460edo: Difference between revisions

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~~The '''~~460 equal divisions of the octave''' ('''460edo'''), or the '''460(-tone) equal temperament''' ('''460tet''', '''460et''') when viewed from a [[regular temperament]] perspective, divides the octave into 460 equal parts of about 2.61 [[cent]]s each.

== Theory ==

460edo is a very strong 19-limit system and is ~~uniquely~~ [[consistent]] to the [[21-odd-limit]], with ~~harmonics~~ of 3 to 19 all tuned flat. It tempers out the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and 65536/65625 in the 7-limit, so that it [[support]]s [[pontiac]]. In the 11-limit it tempers of ~~43923/43904,~~ [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]]; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], 2431/2430, [[2601/2600]] and 4914/4913; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000~~. 460edo supports the 460 & [[1789edo|1789]] temperament in the 2.9.5.7.11.13 subgroup called [[commatose]]~~.

460edo is a very strong 19-limit system and is [[consistency|distinctly consistent]] to the [[21-odd-limit]], with [[harmonic]]s of 3 to 19 all tuned flat.

The equal temperament [[tempering out|tempers out]] the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and [[65536/65625]] in the 7-limit, so that it [[support]]s [[pontiac]], the 171 & 289 temperament. In the 11-limit it tempers of [[3025/3024]] and [[9801/9800]], and 43923/43904; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]], so that it supports [[deca]], the 190 & 270 temperament; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], [[2431/2430]], [[2601/2600]] and [[4914/4913]]; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000.

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 460 factors into ~~22 × 5 × 23~~, 460edo has subset edos 2, 4, 5, 10, 20, 23, 46, 92, 115, and 230.

Since 460 factors into {{factorization|460}}, 460edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 23, 46, 92, 115, and 230 }}.

== Regular temperament properties ==

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| 2.3

| {{monzo| -729 460 }}

| [{{~~val~~| 460 729 }}]

| {{mapping| 460 729 }}

| +0.0681

| 0.0681

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| 2.3.5

| 32805/32768, {{monzo| 6 68 -49 }}

| [{{~~val~~| 460 729 1068 }}]

| {{mapping| 460 729 1068 }}

| +0.0780

| 0.0573

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| 2.3.5.7

| 4375/4374, 32805/32768, {{monzo| -4 -2 -9 10 }}

| [{{~~val~~| 460 729 1068 1291 }}]

| {{mapping| 460 729 1068 1291 }}

| +0.1475

| 0.1303

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| 2.3.5.7.11

| 3025/3024, 4375/4374, 32805/32768, 184877/184320

| [{{~~val~~| 460 729 1068 1291 1591 }}]

| {{mapping| 460 729 1068 1291 1591 }}

| +0.1691

| 0.1243

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| 2.3.5.7.11.13

| 1001/1000, 3025/3024, 4225/4224, 4375/4374, 26411/26364

| [{{~~val~~| 460 729 1068 1291 1591 1702 }}]

| {{mapping| 460 729 1068 1291 1591 1702 }}

| +0.1647

| 0.1139

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| 2.3.5.7.11.13.17

| 833/832, 1001/1000, 1089/1088, 1225/1224, 1701/1700, 4225/4224

| [{{~~val~~| 460 729 1068 1291 1591 1702 1880 }}]

| {{mapping| 460 729 1068 1291 1591 1702 1880 }}

| +0.1624

| 0.1056

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| 2.3.5.7.11.13.17.19

| 833/832, 1001/1000, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1617/1615

| [{{~~val~~| 460 729 1068 1291 1591 1702 1880 1954 }}]

| {{mapping| 460 729 1068 1291 1591 1702 1880 1954 }}

| +0.1457

| 0.1082

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| 566.086 (26.086)

| 238/165 (?)

