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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
451edo shares its [[3/2|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out|tempers out]] 390625000/387420489 ([[quartonic comma]]) in the 5-limit; [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]], providing the [[optimal patent val]] for 5-limit [[quartonic]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 451 factors into | Since 451 factors into {{factorisation|451}}, 451edo has [[11edo]] and [[41edo]] as its subsets. | ||
== 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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| | | 390625000/387420489, {{monzo| -59 5 22 }} | ||
| {{mapping| 451 715 1047 }} | | {{mapping| 451 715 1047 }} | ||
| | | −0.0294 | ||
| 0.2144 | | 0.2144 | ||
| 8.06 | | 8.06 | ||
| Line 53: | Line 54: | ||
=== 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> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |||
| 1 | |||
| 17\451 | |||
| 45.23 | |||
| 250/243 | |||
| [[Quartonic]] (5-limit) | |||
|- | |- | ||
| 1 | | 1 | ||
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| [[Tertiaseptal]] | | [[Tertiaseptal]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
[[Category:Quartonic]] | |||
Latest revision as of 13:32, 13 March 2026
| ← 450edo | 451edo | 452edo → |
451 equal divisions of the octave (abbreviated 451edo or 451ed2), also called 451-tone equal temperament (451tet) or 451 equal temperament (451et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 451 equal parts of about 2.66 ¢ each. Each step represents a frequency ratio of 21/451, or the 451st root of 2.
Theory
451edo shares its fifth with 41edo. Unlike 41, however, 451 is only consistent to the 7-odd-limit, though it has a reasonable approximation up to the 13-limit using the patent val. The equal temperament tempers out 390625000/387420489 (quartonic comma) in the 5-limit; 2401/2400, 65625/65536, 703125/702464, 2100875/2097152, in the 7-limit; 6250/6237, 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and 625/624, 2080/2079, 2200/2197, 4096/4095, 4225/4224, 4459/4455, and 17303/17280 in the 13-limit. It supports tertiaseptal, tertiseptisix, and hemermacomp, providing the optimal patent val for 5-limit quartonic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.48 | -0.50 | -0.31 | -0.54 | +0.27 | -1.19 | +0.49 | -0.34 | +0.13 | -0.91 |
| Relative (%) | +0.0 | +18.2 | -19.0 | -11.7 | -20.4 | +10.2 | -44.6 | +18.5 | -12.6 | +5.1 | -34.3 | |
| Steps (reduced) |
451 (0) |
715 (264) |
1047 (145) |
1266 (364) |
1560 (207) |
1669 (316) |
1843 (39) |
1916 (112) |
2040 (236) |
2191 (387) |
2234 (430) | |
Subsets and supersets
Since 451 factors into 11 × 41, 451edo has 11edo and 41edo as its subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 390625000/387420489, [-59 5 22⟩ | [⟨451 715 1047]] | −0.0294 | 0.2144 | 8.06 |
| 2.3.5.7 | 2401/2400, 65625/65536, 390625000/387420489 | [⟨451 715 1047 126 6]] | +0.0057 | 0.1953 | 7.34 |
| 2.3.5.7.11 | 2401/2400, 6250/6237, 42592/42525, 43923/43904 | [⟨451 715 1047 1266 1560]] | +0.0359 | 0.1849 | 6.95 |
| 2.3.5.7.11.13 | 625/624, 2080/2079, 2200/2197, 2401/2400, 17303/17280 | [⟨451 715 1047 1266 1560 1669]] | +0.0177 | 0.1736 | 6.52 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 17\451 | 45.23 | 250/243 | Quartonic (5-limit) |
| 1 | 29\451 | 77.16 | 256/245 | Tertiaseptal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct