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 ===
Line 9: Line 9:


=== 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
Line 67: Line 61:
=== 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

Latest revision as of 06:26, 21 February 2025

← 469edo 470edo 471edo →
Prime factorization 2 × 5 × 47
Step size 2.55319 ¢ 
Fifth 275\470 (702.128 ¢) (→ 55\94)
Semitones (A1:m2) 45:35 (114.9 ¢ : 89.36 ¢)
Consistency limit 5
Distinct consistency limit 5

470 equal divisions of the octave (abbreviated 470edo or 470ed2), also called 470-tone equal temperament (470tet) or 470 equal temperament (470et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 470 equal parts of about 2.55 ¢ each. Each step represents a frequency ratio of 21/470, or the 470th root of 2.

Theory

470edo shares the fifth with 94edo. Unlike 94edo, however, 470edo is only consistent to the 5-odd-limit. Using the patent val, the equal temperament 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 supports uniwiz and decimetra.

Prime harmonics

Approximation of prime harmonics in 470edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.17 -0.78 -1.17 +0.17 -0.53 -0.27 +1.21 -0.19 -0.64 -1.21
Relative (%) +0.0 +6.8 -30.6 -45.7 +6.7 -20.7 -10.8 +47.4 -7.4 -25.1 -47.2
Steps
(reduced) 		470
(0) 		745
(275) 		1091
(151) 		1319
(379) 		1626
(216) 		1739
(329) 		1921
(41) 		1997
(117) 		2126
(246) 		2283
(403) 		2328
(448)

Subsets and supersets

Since 470 factors into 2 × 5 × 47, 470edo has subset edos 2, 5, 10, 47, 94, and 235.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 1600000/1594323, [-77 -10 40⟩ [⟨470 745 1091]] +0.0759 0.1897 7.43
2.3.5.7 703125/702464, 1500625/1492992, 1600000/1594323 [⟨470 745 1091 1319]] +0.1608 0.2205 8.64
2.3.5.7.11 3025/3024, 4000/3993, 19712/19683, 117649/117128 [⟨470 745 1091 1319 1626]] +0.1187 0.2144 8.40
2.3.5.7.11.13 625/624, 1575/1573, 2080/2079, 13720/13689, 15379/15360 [⟨470 745 1091 1319 1626 1739]] +0.1227 0.1959 7.67
2.3.5.7.11.13.17 595/594, 625/624, 833/832, 1575/1573, 3185/3179, 8624/8619 [⟨470 745 1091 1319 1626 1739 1921]] +0.1148 0.1824 7.14

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 133\470 339.57 243/200 Amity (5-limit)

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

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