457edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|457}} == Theory == 457et tempers out 283115520/282475249, 1220703125/1219784832, 26873856/26796875, 65625/65536 and 200120949/200000000 in the 7..."
 
Cleanup; clarify the title row of the rank-2 temp table
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|457}}
{{EDO intro|457}}
== Theory ==
== Theory ==
457et tempers out 283115520/282475249, 1220703125/1219784832, 26873856/26796875, [[65625/65536]] and 200120949/200000000 in the 7-limit; 95703125/95664294, 100663296/100656875, 161280/161051, 29296875/29218112, 166698/166375, 1953125/1951488, 151263/151250, 2359296/2358125, [[540/539]], 5767168/5764801, 825000/823543, [[8019/8000]], 160083/160000, 16808715/16777216, 539055/537824, 244515348/244140625, 67110351/67108864 and 43923/43904 in the 11-limit.
457edo is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]] is quite large. The equal temperament [[tempering out|tempers out]] [[19683/19600]] and [[65625/65536]] in the 7-limit; [[540/539]], [[8019/8000]], and 43923/43904 in the 11-limit.
===Odd harmonics===
 
=== Odd harmonics ===
{{Harmonics in equal|457}}
{{Harmonics in equal|457}}
===Subsets and supersets===
 
=== Subsets and supersets ===
457edo is the 88th [[prime edo]].  
457edo is the 88th [[prime edo]].  
==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" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|-724 457}}
| {{monzo| -724 457 }}
|{{val|457 724}}
| {{mapping| 457 724 }}
| 0.2716
| 0.2716
| 0.2716
| 0.2716
| 10.34
| 10.34
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-36 11 8}}, {{monzo|-5 31 -19}}
| {{monzo| -36 11 8 }}, {{monzo| -5 31 -19 }}
|{{val|457 724 1061}}
| {{mapping| 457 724 1061 }}
| 0.2267
| 0.2267
| 0.2307
| 0.2307
| 8.79
| 8.79
|-
|-
|2.3.5.7
| 2.3.5.7
|19683/19600, 65625/65536, 7381125/7340032
| 19683/19600, 65625/65536, 7381125/7340032
|{{val|457 724 1061 1283}}
| {{mapping| 457 724 1061 1283 }}
| 0.1609
| 0.1609
| 0.2300
| 0.2300
| 8.76
| 8.76
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|540/539, 8019/8000, 19683/19600, 43923/43904
| 540/539, 8019/8000, 19683/19600, 43923/43904
|{{val|457 724 1061 1283 1581}}
| {{mapping| 457 724 1061 1283 1581 }}
| 0.1227
| 0.1227
| 0.2194
| 0.2194
| 8.36
| 8.36
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|540/539, 1716/1715, 4225/4224, 41067/40960, 43940/43923
| 540/539, 1716/1715, 4225/4224, 41067/40960, 43940/43923
|{{val|457 724 1061 1283 1581 1691}}
| {{mapping| 457 724 1061 1283 1581 1691 }}
| 0.1142
| 0.1142
| 0.2012
| 0.2012
| 7.66
| 7.66
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|936/935, 1089/1088, 1275/1274, 1575/1573, 2601/2600, 4225/4224
| 936/935, 1089/1088, 1275/1274, 1575/1573, 2601/2600, 4225/4224
|{{val|457 724 1061 1283 1581 1691 1868}}
| {{mapping| 457 724 1061 1283 1581 1691 1868 }}
| 0.0952
| 0.0952
| 0.1920
| 0.1920
Line 65: Line 69:
|+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
|-
|-
|1
| 1
|10\457
| 10\457
|26.258
| 26.258
|49/48
| 49/48
|[[Sfourth]]
| [[Sfourth]]
|-
|-
|1
| 1
|136\457
| 136\457
|357.11
| 357.11
|49/40
| 49/40
|[[Dodifo]]
| [[Dodifo]]
|-
|-
|1
| 1
|213\457
| 213\457
|559.30
| 559.30
|864/625
| 864/625
|[[Tritriple]]
| [[Tritriple]]
|}
|}
<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 07:12, 3 November 2023

← 456edo 457edo 458edo →
Prime factorization 457 (prime)
Step size 2.62582 ¢ 
Fifth 267\457 (701.094 ¢)
Semitones (A1:m2) 41:36 (107.7 ¢ : 94.53 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

457edo is consistent to the 7-odd-limit, but the error of harmonic 3 is quite large. The equal temperament tempers out 19683/19600 and 65625/65536 in the 7-limit; 540/539, 8019/8000, and 43923/43904 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 457edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.86 -0.32 +0.10 +0.90 +0.10 -0.27 -1.18 +0.08 -0.80 -0.76 -0.70
Relative (%) -32.8 -12.1 +3.9 +34.4 +4.0 -10.1 -44.9 +2.9 -30.3 -28.9 -26.8
Steps
(reduced) 		724
(267) 		1061
(147) 		1283
(369) 		1449
(78) 		1581
(210) 		1691
(320) 		1785
(414) 		1868
(40) 		1941
(113) 		2007
(179) 		2067
(239)

Subsets and supersets

457edo is the 88th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-724 457⟩ [⟨457 724]] 0.2716 0.2716 10.34
2.3.5 [-36 11 8⟩, [-5 31 -19⟩ [⟨457 724 1061]] 0.2267 0.2307 8.79
2.3.5.7 19683/19600, 65625/65536, 7381125/7340032 [⟨457 724 1061 1283]] 0.1609 0.2300 8.76
2.3.5.7.11 540/539, 8019/8000, 19683/19600, 43923/43904 [⟨457 724 1061 1283 1581]] 0.1227 0.2194 8.36
2.3.5.7.11.13 540/539, 1716/1715, 4225/4224, 41067/40960, 43940/43923 [⟨457 724 1061 1283 1581 1691]] 0.1142 0.2012 7.66
2.3.5.7.11.13.17 936/935, 1089/1088, 1275/1274, 1575/1573, 2601/2600, 4225/4224 [⟨457 724 1061 1283 1581 1691 1868]] 0.0952 0.1920 7.31

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 10\457 26.258 49/48 Sfourth
1 136\457 357.11 49/40 Dodifo
1 213\457 559.30 864/625 Tritriple

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

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