460edo: Difference between revisions

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The ''460 equal division'' divides the octave into 460 equal parts of 2.609 cents each. It is a very strong 19-limit system and is uniquely [[consistent|consistent]] to the 21-limit. It tempers out the schisma, 32805/32768, in the 5-limit and 4375/4374 and 65536/65625 in the 7-limit, so that it supports [[Schismatic_family#Pontiac|pontiac temperament]]. 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|optimal patent val]] for various temperaments such as the rank five temperament tempering out 833/832 and 1001/1000.
{{Infobox ET}}
{{ED intro}}
 
== 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 ===
{{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" | [[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

Latest revision as of 14:59, 20 February 2025

← 459edo 460edo 461edo →
Prime factorization 22 × 5 × 23
Step size 2.6087 ¢ 
Fifth 269\460 (701.739 ¢)
Semitones (A1:m2) 43:35 (112.2 ¢ : 91.3 ¢)
Consistency limit 21
Distinct consistency limit 21

460 equal divisions of the octave (abbreviated 460edo or 460ed2), also called 460-tone equal temperament (460tet) or 460 equal temperament (460et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 460 equal parts of about 2.61 ¢ each. Each step represents a frequency ratio of 21/460, or the 460th root of 2.

Theory

460edo is a very strong 19-limit system and is distinctly 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 supports 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

Approximation of prime harmonics in 460edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.22 -0.23 -1.00 -0.88 -0.53 -0.61 -0.12 +0.42 +0.86 +0.18
Relative (%) +0.0 -8.3 -8.7 -38.3 -33.9 -20.2 -23.3 -4.7 +16.2 +32.9 +7.0
Steps
(reduced) 		460
(0) 		729
(269) 		1068
(148) 		1291
(371) 		1591
(211) 		1702
(322) 		1880
(40) 		1954
(114) 		2081
(241) 		2235
(395) 		2279
(439)

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.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-729 460⟩ [⟨460 729]] +0.0681 0.0681 2.61
2.3.5 32805/32768, [6 68 -49⟩ [⟨460 729 1068]] +0.0780 0.0573 2.20
2.3.5.7 4375/4374, 32805/32768, [-4 -2 -9 10⟩ [⟨460 729 1068 1291]] +0.1475 0.1303 4.99
2.3.5.7.11 3025/3024, 4375/4374, 32805/32768, 184877/184320 [⟨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 [⟨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 [⟨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 [⟨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 422 in either subgroup, and is bettered by 494 in the 17-limit, and 525 in the 19-limit.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(17\460) 		315.65
(44.35) 		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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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