460edo: Difference between revisions

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== Theory ==

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.

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.

~~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.

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 ===

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== Regular temperament properties ==

{~~{comma basis begin}}~~

{| 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

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| 0.1082

| 4.15

~~{{comma basis end}~~}

|}

* 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 ===

{{rank-2 ~~begin}}~~

{| 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

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| 238/165<br />(?)

| [[Soviet ferris wheel]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|460}}
+{{ED intro}}
 == Theory ==
-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.
+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.
-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.
+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 ===
@@ Line 14: / Line 14: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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
@@ Line 64: / Line 73: @@
 | 0.1082
 | 4.15
-{{comma basis end}}
+|}
 * 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 ===
-{{rank-2 begin}}
+{| 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
@@ Line 105: / Line 121: @@
 | 238/165<br />(?)
 | [[Soviet ferris wheel]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct