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470edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|470}}
{{ED intro}}


== Theory ==
== Theory ==
470et is enfactored in the [[3-limit]] and only consistent to the [[5-odd-limit]]. Using the patent val, it tempers out [[703125/702464]], 359661568/358722675, 5250987/5242880 and 200120949/200000000 in the 7-limit; 161280/161051, 820125/819896, 29296875/29218112, [[4000/3993]], 22478848/22460625, 759375/758912, [[6250/6237]], [[200704/200475]], 352947/352000, [[19712/19683]], [[3025/3024]], 16808715/16777216, [[532400/531441]] and [[1771561/1771470]] in the 11-limit. It [[support]]s [[uniwiz]] and [[decimetra]].
470edo shares the [[perfect fifth|fifth]] with [[94edo]]. Unlike 94edo, however, 470edo is only [[consistent]] to the [[5-odd-limit]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[703125/702464]], 823543/820125, and 1500625/1492992 in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], [[19712/19683]], and 117649/117128 in the 11-limit. It [[support]]s [[uniwiz]] and [[decimetra]].


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
470 factors into 2 × 5 × 47, with subset edos {{EDOs|2, 5, 10, 47, 94, and 235}}.
Since 470 factors into {{factorisation|470}}, 470edo has subset edos {{EDOs| 2, 5, 10, 47, 94, and 235 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|149 -94}}
! rowspan="2" | [[Comma list]]
|{{mapping|470 745}}
! rowspan="2" | [[Mapping]]
| -0.0545
! rowspan="2" | Optimal<br />8ve stretch (¢)
| 0.0545
! colspan="2" | Tuning error
| 2.13
|-
|-
|2.3.5
! [[TE error|Absolute]] (¢)
|1600000/1594323, {{monzo|-77 -10 40}}
! [[TE simple badness|Relative]] (%)
|{{mapping|470 745 1091}}
|-
| 2.3.5
| 1600000/1594323, {{monzo|-77 -10 40}}
| {{mapping| 470 745 1091 }}
| +0.0759
| +0.0759
| 0.1897
| 0.1897
| 7.43
| 7.43
|-
|-
|2.3.5.7
| 2.3.5.7
|1500625/1492992, 703125/702464, 1600000/1594323
| 703125/702464, 1500625/1492992, 1600000/1594323
|{{mapping|470 745 1091 1319}}
| {{mapping| 470 745 1091 1319 }}
| +0.1608
| +0.1608
| 0.2205
| 0.2205
| 8.64
| 8.64
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 6250/6237, 420175/418176, 759375/758912
| 3025/3024, 4000/3993, 19712/19683, 117649/117128
|{{mapping|470 745 1091 1319 1626}}
| {{mapping| 470 745 1091 1319 1626 }}
| +0.1187
| +0.1187
| 0.2144
| 0.2144
| 8.40
| 8.40
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|1575/1573, 2080/2079, 625/624, 13720/13689, 218491/217800
| 625/624, 1575/1573, 2080/2079, 13720/13689, 15379/15360
|{{mapping|470 745 1091 1319 1626 1739}}
| {{mapping| 470 745 1091 1319 1626 1739 }}
| +0.1227
| +0.1227
| 0.1959
| 0.1959
| 7.67
| 7.67
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|595/594, 1575/1573, 2080/2079, 12376/12375, 3185/3179, 8624/8619
| 595/594, 625/624, 833/832, 1575/1573, 3185/3179, 8624/8619
|{{mapping|470 745 1091 1319 1626 1739 1921}}
| {{mapping| 470 745 1091 1319 1626 1739 1921 }}
| +0.1148
| +0.1148
| 0.1824
| 0.1824
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| 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 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|133\470
| 133\470
|339.57
| 339.57
|243/200
| 243/200
|[[Amity]]
| [[Amity]] (5-limit)
|}
|}
 
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
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