# 1000edo

 ← 999edo 1000edo 1001edo →
Prime factorization 23 × 53
Step size 1.2¢
Fifth 585\1000 (702¢) (→117\200)
Semitones (A1:m2) 95:75 (114¢ : 90¢)
Consistency limit 9
Distinct consistency limit 9

1000 equal divisions of the octave (abbreviated 1000edo or 1000ed2), also called 1000-tone equal temperament (1000tet) or 1000 equal temperament (1000et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1000 equal parts of exactly 1.2 ¢ each. Each step represents a frequency ratio of 21/1000, or the 1000th root of 2.

1000edo is notable mostly because it is the equal division corresponding to millioctaves.

## Theory

1000edo is related to 200edo, but the patent vals differ on the mapping for 5 and 7. In the 5-limit, it tempers out [38 -2 -15 (luna comma) and [-17 62 -35 (senior comma). In the 7-limit, it tempers out 4375/4374, 201768035/201326592, and 165288374272/164794921875, leading to the lunatic temperament and seniority temperament. It also tempers out 3025/3024, 9801/9800, and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, 4459/4455, and 10648/10647 in the 13-limit, leading to the deca temperament and donar temperament.

### Prime harmonics

Approximation of prime harmonics in 1000edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.045 +0.086 -0.426 -0.518 -0.528 -0.555 +0.087 +0.526 +0.023 -0.236
Relative (%) +0.0 +3.7 +7.2 -35.5 -43.2 -44.0 -46.3 +7.2 +43.8 +1.9 -19.6
Steps
(reduced)
1000
(0)
1585
(585)
2322
(322)
2807
(807)
3459
(459)
3700
(700)
4087
(87)
4248
(248)
4524
(524)
4858
(858)
4954
(954)

### Subsets and supersets

1000edo carries the interval size measure millioctave. Since 1000 factors into 23 × 53, 1000edo has subset edos 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, and 500.

2000edo, which doubles 1000edo, is consistent in the 29-odd-limit and thus provides good corrections for harmonics 7, 11, 13, 17, and 23.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [317 -200 [1000 1585]] -0.0142 0.0142 1.18
2.3.5 [38 -2 -15, [55 -64 20 [1000 1585 2322]] -0.0219 0.0159 1.33
2.3.5.7 4375/4374, 201768035/201326592, [12 -3 -14 9 [1000 1585 2322 2807]] +0.0215 0.0764 6.37
2.3.5.7.11 3025/3024, 4375/4374, 391314/390625, [-32 13 1 2 1 [1000 1585 2322 2807 3459]] +0.0472 0.0854 7.12
2.3.5.7.11.13 1001/1000, 3025/3024, 4225/4224, 4375/4374, 708883245/708837376 [1000 1585 2322 2807 3459 3700]] +0.0631 0.0857 7.14

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 161\1000 193.200 262144/234375 Lunatic (7-limit)
1 269\1000 322.800 3087/2560 Seniority
4 317\1000
(67\1000)
380.400
(80.400)
5103/4096
(22/21)
Quasithird
10 263\1000
(37\1000)
315.600
(44.400)
6/5
(15/14)
Deca
25 301\1000
(21\1000)
361.200
(25.200)
[54 13 -32⟩
(?)
Manganese

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

## Music

Xotla
• "Moongazing" from Lessor Groove (2020) Bandcamp | YouTube – atmospheric-electro, luna[25] in 1000edo