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450edo: Difference between revisions

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== 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 ===
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=== 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 ==
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! 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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| -713 450 }}
| {{Monzo| -713 450 }}
| {{mapping| 450 713 }}
| {{Mapping| 450 713 }}
| +0.1961
| +0.1961
| 0.1961
| 0.1961
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|-
|-
| 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
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| 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
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|+ 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
|-
|-
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|-
|-
| 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 distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
 
[[Category:Decile]]
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