449edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
|-
! 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
| 2.9
Line 31: Line 22:
| 0.1249
| 0.1249
| 4.67
| 4.67
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 48: Line 32:
| 243\200
| 243\200
| [[Amity]] (7-limit)
| [[Amity]] (7-limit)
|}
{{rank-2 end}}
{{orf}}
{{orf}}


Revision as of 01:41, 16 November 2024

← 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

Template:EDO intro

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

Template:Comma basis begin |- | 2.9 | [-1423 449⟩ | [⟨449 1423]] | 0.1249 | 0.1249 | 4.67 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 127\449 | 339.421 | 243\200 | Amity (7-limit) Template:Rank-2 end Template:Orf

Music

Francium
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