450edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 34: | Line 43: | ||
| 0.2192 | | 0.2192 | ||
| 8.22 | | 8.22 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 2 | | 2 | ||
| Line 50: | Line 66: | ||
| 4/3<br />(81/80) | | 4/3<br />(81/80) | ||
| [[Pental (temperament)|Pental]] (5-limit) | | [[Pental (temperament)|Pental]] (5-limit) | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
Revision as of 13:15, 16 November 2024
| ← 449edo | 450edo | 451edo → |
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 | 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 it is distinct