1000edo: Difference between revisions

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

1000edo's step size is called a '''millioctave''' when used as an [[interval size unit]].

== Theory ==

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 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.

=== Prime harmonics ===

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=== Subsets and supersets ===

1000edo ~~carries the interval size measure ''millioctave'' and~~ has subset edos {{EDOs| 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500 }}.

Since 1000 factors into 23 × 53, 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 ~~the 2.~~7.11.13.17.23 ~~subgroup~~.

[[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"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| {{monzo| 317 -200 }}

| {{mapping| 1000 1585 }}

| -0.0142

| −0.0142

| 0.0142

| 1.18

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| {{monzo| 38 -2 -15 }}, {{monzo| 55 -64 20 }}

| {{mapping| 1000 1585 2322 }}

| -0.0219

| −0.0219

| 0.0159

| 1.33

|-

| 2.3.5.7

| 4375/4374, ~~4202539929~~/~~4194304000~~, ~~578509309952/576650390625~~

| 4375/4374, 201768035/201326592, {{monzo| 12 -3 -14 9 }}

| {{mapping| 1000 1585 2322 2807 }}

| +0.0215

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|-

| 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 }}

| {{mapping| 1000 1585 2322 2807 3459 }}

| +0.0472

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|-

| 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

| {{mapping| 1000 1585 2322 2807 3459 3700 }}

| +0.0631

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| 7.14

|}

=== 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

|-

|1

| 1

|161\1000

| 161\1000

|193.200

| 193.200

|262144/234375

| 262144/234375

|[[~~Luna~~]]

| [[Lunatic]] (7-limit)

|-

| 1

| 269\1000

| 322.800

| 3087/2560

| [[Seniority]]

|-

|1

| 4

|~~269~~\1000

| 317\1000 (67\1000)

|~~322~~.~~800~~

| 380.400 (80.400)

|~~{{monzo|-6 23 -13}}~~

| 5103/4096 (22/21)

|[[~~Senior]] / [[seniority~~]]

| [[Quasithird]]

|-

|4

| 10

|~~317~~\1000 (67\1000)

| 263\1000 (37\1000)

|~~380~~.~~400~~ (80.400)

| 315.600 (44.400)

|~~5103~~/~~4096~~ (22/21)

| 6/5 (15/14)

|[[~~Quasithird~~]]

| [[Deca]]

|-

|10

| 25

|~~263~~\1000 (37\1000)

| 301\1000 (21\1000)

|~~315~~.~~600~~ (44.~~400~~)

| 361.200 (25.200)

|~~6/5~~ (~~15/14~~)

| [54 13 -32⟩ (?)

|[[~~Deca~~]]

| [[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 ==

; [[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, ~~luna~~[25] in 1000edo

* "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:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1000}}
+{{ED intro}}
-edo is notable mostly because it is the equal division corresponding to millioctaves.
+edo's step size is called a '''millioctave''' when used as an [[interval size unit]].
 == Theory ==
-edo 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.
+edo 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.
 === Prime harmonics ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-edo carries the interval size measure ''millioctave'' and has subset edos {{EDOs| 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500 }}.
+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 the 2.7.11.13.17.23 subgroup.
+[[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"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 29: / Line 30: @@
 | {{monzo| 317 -200 }}
 | {{mapping| 1000 1585 }}
-| -0.0142
+| −0.0142
 | 0.0142
 | 1.18
@@ Line 36: / Line 37: @@
 | {{monzo| 38 -2 -15 }}, {{monzo| 55 -64 20 }}
 | {{mapping| 1000 1585 2322 }}
-| -0.0219
+| −0.0219
 | 0.0159
 | 1.33
 |-
 | 2.3.5.7
-| 4375/4374, 4202539929/4194304000, 578509309952/576650390625
+| 4375/4374, 201768035/201326592, {{monzo| 12 -3 -14 9 }}
 | {{mapping| 1000 1585 2322 2807 }}
 | +0.0215
@@ Line 48: / Line 49: @@
 |-
 | 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 }}
 | {{mapping| 1000 1585 2322 2807 3459 }}
 | +0.0472
@@ Line 55: / Line 56: @@
 |-
 | 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
 | {{mapping| 1000 1585 2322 2807 3459 3700 }}
 | +0.0631
@@ Line 61: / Line 62: @@
 | 7.14
 |}
 === 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
 |-
-|1
+| 1
-|161\1000
+| 161\1000
-|193.200
+| 193.200
-|262144/234375
+| 262144/234375
-|[[Luna]]
+| [[Lunatic]] (7-limit)
+|-
+| 1
+| 269\1000
+| 322.800
+| 3087/2560
+| [[Seniority]]
 |-
-|1
+| 4
-|269\1000
+| 317\1000<br />(67\1000)
-|322.800
+| 380.400<br />(80.400)
-|{{monzo|-6 23 -13}}
+| 5103/4096<br />(22/21)
-|[[Senior]] / [[seniority]]
+| [[Quasithird]]
 |-
-|4
+| 10
-|317\1000<br>(67\1000)
+| 263\1000<br />(37\1000)
-|380.400<br>(80.400)
+| 315.600<br />(44.400)
-|5103/4096<br>(22/21)
+| 6/5<br />(15/14)
-|[[Quasithird]]
+| [[Deca]]
 |-
-|10
+| 25
-|263\1000<br>(37\1000)
+| 301\1000<br />(21\1000)
-|315.600<br>(44.400)
+| 361.200<br />(25.200)
-|6/5<br>(15/14)
+| [54 13 -32⟩<br />(?)
-|[[Deca]]
+| [[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 ==
+; [[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, luna[25] in 1000edo
+* "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:Listen]]