1000edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''1000 edo''' divides the octave in 1000 equal parts of exactly 1.2 [[cent]]s, or 1 [[millioctave]] each. It is notable mostly because it is the equal division corresponding to millioctaves.
{{ED intro}}


1000edo is related to 200edo, but the [[patent val]]s differ on the mapping for 5 and 7. In the [[5-limit]], it tempers out luna comma, 274877906944/274658203125 and senior comma, {{Monzo| -17 62 -35 }}. 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]].  
1000edo's step size is called a '''millioctave''' when used as an [[interval size unit]].


{{primes in edo|1000|columns=11}}
== Theory ==
1000edo is related to [[200edo]], but the [[patent val]]s differ on the mapping for [[5/1|5]] and [[7/1|7]]. In the [[5-limit]], it [[tempering out|tempers out]] {{monzo| 38 -2 -15 }} ([[luna comma]]) and {{monzo| -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.


==Regular temperament properties==
=== Prime harmonics ===
{{Harmonics in equal|1000}}
 
=== Subsets and supersets ===
Since 1000 factors into 2<sup>3</sup> × 5<sup>3</sup>, 1000edo has subset edos {{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, [[11/1|11]], [[13/1|13]], [[17/1|17]], and [[23/1|23]].
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|2.3
! [[TE error|Absolute]] (¢)
|{{monzo|317 -200}}
! [[TE simple badness|Relative]] (%)
|{{val|1000 1585}}
|-
| -0.0142
| 2.3
| {{monzo| 317 -200 }}
| {{mapping| 1000 1585 }}
| −0.0142
| 0.0142
| 0.0142
| 1.18
| 1.18
|-
|-
|2.3.5
| 2.3.5
|{{monzo|38 -2 -15}}, {{monzo|55 -64 20}}
| {{monzo| 38 -2 -15 }}, {{monzo| 55 -64 20 }}
|{{val|1000 1585 2322}}
| {{mapping| 1000 1585 2322 }}
| -0.0219
| −0.0219
| 0.0159
| 0.0159
| 1.33
| 1.33
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, 4202539929/4194304000, 578509309952/576650390625
| 4375/4374, 201768035/201326592, {{monzo| 12 -3 -14 9 }}
|{{val|1000 1585 2322 2807}}
| {{mapping| 1000 1585 2322 2807 }}
| +0.0215
| +0.0215
| 0.0764
| 0.0764
| 6.37
| 6.37
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 4375/4374, 422576/421875, 5907360375/5905580032
| 3025/3024, 4375/4374, 391314/390625, {{monzo| -32 13 1 2 1 }}
|{{val|1000 1585 2322 2807 3459}}
| {{mapping| 1000 1585 2322 2807 3459 }}
| +0.0472
| +0.0472
| 0.0854
| 0.0854
| 7.12
| 7.12
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|1001/1000, 3025/3024, 4459/4455, 43904/43875, 708883245/708837376
| 1001/1000, 3025/3024, 4225/4224, 4375/4374, 708883245/708837376
|{{val|1000 1585 2322 2807 3459 3700}}
| {{mapping| 1000 1585 2322 2807 3459 3700 }}
| +0.0631
| +0.0631
| 0.0857
| 0.0857
| 7.14
| 7.14
|}
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 161\1000
| 193.200
| 262144/234375
| [[Lunatic]] (7-limit)
|-
| 1
| 269\1000
| 322.800
| 3087/2560
| [[Seniority]]
|-
| 4
| 317\1000<br />(67\1000)
| 380.400<br />(80.400)
| 5103/4096<br />(22/21)
| [[Quasithird]]
|-
| 10
| 263\1000<br />(37\1000)
| 315.600<br />(44.400)
| 6/5<br />(15/14)
| [[Deca]]
|-
| 25
| 301\1000<br />(21\1000)
| 361.200<br />(25.200)
| [54 13 -32⟩<br />(?)
| [[Manganese]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Music ==
== Music ==
* [https://www.youtube.com/watch?v=lTT3QGTngIs Dream Up (Demo Version)] by [[Sevish]]
; [[Francium]]
* "Unsuspecting Tyrant Double Decker Beef Fort" from ''Unsuspecting Tyrant Double Decker Beef Fort'' (2026) – [https://open.spotify.com/track/0pblUR02SXzAHot04qdA10 Spotify] | [https://francium223.bandcamp.com/track/unsuspecting-tyrant-double-decker-beef-fort Bandcamp] | [https://www.youtube.com/watch?v=BrZ07LxWCvQ YouTube] – in Deca, 1000edo tuning
 
; [[Xotla]]
* "Moongazing" from ''Lessor Groove'' (2020) [https://xotla.bandcamp.com/track/moongazing-luna-25 Bandcamp] | [https://www.youtube.com/watch?v=EWuVnLOcaRg YouTube] – atmospheric-electro, in Luna[25], 1000edo tuning


[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
[[Category:Listen]]

Latest revision as of 13:31, 13 March 2026

← 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's step size is called a millioctave when used as an interval size unit.

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

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 distinct

Music

Francium
  • "Unsuspecting Tyrant Double Decker Beef Fort" from Unsuspecting Tyrant Double Decker Beef Fort (2026) – Spotify | Bandcamp | YouTube – in Deca, 1000edo tuning
Xotla
  • "Moongazing" from Lessor Groove (2020) Bandcamp | YouTube – atmospheric-electro, in Luna[25], 1000edo tuning
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