Xenharmonic Wiki

451edo: Difference between revisions

VisualWikitext
Francium (talk | contribs)
4,862 edits
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..."
 
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"
 
(6 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|451}}
{{ED intro}}


== 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]].
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 ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
451 factors into 11 × 41, with [[11edo]] and [[41edo]] as its subset edos.
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" |[[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|65 -41}}
! rowspan="2" | [[Comma list]]
|{{mapping|451 715}}
! rowspan="2" | [[Mapping]]
| -0.1527
! rowspan="2" | Optimal<br />8ve stretch (¢)
| 0.1527
! colspan="2" | Tuning error
| 5.74
|-
|-
|2.3.5
! [[TE error|Absolute]] (¢)
|{{monzo|3 -18 11}}, {{monzo|-59 5 22}}
! [[TE simple badness|Relative]] (%)
|{{mapping|451 715 1047}}
|-
| -0.0294
| 2.3.5
| 390625000/387420489, {{monzo| -59 5 22 }}
| {{mapping| 451 715 1047 }}
| &minus;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
Line 60: 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>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|29\451
| 17\451
|77.16
| 45.23
|256/245
| 250/243
|[[Tertiaseptal]]
| [[Quartonic]] (5-limit)
|-
| 1
| 29\451
| 77.16
| 256/245
| [[Tertiaseptal]]
|}
|}
<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


<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Quartonic]]
Retrieved from "https://en.xen.wiki/w/451edo"