2000edo: Difference between revisions

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{{~~EDO~~ intro~~|2000~~}} It 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, the first edo to do so being [[2460edo]].

~~2000~~ = ~~24 × 53; some of its divisors are~~ [[~~10edo~~|10]], [[~~16edo|16~~]], [[~~25edo|25~~]], [[~~50edo~~|50]], [[~~80edo|80~~]], [[~~100edo|100~~]], [[~~125edo~~|~~125~~]] and [[~~200edo|200~~]]~~. Also there is~~ the ~~1000 division of [[millioctave]]s~~, ~~where~~ it ~~might be argued that cutting these in half makes for a better system~~.

== Theory ==

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 ===

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

[[~~Category:Equal divisions~~ of the ~~octave|####~~]]

=== Subsets and supersets ===

[[Category:~~29-limit~~]]

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 ==

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

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | 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]]

|}

<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]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|2000}} It 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, the first edo to do so being [[2460edo]].
+{{ED intro}}
-= 2<sup>4</sup> × 5<sup>3</sup>; some of its divisors are [[10edo|10]], [[16edo|16]], [[25edo|25]], [[50edo|50]], [[80edo|80]], [[100edo|100]], [[125edo|125]] and [[200edo|200]]. Also there is the 1000 division of [[millioctave]]s, where it might be argued that cutting these in half makes for a better system.
+== Theory ==
+edo 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 ===
-{{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)}}
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
+=== Subsets and supersets ===
-[[Category:29-limit]]
+= {{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 ==
+edo has the smallest relative error than any previous equal temperaments in the 19-limit. It is only bettered by [[2460edo]].
+=== 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
+|-
+| 20
+| 287\2000<br />(87\2000)
+| 172.2<br />(52.2)
+| 169/153<br />(?)
+| [[Calcium]]
+|-
+|25
+|301\2000<br />(1\2000)
+|180.6<br />(0.6)
+|272/245<br />(?)
+|[[Hemimanganese]]
+|-
+| 80
+| 619\2000<br />(19\2000)
+| 371.4<br />(11.4)
+| 2275/1836<br />(?)
+| [[Mercury]]
+|}
+<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]]