449edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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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]] (%) | |||
|- | |- | ||
| 2.9 | | 2.9 | ||
| {{monzo| -1423 449 }} | | {{monzo| -1423 449 }} | ||
| {{mapping| 449 1423 }} | | {{mapping| 449 1423 }} | ||
| 0.1249 | | +0.1249 | ||
| 0.1249 | | 0.1249 | ||
| 4.67 | | 4.67 | ||
|} | |||
=== 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 | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 32: | Line 48: | ||
| 243\200 | | 243\200 | ||
| [[Amity]] (7-limit) | | [[Amity]] (7-limit) | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Music == | == Music == | ||
Latest revision as of 12:54, 21 February 2025
| ← 448edo | 449edo | 450edo → |
449 equal divisions of the octave (abbreviated 449edo or 449ed2), also called 449-tone equal temperament (449tet) or 449 equal temperament (449et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 449 equal parts of about 2.67 ¢ each. Each step represents a frequency ratio of 21/449, or the 449th root of 2.
Theory
449edo is consistent to the 7-odd-limit, but the errors of harmonics 3, 5, and 7 are all quite large, giving us the option of treating it as a full 7-limit temperament, or a 2.9.15.21.11.13 subgroup temperament.
Using the patent val, the equal temperament tempers out 4375/4374 and 26873856/26796875 in the 7-limit; 41503/41472, 160083/160000, 539055/537824, 805255/802816, and 825000/823543 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.94 | +1.21 | +1.33 | -0.79 | -0.76 | -1.33 | -0.52 | -0.72 | -0.85 | -0.40 | -0.21 |
| Relative (%) | +35.2 | +45.4 | +49.8 | -29.6 | -28.5 | -49.7 | -19.4 | -27.1 | -31.9 | -15.1 | -7.9 | |
| Steps (reduced) |
712 (263) |
1043 (145) |
1261 (363) |
1423 (76) |
1553 (206) |
1661 (314) |
1754 (407) |
1835 (39) |
1907 (111) |
1972 (176) |
2031 (235) | |
Subsets and supersets
449edo is the 87th prime edo. 898edo, which doubles it, gives a good correction to the harmonic 3, 5 and 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1423 449⟩ | [⟨449 1423]] | +0.1249 | 0.1249 | 4.67 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 127\449 | 339.421 | 243\200 | Amity (7-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Music
- Little Victorious Dance (2023)