# 1051edo

 ← 1050edo 1051edo 1052edo →
Prime factorization 1051 (prime)
Step size 1.14177¢
Fifth 615\1051 (702.188¢)
Semitones (A1:m2) 101:78 (115.3¢ : 89.06¢)
Consistency limit 3
Distinct consistency limit 3

1051 equal divisions of the octave (abbreviated 1051edo or 1051ed2), also called 1051-tone equal temperament (1051tet) or 1051 equal temperament (1051et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1051 equal parts of about 1.14 ¢ each. Each step represents a frequency ratio of 21/1051, or the 1051st root of 2.

## Theory

1051edo only has a consistency limit of 3 and does poorly with approximating the harmonic 5. However, it has a reasonable representation of the 2.3.7.11.17.19 subgroup.

Assume the patent val, 1051et tempers out 2460375/2458624 in the 7-limit; 820125/819896, 2097152/2096325, 514714375/514434888, 180224/180075, 184549376/184528125, 43923/43904 and 20614528/20588575 in the 11-limit.

### Odd harmonics

Approximation of odd harmonics in 1051edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.233 -0.396 +0.537 +0.467 +0.157 -0.185 -0.162 +0.087 +0.489 -0.372 -0.301
Relative (%) +20.4 -34.6 +47.0 +40.9 +13.7 -16.2 -14.2 +7.7 +42.8 -32.6 -26.4
Steps
(reduced)
1666
(615)
2440
(338)
2951
(849)
3332
(179)
3636
(483)
3889
(736)
4106
(953)
4296
(92)
4465
(261)
4616
(412)
4754
(550)

### Subsets and supersets

1051edo is the 177th prime edo. 2102edo, which doubles it, gives a good correction to the harmonic 5 and 7. 4212edo, which quadruples it, gives a good correction to the harmonic 3.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [1666 -1051 [1051 1666]] -0.0736 0.0736 6.45
2.3.5 [-68 18 17, [-26 -29 31 [1051 1666 2440]] (1051) +0.0077 0.1298 11.4
2.3.5 [40 7 -22, [63 -50 7 [1051 1666 2441]] (1051c) -0.1562 0.1313 11.5

Francium