472edo: Difference between revisions

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~~'''472edo''' is the [[EDO|equal division of the octave]] into 472 parts of 2.54237 [[cent]]s each.~~

472edo is [[~~consistent]] to the [[11-odd-limit]]. It is [[Enfactoring~~|enfactored]] in the 5-limit, with the same tuning as 118edo, defined by tempering out the [[schisma]] and the [[parakleisma]]~~. In the 7-limit, it tempers out [[2401/2400]], 2460375/2458624~~, ~~and 30623756184/30517578125; in~~ the ~~11-limit,~~ [[~~9801/9800~~]]~~, 46656/46585, 117649/117612, and 234375/234256~~ , [[~~Support|supporting~~]] the [[~~Breedsmic temperaments #Maviloid|maviloid~~]] ~~temperament,~~ the [[Schismatic family #Bisesqui|bisesqui temperament]], and the [[Schismatic family #Octant|octant temperament]]. Using the [[patent val]], it tempers out [[729/728]], [[1575/1573]], [[2200/2197]], [[4096/4095]], and 21168/21125 in the 13-~~limit, so it also supports the 13~~-limit ~~octant~~.

== Theory ==

472edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]], but the approximation to higher harmonics are much improved. It is a [[zeta peak integer edo]], [[consistent]] to the [[11-odd-limit]] or the no-13 [[29-odd-limit]].

~~It is a~~ [[~~zeta peak integer edo~~]].

In the 7-limit, the equal temperament tempers out [[2401/2400]], 2460375/2458624, and 30623756184/30517578125; in the 11-limit, [[9801/9800]], 46656/46585, 117649/117612, and 234375/234256, [[support]]ing the [[maviloid]] temperament, the [[Schismatic family #Bisesqui|bisesqui]] temperament, and the [[octant]] temperament. Using the [[patent val]], it tempers out [[729/728]], [[1575/1573]], [[2200/2197]], [[4096/4095]], and 21168/21125 in the 13-limit, so it also supports the 13-limit octant.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 472 factors into {{factorization|472}}, 472edo has subset edos {{EDOs| 2, 4, 8, 59, 118, and 236 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

|-

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

! 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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| 729/728, 1575/1573, 2200/2197, 2401/2400, 4096/4095

| [{{val| 472 748 1096 1325 1633 1747 }}]

| -0.0341

| −0.0341

| 0.1365

| 5.37

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{| 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 ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

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| 1

| 205\472

| ~~498~~.31

| 521.19

| 875/648

| [[Maviloid]]

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|-

| 8

| 196\472 (19\472)

| 196\472 (19\472)

| 498.31 (48.31)

| 498.31 (48.31)

| 4/3 (36/35)

| 4/3 (36/35)

| [[Octant]]

|}

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

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

[[~~Category:Zeta~~]]

@@ Line 1: / Line 1: @@
-'''472edo''' is the [[EDO|equal division of the octave]] into 472 parts of 2.54237 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
-edo is [[consistent]] to the [[11-odd-limit]]. It is [[Enfactoring|enfactored]] in the 5-limit, with the same tuning as 118edo, defined by tempering out the [[schisma]] and the [[parakleisma]]. In the 7-limit, it tempers out [[2401/2400]], 2460375/2458624, and 30623756184/30517578125; in the 11-limit, [[9801/9800]], 46656/46585, 117649/117612, and 234375/234256 , [[Support|supporting]] the [[Breedsmic temperaments #Maviloid|maviloid]] temperament, the [[Schismatic family #Bisesqui|bisesqui temperament]], and the [[Schismatic family #Octant|octant temperament]]. Using the [[patent val]], it tempers out [[729/728]], [[1575/1573]], [[2200/2197]], [[4096/4095]], and 21168/21125 in the 13-limit, so it also supports the 13-limit octant.
+== Theory ==
+edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]], but the approximation to higher harmonics are much improved. It is a [[zeta peak integer edo]], [[consistent]] to the [[11-odd-limit]] or the no-13 [[29-odd-limit]].
-It is a [[zeta peak integer edo]].
+In the 7-limit, the equal temperament tempers out [[2401/2400]], 2460375/2458624, and 30623756184/30517578125; in the 11-limit, [[9801/9800]], 46656/46585, 117649/117612, and 234375/234256, [[support]]ing the [[maviloid]] temperament, the [[Schismatic family #Bisesqui|bisesqui]] temperament, and the [[octant]] temperament. Using the [[patent val]], it tempers out [[729/728]], [[1575/1573]], [[2200/2197]], [[4096/4095]], and 21168/21125 in the 13-limit, so it also supports the 13-limit octant.
 === Prime harmonics ===
 {{Harmonics in equal|472}}
+=== Subsets and supersets ===
+Since 472 factors into {{factorization|472}}, 472edo has subset edos {{EDOs| 2, 4, 8, 59, 118, and 236 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 36: / Line 42: @@
 | 729/728, 1575/1573, 2200/2197, 2401/2400, 4096/4095
 | [{{val| 472 748 1096 1325 1633 1747 }}]
-| -0.0341
+| −0.0341
 | 0.1365
 | 5.37
@@ Line 45: / Line 51: @@
 {| 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 octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 66: / Line 73: @@
 | 1
 | 205\472
-| 498.31
+| 521.19
 | 875/648
 | [[Maviloid]]
@@ Line 77: / Line 84: @@
 |-
 | 8
-| 196\472<br>(19\472)
+| 196\472<br />(19\472)
-| 498.31<br>(48.31)
+| 498.31<br />(48.31)
-| 4/3<br>(36/35)
+| 4/3<br />(36/35)
 | [[Octant]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-[[Category:Equal divisions of the octave]]
-[[Category:Zeta]]