| [[Soviet ~~Ferris~~ wheel]]

| [[Soviet ferris wheel]]

|}

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''460 equal divisions of the octave''' ('''460edo'''), or the '''460(-tone) equal temperament''' ('''460tet''', '''460et''') when viewed from a [[regular temperament]] perspective, divides the octave into 460 equal parts of about 2.61 [[cent]]s each.
+{{EDO intro|460}}
 == Theory ==
-edo is a very strong 19-limit system and is uniquely [[consistent]] to the [[21-odd-limit]], with harmonics of 3 to 19 all tuned flat. It tempers out the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and 65536/65625 in the 7-limit, so that it [[support]]s [[pontiac]]. In the 11-limit it tempers of 43923/43904, [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]]; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], 2431/2430, [[2601/2600]] and 4914/4913; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000. 460edo supports the 460 & [[1789edo|1789]] temperament in the 2.9.5.7.11.13 subgroup called [[commatose]].
+edo is a very strong 19-limit system and is [[consistency|distinctly consistent]] to the [[21-odd-limit]], with [[harmonic]]s of 3 to 19 all tuned flat.
+The equal temperament [[tempering out|tempers out]] the [[schisma]], 32805/32768, in the 5-limit and [[4375/4374]] and [[65536/65625]] in the 7-limit, so that it [[support]]s [[pontiac]], the 171 & 289 temperament. In the 11-limit it tempers of [[3025/3024]] and [[9801/9800]], and 43923/43904; in the 13-limit [[1001/1000]], [[4225/4224]] and [[10648/10647]], so that it supports [[deca]], the 190 & 270 temperament; in the 17-limit [[833/832]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2058/2057]], [[2431/2430]], [[2601/2600]] and [[4914/4913]]; and in the 19-limit 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the [[optimal patent val]] for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000.
 === Prime harmonics ===
@@ Line 9: / Line 11: @@
 === Subsets and supersets ===
-Since 460 factors into 2<sup>2</sup> × 5 × 23, 460edo has subset edos 2, 4, 5, 10, 20, 23, 46, 92, 115, and 230.
+Since 460 factors into {{factorization|460}}, 460edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 23, 46, 92, 115, and 230 }}.
 == Regular temperament properties ==
@@ Line 24: / Line 26: @@
 | 2.3
 | {{monzo| -729 460 }}
-| [{{val| 460 729 }}]
+| {{mapping| 460 729 }}
 | +0.0681
 | 0.0681
@@ Line 31: / Line 33: @@
 | 2.3.5
 | 32805/32768, {{monzo| 6 68 -49 }}
-| [{{val| 460 729 1068 }}]
+| {{mapping| 460 729 1068 }}
 | +0.0780
 | 0.0573
@@ Line 38: / Line 40: @@
 | 2.3.5.7
 | 4375/4374, 32805/32768, {{monzo| -4 -2 -9 10 }}
-| [{{val| 460 729 1068 1291 }}]
+| {{mapping| 460 729 1068 1291 }}
 | +0.1475
 | 0.1303
@@ Line 45: / Line 47: @@
 | 2.3.5.7.11
 | 3025/3024, 4375/4374, 32805/32768, 184877/184320
-| [{{val| 460 729 1068 1291 1591 }}]
+| {{mapping| 460 729 1068 1291 1591 }}
 | +0.1691
 | 0.1243
@@ Line 52: / Line 54: @@
 | 2.3.5.7.11.13
 | 1001/1000, 3025/3024, 4225/4224, 4375/4374, 26411/26364
-| [{{val| 460 729 1068 1291 1591 1702 }}]
+| {{mapping| 460 729 1068 1291 1591 1702 }}
 | +0.1647
 | 0.1139
@@ Line 59: / Line 61: @@
 | 2.3.5.7.11.13.17
 | 833/832, 1001/1000, 1089/1088, 1225/1224, 1701/1700, 4225/4224
-| [{{val| 460 729 1068 1291 1591 1702 1880 }}]
+| {{mapping| 460 729 1068 1291 1591 1702 1880 }}
 | +0.1624
 | 0.1056
@@ Line 66: / Line 68: @@
 | 2.3.5.7.11.13.17.19
 | 833/832, 1001/1000, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1617/1615
-| [{{val| 460 729 1068 1291 1591 1702 1880 1954 }}]
+| {{mapping| 460 729 1068 1291 1591 1702 1880 1954 }}
 | +0.1457
 | 0.1082
@@ Line 76: / Line 78: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 115: / Line 117: @@
 | 566.086<br>(26.086)
 | 238/165<br>(?)
-| [[Soviet Ferris wheel]]
+| [[Soviet ferris wheel]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct