1559edo

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Prime factorization 1559 (prime)
Step size 0.769724¢ 
Fifth 912\1559 (701.988¢)
Semitones (A1:m2) 148:117 (113.9¢ : 90.06¢)
Consistency limit 9
Distinct consistency limit 9

1559 equal divisions of the octave (abbreviated 1559edo or 1559ed2), also called 1559-tone equal temperament (1559tet) or 1559 equal temperament (1559et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1559 equal parts of about 0.77 ¢ each. Each step represents a frequency ratio of 21/1559, or the 1559th root of 2.

Theory

1559edo is consistent to the 9-odd-limit. Using the patent val, it tempers out [38 -2 -15 (luna comma) and [71 -99 37 (raider comma) in the 5-limit; 43046721/43025920, 78125000/78121827 and [32 1 -6 -7 in the 7-limit, 3025/3024, 180224/180075, 50014503/50000000 and 56953125/56942116 in the 11-limit.

Prime harmonics

Approximation of prime harmonics in 1559edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.033 +0.088 +0.257 -0.195 +0.011 -0.273 +0.370 -0.179 +0.314 +0.314
Relative (%) +0.0 +4.3 +11.4 +33.4 -25.4 +1.4 -35.5 +48.1 -23.3 +40.8 +40.8
Steps
(reduced)
1559
(0)
2471
(912)
3620
(502)
4377
(1259)
5393
(716)
5769
(1092)
6372
(136)
6623
(387)
7052
(816)
7574
(1338)
7724
(1488)

Subsets and supersets

1559edo is the 246th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [2471 -1559 [1559 2471]] -0.0106 0.0106 1.38
2.3.5 [38 -2 -15, [71 -99 37 [1559 2471 3620]] -0.0196 0.0155 2.01
2.3.5.7 43046721/43025920, 78125000/78121827, 12884901888/12867859375 [1559 2471 3620 4377]] -0.0376 0.0339 4.40
2.3.5.7.11 3025/3024, 180224/180075, 50014503/50000000, 56953125/56942116 [1559 2471 3620 4377 5393]] -0.0188 0.0483 6.27
2.3.5.7.11.13 1716/1715, 3025/3024, 4096/4095, 492128/492075, 5282739/5281250 [1559 2471 3620 4377 5393 5769]] -0.0162 0.0445 5.78

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 251\1559 193.201 262144/234375 Luna
1 446\1559 343.297 8000/6561 Raider

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Francium
  • "confluenceprettype" from albumwithoutspaces (2024) – Spotify | Bandcamp | YouTube – luna[25] in 1559edo tuning