450edo: Difference between revisions
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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" |
→Regular temperament properties: update |
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! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! 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 }} | ||
| {{ | | {{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 }} | ||
| {{ | | {{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 }} | ||
| +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 | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 62: | Line 62: | ||
|- | |- | ||
| 5 | | 5 | ||
| | | 7\450 | ||
| | | 18.67 | ||
| | | 81/80 | ||
| [[ | | [[Quintile]] (5-limit) | ||
|} | |} | ||
<nowiki />* | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||
Revision as of 05:57, 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 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
| 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
| 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) |