457edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|457}}
{{ED intro}}


== Theory ==
== Theory ==
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.
457edo is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]] is quite large. As an equal temperament, it [[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 ===
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=== 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]]
! 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]] (%)
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| {{monzo| -724 457 }}
| {{monzo| -724 457 }}
| {{mapping| 457 724 }}
| {{mapping| 457 724 }}
| 0.2716
| +0.2716
| 0.2716
| 0.2716
| 10.34
| 10.34
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| {{monzo| -36 11 8 }}, {{monzo| -5 31 -19 }}
| {{monzo| -36 11 8 }}, {{monzo| -5 31 -19 }}
| {{mapping| 457 724 1061 }}
| {{mapping| 457 724 1061 }}
| 0.2267
| +0.2267
| 0.2307
| 0.2307
| 8.79
| 8.79
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| 19683/19600, 65625/65536, 7381125/7340032
| 19683/19600, 65625/65536, 7381125/7340032
| {{mapping| 457 724 1061 1283 }}
| {{mapping| 457 724 1061 1283 }}
| 0.1609
| +0.1609
| 0.2300
| 0.2300
| 8.76
| 8.76
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| 540/539, 8019/8000, 19683/19600, 43923/43904
| 540/539, 8019/8000, 19683/19600, 43923/43904
| {{mapping| 457 724 1061 1283 1581 }}
| {{mapping| 457 724 1061 1283 1581 }}
| 0.1227
| +0.1227
| 0.2194
| 0.2194
| 8.36
| 8.36
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| 540/539, 1716/1715, 4225/4224, 41067/40960, 43940/43923
| 540/539, 1716/1715, 4225/4224, 41067/40960, 43940/43923
| {{mapping| 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
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| 936/935, 1089/1088, 1275/1274, 1575/1573, 2601/2600, 4225/4224
| 936/935, 1089/1088, 1275/1274, 1575/1573, 2601/2600, 4225/4224
| {{mapping| 457 724 1061 1283 1581 1691 1868 }}
| {{mapping| 457 724 1061 1283 1581 1691 1868 }}
| 0.0952
| +0.0952
| 0.1920
| 0.1920
| 7.31
| 7.31
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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
|-
|-
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| [[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
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Music ==
== Music ==
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* "Less and Less" from ''Take Advantage'' (2024) – [https://open.spotify.com/track/5w3K1Q6VhCkUPMXmuh5pVt Spotify] | [https://francium223.bandcamp.com/track/less-and-less Bandcamp] | [https://www.youtube.com/watch?v=y4MGhHE20b4 YouTube]
* "Less and Less" from ''Take Advantage'' (2024) – [https://open.spotify.com/track/5w3K1Q6VhCkUPMXmuh5pVt Spotify] | [https://francium223.bandcamp.com/track/less-and-less Bandcamp] | [https://www.youtube.com/watch?v=y4MGhHE20b4 YouTube]
* "Porcelain Stoneware" from ''Scoop'' (2024) – [https://open.spotify.com/track/055lRdMBKr37MU8HQqlvus Spotify] | [https://francium223.bandcamp.com/track/porcelain-stoneware Bandcamp] | [https://www.youtube.com/watch?v=bzycN6PYMQQ YouTube]
* "Porcelain Stoneware" from ''Scoop'' (2024) – [https://open.spotify.com/track/055lRdMBKr37MU8HQqlvus Spotify] | [https://francium223.bandcamp.com/track/porcelain-stoneware Bandcamp] | [https://www.youtube.com/watch?v=bzycN6PYMQQ YouTube]
[[Category:Listen]]

Latest revision as of 06:09, 21 February 2025

← 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

457 equal divisions of the octave (abbreviated 457edo or 457ed2), also called 457-tone equal temperament (457tet) or 457 equal temperament (457et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 457 equal parts of about 2.63 ¢ each. Each step represents a frequency ratio of 21/457, or the 457th root of 2.

Theory

457edo is consistent to the 7-odd-limit, but the error of harmonic 3 is quite large. As an equal temperament, it 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 distinct

Music

Francium
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