460edo: Difference between revisions

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Cleanup; clarify the title row of the rank-2 temp table; -"commatose" (requires 500+ gensteps), replaced with more notable temps
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{{Infobox ET}}
{{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 ==
== 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 ===
=== Prime harmonics ===
Line 9: Line 11:


=== Subsets and supersets ===
=== 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 ==
== Regular temperament properties ==
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| 2.3
| 2.3
| {{monzo| -729 460 }}
| {{monzo| -729 460 }}
| [{{val| 460 729 }}]
| {{mapping| 460 729 }}
| +0.0681
| +0.0681
| 0.0681
| 0.0681
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| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 6 68 -49 }}
| 32805/32768, {{monzo| 6 68 -49 }}
| [{{val| 460 729 1068 }}]
| {{mapping| 460 729 1068 }}
| +0.0780
| +0.0780
| 0.0573
| 0.0573
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| 2.3.5.7
| 2.3.5.7
| 4375/4374, 32805/32768, {{monzo| -4 -2 -9 10 }}
| 4375/4374, 32805/32768, {{monzo| -4 -2 -9 10 }}
| [{{val| 460 729 1068 1291 }}]
| {{mapping| 460 729 1068 1291 }}
| +0.1475
| +0.1475
| 0.1303
| 0.1303
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| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, 32805/32768, 184877/184320
| 3025/3024, 4375/4374, 32805/32768, 184877/184320
| [{{val| 460 729 1068 1291 1591 }}]
| {{mapping| 460 729 1068 1291 1591 }}
| +0.1691
| +0.1691
| 0.1243
| 0.1243
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 1001/1000, 3025/3024, 4225/4224, 4375/4374, 26411/26364
| 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
| +0.1647
| 0.1139
| 0.1139
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 833/832, 1001/1000, 1089/1088, 1225/1224, 1701/1700, 4225/4224
| 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
| +0.1624
| 0.1056
| 0.1056
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 833/832, 1001/1000, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1617/1615
| 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
| +0.1457
| 0.1082
| 0.1082
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| 566.086<br>(26.086)
| 566.086<br>(26.086)
| 238/165<br>(?)
| 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

Revision as of 07:01, 3 November 2023

← 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

Template:EDO intro

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.

The equal temperament 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

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 it is distinct

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