2000edo: Difference between revisions

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== 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.

~~2000~~ = ~~24 × 53 , and its divisors are~~ {{~~EDOs~~|~~1, 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.~~

=== Prime harmonics ===

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

=== ~~Prime harmonics~~ ===

=== Subsets and supersets ===

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 ==

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=== Rank-2 temperaments ===

{| 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 per 8ve

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

| 20

| 287\2000 (87\2000)

| 287\2000 (87\2000)

| 172.2 (52.2)

| 172.2 (52.2)

| 169/153 (?)

| 169/153 (?)

| [[Calcium]]

|-

|25

|301\2000 (1\2000)

|180.6 (0.6)

|272/245 (?)

|[[Hemimanganese]]

|-

| 80

| 619\2000 (19\2000)

| 619\2000 (19\2000)

| 371.4 (11.4)

| 371.4 (11.4)

| 2275/1836 (?)

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

[[Category:Listen]]

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|2000}}
+{{ED intro}}
 == Theory ==
-edo 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.
+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.
-= 2<sup>4</sup> × 5<sup>3</sup> , and its divisors are {{EDOs|1, 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.
+=== Prime harmonics ===
+{{Harmonics in equal|2000|columns=12}}
+{{Harmonics in equal|2000|start=13|columns=12|collapsed=1|title=Approximation of prime harmonics in 2000edo (continued)}}
-=== Prime harmonics ===
+=== Subsets and supersets ===
-{{Harmonics in equal|2000|columns=13}}
+= {{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 ==
@@ Line 15: / Line 17: @@
 === Rank-2 temperaments ===
 {| 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
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
 | 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]]
+|-
+|25
+|301\2000<br />(1\2000)
+|180.6<br />(0.6)
+|272/245<br />(?)
+|[[Hemimanganese]]
 |-
 | 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]]
 |}
+<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:Equal divisions of the octave|####]] <!-- 4-digit number -->
+[[Category:Listen]]
-[[Category:29-limit]]