.0 -0.3 -30.9 -46.3 +8.4 +49.9 -12.4 +24.4 -25.5 -14.5 -19.4
Steps
(reduced)
2048
(0)
3246
(1198)
4755
(659)
5749
(1653)
7085
(941)
7579
(1435)
8371
(179)
8700
(508)
9264
(1072)
9949
(1757)
10146
(1954)

Subsets and supersets

2048edo is the 11th-power-of-two edo, with subset edos 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024.

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