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 | 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>, | 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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| 81/80 | | 81/80 | ||
| [[Quintile]] (5-limit) | | [[Quintile]] (5-limit) | ||
|- | |||
| 10 | |||
| 7\450 | |||
| 18.67 | |||
| 81/80 | |||
| [[Decile]] | |||
|} | |} | ||
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||
[[Category:Decile]] | |||
Revision as of 06:01, 4 September 2026
| ← 449edo | 450edo | 451edo → |
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
| 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
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 2 | 61\450 | 162.67 | 1125/1024 | Kwazy |
| 5 | 7\450 | 18.67 | 81/80 | Quintile (5-limit) |
| 10 | 7\450 | 18.67 | 81/80 | Decile |