449edo: Difference between revisions

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{{novelty}}{{stub}}{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|449}}
{{ED intro}}
 
== Theory ==
== Theory ==
449et tempers out 26873856/26796875 and [[4375/4374]] in the 7-limit; 100663296/100656875, [[117440512/117406179]], 4302592/4296875, 825000/823543, 85937500/85766121, 160083/160000, [[41503/41472]], 539055/537824 and 805255/802816 in the 11-limit.
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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.9
! rowspan="2" | [[Subgroup]]
|{{monzo|-1423 449}}
! rowspan="2" | [[Comma list]]
|{{val|449 1423}}
! rowspan="2" | [[Mapping]]
| 0.1249
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.9
| {{monzo| -1423 449 }}
| {{mapping| 449 1423 }}
| +0.1249
| 0.1249
| 0.1249
| 4.67
| 4.67
Line 28: Line 35:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! 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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Music ==
== Music ==
*[https://www.youtube.com/watch?v=vLZuStkREoE Little Victorious Dance] by [[User:Francium|Francium]]
; [[Francium]]
* [https://www.youtube.com/watch?v=vLZuStkREoE ''Little Victorious Dance''] (2023)
 
[[Category:Listen]]

Latest revision as of 12:54, 21 February 2025

← 448edo 449edo 450edo →
Prime factorization 449 (prime)
Step size 2.67261 ¢ 
Fifth 263\449 (702.895 ¢)
Semitones (A1:m2) 45:32 (120.3 ¢ : 85.52 ¢)
Dual sharp fifth 263\449 (702.895 ¢)
Dual flat fifth 262\449 (700.223 ¢)
Dual major 2nd 76\449 (203.118 ¢)
Consistency limit 7
Distinct consistency limit 7

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

Approximation of odd harmonics in 449edo
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

Table of rank-2 temperaments by generator
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

Francium
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