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The '''460 equal divisions of the octave''' divides the octave into 460 equal parts of 2.609 cents each.
{{Infobox ET}}
{{ED intro}}


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 temperament|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 ===
=== Prime harmonics ===
{{Harmonics in equal|460}}
{{Harmonics in equal|460}}


[[Category:Equal divisions of the octave]]
=== 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" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -729 460 }}
| {{mapping| 460 729 }}
| +0.0681
| 0.0681
| 2.61
|-
| 2.3.5
| 32805/32768, {{monzo| 6 68 -49 }}
| {{mapping| 460 729 1068 }}
| +0.0780
| 0.0573
| 2.20
|-
| 2.3.5.7
| 4375/4374, 32805/32768, {{monzo| -4 -2 -9 10 }}
| {{mapping| 460 729 1068 1291 }}
| +0.1475
| 0.1303
| 4.99
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 32805/32768, 184877/184320
| {{mapping| 460 729 1068 1291 1591 }}
| +0.1691
| 0.1243
| 4.76
|-
| 2.3.5.7.11.13
| 1001/1000, 3025/3024, 4225/4224, 4375/4374, 26411/26364
| {{mapping| 460 729 1068 1291 1591 1702 }}
| +0.1647
| 0.1139
| 4.36
|-
| 2.3.5.7.11.13.17
| 833/832, 1001/1000, 1089/1088, 1225/1224, 1701/1700, 4225/4224
| {{mapping| 460 729 1068 1291 1591 1702 1880 }}
| +0.1624
| 0.1056
| 4.05
|-
| 2.3.5.7.11.13.17.19
| 833/832, 1001/1000, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1617/1615
| {{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"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 9\460
| 23.48
| 531441/524288
| [[Commatose]]
|-
| 1
| 121\460
| 315.65
| 6/5
| [[Egads]]
|-
| 1
| 191\460
| 498.26
| 4/3
| [[Pontiac]]
|-
| 10
| 121\460<br />(17\460)
| 315.65<br />(44.35)
| 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
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