451edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|451}} == Theory == 451et is consistent to the 7-odd-limit. Using the patent val, it tempers out 52734375/52706752, 703125/702464, 65625/6..."
 
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== Theory ==
== Theory ==
451et is consistent to the [[7-odd-limit]]. Using the patent val, it tempers out 52734375/52706752, [[703125/702464]], [[65625/65536]], [[2401/2400]], [[2100875/2097152]] and 201768035/201326592 in the 7-limit; [[117440512/117406179]], 35156250/35153041, 234375/234256, [[536870912/535869675]], 104857600/104825259, [[131072/130977]], [[6250/6237]], 200704/200475, 42875/42768, 1879453125/1879048192, 1362944/1361367, 42592/42525, 166375/165888, 456533/455625, 3294225/3294172, 43923/43904 and 102487/102400 in the 11-limit. It [[support]]s [[hemermacomp]], [[quartonic]] and [[tertiseptisix]].
451 = 11 × 41, and 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]] [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], and 390625000/387420489 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]].


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
451 factors into 11 × 41, with [[11edo]] and [[41edo]] as its subset edos.
Since 451 factors into 11 × 41, 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|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
| 2.3.5
|{{monzo|65 -41}}
| {{monzo| 3 -18 11 }}, {{monzo| -59 5 22 }}
|{{mapping|451 715}}
| {{mapping| 451 715 1047 }}
| -0.1527
| 0.1527
| 5.74
|-
|2.3.5
|{{monzo|3 -18 11}}, {{monzo|-59 5 22}}
|{{mapping|451 715 1047}}
| -0.0294
| -0.0294
| 0.2144
| 0.2144
| 8.06
| 8.06
|-
|-
|2.3.5.7
| 2.3.5.7
|2401/2400, 703125/702464, 390625000/387420489
| 2401/2400, 65625/65536, 390625000/387420489
|{{mapping|451 715 1047 1266}}
| {{mapping| 451 715 1047 126 6}}
| +0.0057
| +0.0057
| 0.1953
| 0.1953
| 7.34
| 7.34
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|2401/2400, 6250/6237, 42592/42525, 43923/43904
| 2401/2400, 6250/6237, 42592/42525, 43923/43904
|{{mapping|451 715 1047 1266 1560}}
| {{mapping| 451 715 1047 1266 1560 }}
| +0.0359
| +0.0359
| 0.1849
| 0.1849
| 6.95
| 6.95
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|2080/2079, 625/624, 4459/4455, 2200/2197, 20449/20412
| 625/624, 2080/2079, 2200/2197, 2401/2400, 17303/17280
|{{mapping|451 715 1047 1266 1560 1669}}
| {{mapping| 451 715 1047 1266 1560 1669 }}
| +0.0177
| +0.0177
| 0.1736
| 0.1736
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! Temperaments
! Temperaments
|-
|-
|1
| 1
|29\451
| 29\451
|77.16
| 77.16
|256/245
| 256/245
|[[Tertiaseptal]]
| [[Tertiaseptal]]
|}
|}
<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>*</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 10:39, 21 January 2024

← 450edo 451edo 452edo →
Prime factorization 11 × 41
Step size 2.66075 ¢ 
Fifth 264\451 (702.439 ¢) (→ 24\41)
Semitones (A1:m2) 44:33 (117.1 ¢ : 87.8 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

451 = 11 × 41, and 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 2401/2400, 65625/65536, 703125/702464, 2100875/2097152, and 390625000/387420489 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.

Prime harmonics

Approximation of prime harmonics in 451edo
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 [3 -18 11⟩, [-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

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 29\451 77.16 256/245 Tertiaseptal

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

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