2000edo: Difference between revisions

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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2000}}
{{ED intro}}


== Theory ==
== Theory ==
2000edo is distinctly [[consistent]] through the 29-odd-limit and a strong no-31's 41-limit system; the only smaller edo with a smaller [[29-limit]] [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being [[1578edo]]. The only ones superior to it in the [[23-limit]] are 1578 and [[1889edo]], and in the 19-limit, nothing smaller defeats it.  
2000edo is [[consistency|distinctly consistent]] through the [[29-odd-limit]] and a strong no-31's 41-limit system; the only smaller edo with a smaller [[29-limit]] [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being [[1578edo]]. The only ones superior to it in the [[23-limit]] are [[1578edo|1578-]] and [[1889edo]], and in the 19-limit, nothing smaller defeats it.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|2000|columns=13}}
{{Harmonics in equal|2000|columns=12}}
{{Harmonics in equal|2000|start=13|columns=12|collapsed=1|title=Approximation of prime harmonics in 2000edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
2000 = 2<sup>4</sup> × 5<sup>3</sup>, and its divisors are {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000 }}. From these, [[1000edo]] is notable because it carries the interval size measure [[millioctave]]. It is argued that cutting millioctaves in half makes for a better interval measuring system, in light of 2000edo's high consistency limit, which introduces just interval approximations not present in 1000edo. In addition, 2000edo inherits its fifth from [[200edo]], where it is semiconvergent.
2000 = {{factorization|2000}}, and its nontrivial divisors are {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000 }}. From these, [[1000edo]] is notable because it carries the interval size measure [[millioctave]]. It is argued that cutting millioctaves in half makes for a better interval measuring system, in light of 2000edo's high consistency limit, which introduces just interval approximations not present in 1000edo. In addition, 2000edo inherits its fifth from [[200edo]], where it is semiconvergent.


== Regular temperament properties ==
== Regular temperament properties ==
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
| 20
| 20
| 287\2000<br>(87\2000)
| 287\2000<br />(87\2000)
| 172.2<br>(52.2)
| 172.2<br />(52.2)
| 169/153<br>(?)
| 169/153<br />(?)
| [[Calcium]]
| [[Calcium]]
|-
|-
|25
|25
|301\2000<br>(1\2000)
|301\2000<br />(1\2000)
|180.6<br>(0.6)
|180.6<br />(0.6)
|272/245<br>(?)
|272/245<br />(?)
|[[Hemimanganese]]
|[[Hemimanganese]]
|-
|-
| 80
| 80
| 619\2000<br>(19\2000)
| 619\2000<br />(19\2000)
| 371.4<br>(11.4)
| 371.4<br />(11.4)
| 2275/1836<br>(?)
| 2275/1836<br />(?)
| [[Mercury]]
| [[Mercury]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<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 ==
; [[Eliora]]
* ''[https://www.youtube.com/watch?v=gM4dfrF5wPg Fugue, but Not (in A Mercury & Bidia)]'' (2024)
 
[[Category:Listen]]

Latest revision as of 13:31, 13 March 2026

← 1999edo 2000edo 2001edo →
Prime factorization 24 × 53
Step size 0.6 ¢ 
Fifth 1170\2000 (702 ¢) (→ 117\200)
Semitones (A1:m2) 190:150 (114 ¢ : 90 ¢)
Consistency limit 29
Distinct consistency limit 29

2000 equal divisions of the octave (abbreviated 2000edo or 2000ed2), also called 2000-tone equal temperament (2000tet) or 2000 equal temperament (2000et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2000 equal parts of exactly 0.6 ¢ each. Each step represents a frequency ratio of 21/2000, or the 2000th root of 2.

Theory

2000edo is distinctly consistent through the 29-odd-limit and a strong no-31's 41-limit system; the only smaller edo with a smaller 29-limit relative error being 1578edo. The only ones superior to it in the 23-limit are 1578- and 1889edo, and in the 19-limit, nothing smaller defeats it.

Prime harmonics

Approximation of prime harmonics in 2000edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.000 +0.045 +0.086 +0.174 +0.082 +0.072 +0.045 +0.087 -0.074 +0.023 -0.236 +0.056
Relative (%) +0.0 +7.5 +14.4 +29.0 +13.7 +12.1 +7.4 +14.5 -12.4 +3.8 -39.3 +9.3
Steps
(reduced) 		2000
(0) 		3170
(1170) 		4644
(644) 		5615
(1615) 		6919
(919) 		7401
(1401) 		8175
(175) 		8496
(496) 		9047
(1047) 		9716
(1716) 		9908
(1908) 		10419
(419)
Approximation of prime harmonics in 2000edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) -0.062 +0.282 -0.107 +0.095 -0.172 -0.285 -0.107 -0.297 +0.211 +0.263 -0.047 -0.280
Relative (%) -10.4 +47.0 -17.8 +15.9 -28.6 -47.5 -17.8 -49.4 +35.1 +43.9 -7.9 -46.7
Steps
(reduced) 		10715
(715) 		10853
(853) 		11109
(1109) 		11456
(1456) 		11765
(1765) 		11861
(1861) 		12132
(132) 		12299
(299) 		12380
(380) 		12608
(608) 		12750
(750) 		12951
(951)

Subsets and supersets

2000 = 24 × 53, and its nontrivial divisors are 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000. From these, 1000edo is notable because it carries the interval size measure millioctave. It is argued that cutting millioctaves in half makes for a better interval measuring system, in light of 2000edo's high consistency limit, which introduces just interval approximations not present in 1000edo. In addition, 2000edo inherits its fifth from 200edo, where it is semiconvergent.

Regular temperament properties

2000edo has the smallest relative error than any previous equal temperaments in the 19-limit. It is only bettered by 2460edo.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
20 287\2000
(87\2000) 		172.2
(52.2) 		169/153
(?) 		Calcium
25 301\2000
(1\2000) 		180.6
(0.6) 		272/245
(?) 		Hemimanganese
80 619\2000
(19\2000) 		371.4
(11.4) 		2275/1836
(?) 		Mercury

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

Music

Eliora
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