450edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|450}} == Theory == 450et is consistent to the 7-odd-limit. It can be considered for the 2.3.5.7.13.17.29.31.37 subgroup, tempering out 651/..."
 
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}}
{{EDO intro|450}}
{{ED intro}}


== Theory ==
== Theory ==
450et is consistent to the [[7-odd-limit]]. It can be considered for the 2.3.5.7.13.17.29.31.37 [[subgroup]], tempering 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 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]].


=== Odd harmonics ===
=== Odd harmonics ===
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== 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|-713 450}}
! rowspan="2" | [[Comma list]]
|{{mapping|450 713}}
! rowspan="2" | [[Mapping]]
| 0.1961
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -713 450 }}
| {{mapping| 450 713 }}
| +0.1961
| 0.1961
| 0.1961
| 7.35
| 7.35
|-
|-
|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
| 8.60
| 8.60
|-
|-
|2.3.5.7
| 2.3.5.7
|321489/320000, 235298/234375, 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
| 8.22
| 8.22
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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
|-
|-
|2
| 2
|61\450
| 61\450
|162.67
| 162.67
|1125/1024
| 1125/1024
|[[Kwazy]]
| [[Kwazy]]
|-
|-
|5
| 5
|187\450<br>(7\450)
| 187\450<br />(7\450)
|498.67<br>(18.67)
| 498.67<br />(18.67)
|4/3<br>(81/80)
| 4/3<br />(81/80)
|[[Pental]]
| [[Pental (temperament)|Pental]] (5-limit)
|}
|}
 
<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

Latest revision as of 13:33, 13 March 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 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. It supports decal and varuna.

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

450 factors into 2 × 32 × 52, with 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
5 187\450
(7\450) 		498.67
(18.67) 		4/3
(81/80) 		Pental (5-limit)

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

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