400edo: Difference between revisions

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== Theory ==

400edo is [[consistent]] in the [[21-odd-limit]]. It tempers out the unidecma, {{monzo| -7 22 -12 }}, and the quintosec comma, {{monzo| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], 4914/4913 and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.

400edo is a strong 17- and 19-limit system, [[consistency|distinctly and purely consistent]] to the [[21-odd-limit]]. It shares its excellent [[harmonic]] [[3/1|3]] with [[200edo]], which is a semiconvergent, while correcting the higher harmonics to near-just qualities.

As an equal temperament, it [[tempering out|tempers out]] the unidecma, {{monzo| -7 22 -12 }}, and the quintosec comma, {{monzo| 47 -15 -10 }}, in the [[5-limit]]; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the [[7-limit]]; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the [[11-limit]]; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the [[13-limit]], [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.

=== Prime harmonics ===

{{Harmonics in equal|400|columns=13|start=14|collapsed=true|title=Approximation of prime harmonics in 400edo (continued)}}

=== Subsets and supersets ===

400 factors into 24 × 52, ~~with~~ subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}. ~~Notably~~, ~~200edo contains a semiconvergent approximation to 3/2~~.

Since 400 factors into 24 × 52, 400edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}.

Of edos that are a multiple of 400, {{EDOs| 1600 and 2000}} are notable for their high consistency limits, as [[Interval size measure|interval size measures]], and perhaps as ways of tuning various temperaments.

== Interval table ==

~~{{Interval table}}~~

=== All intervals ===

See [[Table of 400edo intervals]].

=== Selected intervals ===

{| class="wikitable center-1"

|+

|-

! Step

! Eliora's ~~Naming System~~

! Eliora's naming system

! Associated ratio

|-

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== 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

|-

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| 2.3.5

| {{monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}

| [{{~~val~~| 400 634 929 }}]

| {{mapping| 400 634 929 }}

| -0.1080

| −0.1080

| 0.1331

| 4.44

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| 2.3.5.7

| 2401/2400, 1959552/1953125, 14348907/14336000

| [{{~~val~~| 400 634 929 1123 }}]

| {{mapping| 400 634 929 1123 }}

| -0.0965

| −0.0965

| 0.1170

| 3.90

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| 2.3.5.7.11

| 2401/2400, 5632/5625, 9801/9800, 46656/46585

| [{{~~val~~| 400 634 929 1123 1384 }}]

| {{mapping| 400 634 929 1123 1384 }}

| -0.1166

| −0.1166

| 0.1121

| 3.74

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| 2.3.5.7.11.13

| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325

| [{{~~val~~| 400 634 929 1123 1384 1480 }}]

| {{mapping| 400 634 929 1123 1384 1480 }}

| -0.0734

| −0.0734

| 0.1407

| 4.69

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| 2.3.5.7.11.13.17

| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095

| [{{~~val~~| 400 634 929 1123 1384 1480 1635 }}]

| {{mapping| 400 634 929 1123 1384 1480 1635 }}

| -0.0645

| −0.0645

| 0.1321

| 4.40

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| 2.3.5.7.11.13.17.19

| 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715

| [{{~~val~~| 400 634 929 1123 1384 1480 1635 1699 }}]

| {{mapping| 400 634 929 1123 1384 1480 1635 1699 }}

| -0.0413

| −0.0413

| 0.1380

| 4.60

|}

* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo|354]] in the 17-limit, and [[311edo|311]] in the 19-limit; it is bettered by [[422edo|422]] in either subgroup.

=== 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*

! ~~Temperaments~~

! Associated ratio*

! Temperament

|-

| 1

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|-

| 5

| 123\400 (37\400)

| 123\400 (37\400)

| 369.00 (111.00)

| 369.00 (111.00)

| 1024/891 (16/15)

| 1024/891 (16/15)

| [[Quintosec]]

|-

| 10

| 83\400 (3\400)

| 83\400 (3\400)

| 249.00 (9.00)

| 249.00 (9.00)

| 15/13 (176/175)

| 15/13 (176/175)

| [[Decoid]]

|-

| 80

| 166\400 (1\400)

| 166\400 (1\400)

| 498.00 (3.00)

| 498.00 (3.00)

| 4/3 (245/243)

| 4/3 (245/243)

| [[Octogintic]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==

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== Music ==

; [[Eliora]]

* [https://www.youtube.com/watch?v=av_RLK68ZUY ''Etude in Monzismic'' ~~(Op. 2, No. 3)~~] (2023)

* [https://www.youtube.com/watch?v=av_RLK68ZUY ''Etude in Monzismic''] (2023)

; [[Francium]]

* [https://www.youtube.com/watch?v=aTo2zfCWP9M ''thank you all''] (2023)

