450edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,231 edits
m Partial undo
m →Regular temperament properties: quintile isn't defined beyond the 5-limit
 
(5 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|450}}
{{ED intro}}


== Theory ==
== Theory ==
450edo is [[consistent]] to the [[7-odd-limit]]. It can be considered for the 2.3.5.7.13.17.29.31.37 [[subgroup]], where it [[tempering out|tempers out]] 651/650, 1666/1665, 1887/1885, 2016/2015, 2295/2294, 5916/5915, 4901/4900 and 14229/14210. It [[support]]s [[decal]] and [[varuna]].
450edo is [[consistent]] to the [[7-odd-limit]]. It tempers out the [[triwellisma]] and the [[varunisma]] in the [[7-limit]], [[support]]ing and providing the [[optimal patent val]] for 7-limit [[decile]]. It can be considered for the 2.3.5.7.13.17.29.31.37 [[subgroup]], where it [[tempering out|tempers out]] 651/650, 1666/1665, 1887/1885, 2016/2015, 2295/2294, 5916/5915, 4901/4900 and 14229/14210.  


=== Odd harmonics ===
=== Odd harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
450 factors into 2 × 3<sup>2</sup> × 5<sup>2</sup>, with subset edos {{EDOs|2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, and 225}}.
Since 450 factors into {{nowrap| 2 × 3<sup>2</sup> × 5<sup>2</sup> }}, 450edo has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, and 225 }}.


== Regular temperament properties ==
== Regular temperament properties ==
Line 17: Line 17:
! rowspan="2" | [[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
|-
|-
Line 24: Line 24:
|-
|-
| 2.3
| 2.3
| {{monzo| -713 450 }}
| {{Monzo| -713 450 }}
| {{mapping| 450 713 }}
| {{Mapping| 450 713 }}
| +0.1961
| +0.1961
| 0.1961
| 0.1961
Line 31: Line 31:
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| -28 25 -5 }}, {{monzo| 25 15 -21 }}
| {{Monzo| -28 25 -5 }}, {{monzo| 25 15 -21 }}
| {{mapping| 450 713 1045 }}
| {{Mapping| 450 713 1045 }}
| +0.0800
| +0.0800
| 0.2293
| 0.2293
Line 39: Line 39:
| 2.3.5.7
| 2.3.5.7
| 235298/234375, 321489/320000, 26873856/26796875
| 235298/234375, 321489/320000, 26873856/26796875
| {{mapping| 450 713 1045 1263 }}
| {{Mapping| 450 713 1045 1263 }}
| +0.1336
| +0.1336
| 0.2192
| 0.2192
Line 49: Line 49:
|+ style="font-size: 105%;" | 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
|-
|-
Line 62: Line 62:
|-
|-
| 5
| 5
| 187\450<br />(7\450)
| 7\450
| 498.67<br />(18.67)
| 18.67
| 4/3<br />(81/80)
| 81/80
| [[Pental (temperament)|Pental]] (5-limit)
| [[Quintile]]
|-
| 10
| 7\450
| 18.67
| 81/80
| [[Decile]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
 
[[Category:Decile]]

Latest revision as of 06:16, 4 September 2026

← 449edo 450edo 451edo →
Prime factorization 2 × 32 × 52
Step size 2.66667 ¢ 
Fifth 263\450 (701.333 ¢)
Semitones (A1:m2) 41:35 (109.3 ¢ : 93.33 ¢)
Consistency limit 7
Distinct consistency limit 7

450 equal divisions of the octave (abbreviated 450edo or 450ed2), also called 450-tone equal temperament (450tet) or 450 equal temperament (450et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 450 equal parts of about 2.67 ¢ each. Each step represents a frequency ratio of 21/450, or the 450th root of 2.

Theory

450edo is consistent to the 7-odd-limit. It tempers out the triwellisma and the varunisma in the 7-limit, supporting and providing the optimal patent val for 7-limit decile. It can be considered for the 2.3.5.7.13.17.29.31.37 subgroup, where it tempers out 651/650, 1666/1665, 1887/1885, 2016/2015, 2295/2294, 5916/5915, 4901/4900 and 14229/14210.

Odd harmonics

Approximation of odd harmonics in 450edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.62 +0.35 -0.83 -1.24 +0.68 -0.53 -0.27 -0.96 +1.15 +1.22 +1.06
Relative (%) -23.3 +13.2 -31.0 -46.6 +25.6 -19.8 -10.1 -35.8 +43.3 +45.7 +39.7
Steps
(reduced) 		713
(263) 		1045
(145) 		1263
(363) 		1426
(76) 		1557
(207) 		1665
(315) 		1758
(408) 		1839
(39) 		1912
(112) 		1977
(177) 		2036
(236)

Subsets and supersets

Since 450 factors into 2 × 32 × 52, 450edo has subset edos 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, and 225.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-713 450⟩ [⟨450 713]] +0.1961 0.1961 7.35
2.3.5 [-28 25 -5⟩, [25 15 -21⟩ [⟨450 713 1045]] +0.0800 0.2293 8.60
2.3.5.7 235298/234375, 321489/320000, 26873856/26796875 [⟨450 713 1045 1263]] +0.1336 0.2192 8.22

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
2 61\450 162.67 1125/1024 Kwazy
5 7\450 18.67 81/80 Quintile
10 7\450 18.67 81/80 Decile

* In minimal-generator form

Retrieved from "https://en.xen.wiki/index.php?title=450edo&oldid=236690"