444edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
It's enfactored in the 3-limit
ArrowHead294 (talk | contribs)
15,215 edits
mNo edit summary
Line 12: Line 12:


== 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]] (%)
|-
|-
| 2.3.5
| 2.3.5
Line 28: Line 20:
| 0.2071
| 0.2071
| 7.66
| 7.66
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 50: Line 36:
| 4/3<br>(35/33)
| 4/3<br>(35/33)
| [[Undim]] (444d)
| [[Undim]] (444d)
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 01:32, 16 November 2024

← 443edo 444edo 445edo →
Prime factorization 22 × 3 × 37
Step size 2.7027 ¢ 
Fifth 260\444 (702.703 ¢) (→ 65\111)
Semitones (A1:m2) 44:32 (118.9 ¢ : 86.49 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

444edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. 444 = 4 × 111, and its harmonic 3 derives from 111edo. Using the patent val, the equal temperament tempers out 250047/250000, 29360128/29296875, 67108864/66976875 and in the 7-limit; 3025/3024, 5632/5625, 42592/42525, 102487/102400, 131072/130977, 160083/160000, 172032/171875, 322102/321489, 391314/390625 and 1771561/1769472 in the 11-limit. It supports the magnesium temperament.

Odd harmonics

Approximation of odd harmonics in 444edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.75 +0.17 -1.26 -1.21 +0.03 +0.01 +0.92 +0.45 -0.22 -0.51 -1.25
Relative (%) +27.7 +6.4 -46.6 -44.7 +1.2 +0.5 +34.1 +16.6 -8.0 -18.9 -46.2
Steps
(reduced) 		704
(260) 		1031
(143) 		1246
(358) 		1407
(75) 		1536
(204) 		1643
(311) 		1735
(403) 		1815
(39) 		1886
(110) 		1950
(174) 		2008
(232)

Subsets and supersets

Since 444 factors into 22 × 3 × 37, 444edo has subset edos 2, 3, 4, 6, 12, 37, 74, 111, 148, and 222. 1332edo, which triples it, gives a good correction to the harmonic 7.

Regular temperament properties

Template:Comma basis begin |- | 2.3.5 | [41 -20 -4⟩, [-29 -11 20⟩ | [⟨444 704 1031]] | -0.1821 | 0.2071 | 7.66 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 13\444 | 35.14 | 1990656/1953125 | Gammic (5-limit) |- | 4 | 184\444
(38\444) | 497.30
(102.70) | 4/3
(35/33) | Undim (444d) Template:Rank-2 end Template:Orf

Retrieved from "https://en.xen.wiki/index.php?title=444edo&oldid=165633"