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|400}}
+{{ED intro}}
 == Theory ==
-edo is [[consistent]] in the [[21-odd-limit]]. It tempers out the unidecma, {{monzo| -7 22 -12 }}, and the quintosec comma, {{monzo| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], 4914/4913 and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.
+edo is a strong 17- and 19-limit system, [[consistency|distinctly and purely consistent]] to the [[21-odd-limit]]. It shares its excellent [[harmonic]] [[3/1|3]] with [[200edo]], which is a semiconvergent, while correcting the higher harmonics to near-just qualities.
+As an equal temperament, it [[tempering out|tempers out]] the unidecma, {{monzo| -7 22 -12 }}, and the quintosec comma, {{monzo| 47 -15 -10 }}, in the [[5-limit]]; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the [[7-limit]]; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the [[11-limit]]; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the [[13-limit]], [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.
 === Prime harmonics ===
 {{Harmonics in equal|400|columns=13}}
+{{Harmonics in equal|400|columns=13|start=14|collapsed=true|title=Approximation of prime harmonics in 400edo (continued)}}
 === Subsets and supersets ===
-factors into  2<sup>4</sup> × 5<sup>2</sup>, with subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}. Notably, 200edo contains a semiconvergent approximation to 3/2.
+Since 400 factors into 2<sup>4</sup> × 5<sup>2</sup>, 400edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}.
+Of edos that are a multiple of 400, {{EDOs| 1600 and 2000}} are notable for their high consistency limits, as [[Interval size measure|interval size measures]], and perhaps as ways of tuning various temperaments.
 == Interval table ==
-{{Interval table}}
+=== All intervals ===
+See [[Table of 400edo intervals]].
 === Selected intervals ===
 {| class="wikitable center-1"
-|+
+|-
 ! Step
-! Eliora's Naming System
+! Eliora's naming system
 ! Associated ratio
 |-
@@ Line 76: / Line 82: @@
 == 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
 |-
@@ Line 87: / Line 94: @@
 | 2.3.5
 | {{monzo| -7 22 -12 }}, {{monzo| 47 -15 -10 }}
-| [{{val| 400 634 929 }}]
+| {{mapping| 400 634 929 }}
-| -0.1080
+| &minus;0.1080
 | 0.1331
 | 4.44
@@ Line 94: / Line 101: @@
 | 2.3.5.7
 | 2401/2400, 1959552/1953125, 14348907/14336000
-| [{{val| 400 634 929 1123 }}]
+| {{mapping| 400 634 929 1123 }}
-| -0.0965
+| &minus;0.0965
 | 0.1170
 | 3.90
@@ Line 101: / Line 108: @@
 | 2.3.5.7.11
 | 2401/2400, 5632/5625, 9801/9800, 46656/46585
-| [{{val| 400 634 929 1123 1384 }}]
+| {{mapping| 400 634 929 1123 1384 }}
-| -0.1166
+| &minus;0.1166
 | 0.1121
 | 3.74
@@ Line 108: / Line 115: @@
 | 2.3.5.7.11.13
 | 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325
-| [{{val| 400 634 929 1123 1384 1480 }}]
+| {{mapping| 400 634 929 1123 1384 1480 }}
-| -0.0734
+| &minus;0.0734
 | 0.1407
 | 4.69
@@ Line 115: / Line 122: @@
 | 2.3.5.7.11.13.17
 | 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095
-| [{{val| 400 634 929 1123 1384 1480 1635 }}]
+| {{mapping| 400 634 929 1123 1384 1480 1635 }}
-| -0.0645
+| &minus;0.0645
 | 0.1321
 | 4.40
@@ Line 122: / Line 129: @@
 | 2.3.5.7.11.13.17.19
 | 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715
-| [{{val| 400 634 929 1123 1384 1480 1635 1699 }}]
+| {{mapping| 400 634 929 1123 1384 1480 1635 1699 }}
-| -0.0413
+| &minus;0.0413
 | 0.1380
 | 4.60
 |}
+* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo|354]] in the 17-limit, and [[311edo|311]] in the 19-limit; it is bettered by [[422edo|422]] in either subgroup.
 === 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*
-! Temperaments
+! Associated<br />ratio*
+! Temperament
 |-
 | 1
@@ Line 174: / Line 183: @@
 |-
 | 5
-| 123\400<br>(37\400)
+| 123\400<br />(37\400)
-| 369.00<br>(111.00)
+| 369.00<br />(111.00)
-| 1024/891<br>(16/15)
+| 1024/891<br />(16/15)
 | [[Quintosec]]
 |-
 | 10
-| 83\400<br>(3\400)
+| 83\400<br />(3\400)
-| 249.00<br>(9.00)
+| 249.00<br />(9.00)
-| 15/13<br>(176/175)
+| 15/13<br />(176/175)
 | [[Decoid]]
 |-
 | 80
-| 166\400<br>(1\400)
+| 166\400<br />(1\400)
-| 498.00<br>(3.00)
+| 498.00<br />(3.00)
-| 4/3<br>(245/243)
+| 4/3<br />(245/243)
 | [[Octogintic]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 203: / Line 213: @@
 == Music ==
 ; [[Eliora]]
-* [https://www.youtube.com/watch?v=av_RLK68ZUY ''Etude in Monzismic'' (Op. 2, No. 3)] (2023)
+* [https://www.youtube.com/watch?v=av_RLK68ZUY ''Etude in Monzismic''] (2023)
 ; [[Francium]]
 * [https://www.youtube.com/watch?v=aTo2zfCWP9M ''thank you all''] (2023)
+[[Category:Listen]]