460edo: Difference between revisions

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~~The '''460 equal divisions of the octave''' divides the octave into 460 equal parts of 2.609 cents each.~~

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 ~~five~~ temperament tempering out 833/832 and 1001/1000.

== Theory ==

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

It [[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 {{nowrap|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 {{nowrap|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 ===

=== Subsets and supersets ===

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 ==

{| 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

|-

Line 19:

Line 27:

| 2.3

| {{monzo| -729 460 }}

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

| {{mapping| 460 729 }}

| +0.0681

| 0.0681

Line 26:

Line 34:

| 2.3.5

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

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

| {{mapping| 460 729 1068 }}

| +0.0780

| 0.0573

Line 33:

Line 41:

| 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 40:

Line 48:

| 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 47:

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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

Line 54:

Line 62:

| 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

| 4.15

|}

* 460et has lower absolute errors in the 17- and 19-limit than any previous equal temperaments. It beats [[422edo|422]] in either subgroup, and is bettered by [[494edo|494]] in the 17-limit, and [[525edo|525]] in the 19-limit.

=== 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 ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 9\460

| 23.48

| 531441/524288

| [[Commatose]]

|-

| 1

Line 86:

Line 102:

| 498.26

| 4/3

| [[~~Helmholtz]] / [[pontiac~~]]

| [[Pontiac]]

|-

| 10

| 121\460 (17\460)

| 121\460 (17\460)

| 315.65 (44.35)

| 315.65 (44.35)

| 6/5 (40/39)

| 6/5 (40/39)

| [[Deca]]

|-

| 20

| 66\460 (20\460)

| 172.173 (52.173)

| 169/153 (?)

| [[Calcium]]

|-

| 20

| 217\460 (10\460)

| 566.086 (26.086)

| 238/165 (?)

| [[Soviet ferris wheel]]

|}

<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]]

@@ Line 1: / Line 1: @@
-The '''460 equal divisions of the octave''' divides the octave into 460 equal parts of 2.609 cents each.
+{{Infobox ET}}
+{{ED intro}}
-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 five temperament tempering out 833/832 and 1001/1000.
+== Theory ==
+edo is a very strong 19-limit system and is [[distinctly consistent]] to the [[21-odd-limit]], with [[harmonic]]s of 3 to 19 all tuned flat.
+It [[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 {{nowrap|171 &amp; 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 {{nowrap|190 &amp; 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 ===
 {{Harmonics in equal|460}}
+=== Subsets and supersets ===
+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 ==
 {| 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 19: / Line 27: @@
 | 2.3
 | {{monzo| -729 460 }}
-| [{{val| 460 729 }}]
+| {{mapping| 460 729 }}
 | +0.0681
 | 0.0681
@@ Line 26: / Line 34: @@
 | 2.3.5
 | 32805/32768, {{monzo| 6 68 -49 }}
-| [{{val| 460 729 1068 }}]
+| {{mapping| 460 729 1068 }}
 | +0.0780
 | 0.0573
@@ Line 33: / Line 41: @@
 | 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 40: / Line 48: @@
 | 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 47: / Line 55: @@
 | 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 54: / Line 62: @@
 | 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 61: / Line 69: @@
 | 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
 | 4.15
 |}
+* 460et has lower absolute errors in the 17- and 19-limit than any previous equal temperaments. It beats [[422edo|422]] in either subgroup, and is bettered by [[494edo|494]] in the 17-limit, and [[525edo|525]] in the 19-limit.
 === 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 octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
+|-
+| 1
+| 9\460
+| 23.48
+| 531441/524288
+| [[Commatose]]
 |-
 | 1
@@ Line 86: / Line 102: @@
 | 498.26
 | 4/3
-| [[Helmholtz]] / [[pontiac]]
+| [[Pontiac]]
 |-
 | 10
-| 121\460<br>(17\460)
+| 121\460<br />(17\460)
-| 315.65<br>(44.35)
+| 315.65<br />(44.35)
-| 6/5<br>(40/39)
+| 6/5<br />(40/39)
 | [[Deca]]
+|-
+| 20
+| 66\460<br />(20\460)
+| 172.173<br />(52.173)
+| 169/153<br />(?)
+| [[Calcium]]
+|-
+| 20
+| 217\460<br />(10\460)
+| 566.086<br />(26.086)
+| 238/165<br />(?)
+| [[Soviet ferris wheel]]
 |}
+<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]]