449edo: Difference between revisions
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Cleanup; clarify the title row of the rank-2 temp table |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|449}} | {{EDO intro|449}} | ||
== Theory == | == Theory == | ||
449edo is [[consistent]] to the [[7-odd-limit]], but the errors of [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|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. | |||
===Odd harmonics=== | |||
Using the [[patent val]], the equal temperament [[tempering out|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 === | |||
{{Harmonics in equal|449}} | {{Harmonics in equal|449}} | ||
===Subsets and supersets=== | |||
=== Subsets and supersets === | |||
449edo is the 87th [[prime edo]]. 898edo, which doubles it, gives a good correction to the harmonic 3, 5 and 7. | 449edo is the 87th [[prime edo]]. 898edo, which doubles it, gives a good correction to the harmonic 3, 5 and 7. | ||
==Regular temperament properties== | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|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 | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.9 | | 2.9 | ||
|{{monzo|-1423 449}} | | {{monzo| -1423 449 }} | ||
|{{ | | {{mapping| 449 1423 }} | ||
| 0.1249 | | 0.1249 | ||
| 0.1249 | | 0.1249 | ||
| Line 30: | Line 36: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|127\449 | | 127\449 | ||
|339.421 | | 339.421 | ||
|243\200 | | 243\200 | ||
|[[Amity]] (7-limit) | | [[Amity]] (7-limit) | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Music == | == Music == | ||
*[https://www.youtube.com/watch?v=vLZuStkREoE Little Victorious Dance] | ; [[User:Francium|Francium]] | ||
* [https://www.youtube.com/watch?v=vLZuStkREoE ''Little Victorious Dance''] (2023) | |||
Revision as of 07:41, 3 November 2023
| ← 448edo | 449edo | 450edo → |
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 it is distinct
Music
- Little Victorious Dance (2023)