444edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|444}}
{{ED intro}}


== Theory ==
== Theory ==
444et is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. Using the [[patent val]], the equal temperament [[tempering out|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 [[support]]s the [[magnesium]] temperament.
444edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. Since {{nowrap|444 {{=}} 4 × 111}}, its harmonic [[3/1|3]] derives from [[111edo]]. Using the [[patent val]], the equal temperament [[tempering out|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 [[support]]s the [[magnesium]] temperament.


=== Odd harmonics ===
=== Odd harmonics ===
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== 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]]
! 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]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 444 704 }}
| -0.2359
| 0.2358
| 8.72
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 41 -20 -4 }}, {{monzo| -29 -11 20 }}
| {{monzo| 41 -20 -4 }}, {{monzo| -29 -11 20 }}
| {{mapping| 444 704 1031 }}
| {{mapping| 444 704 1031 }}
| -0.1821
| −0.1821
| 0.2071
| 0.2071
| 7.66
| 7.66
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=== 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
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| 497.30<br>(102.70)
| 497.30<br>(102.70)
| 4/3<br>(35/33)
| 4/3<br>(35/33)
| [[Undim]]
| [[Undim]] (444d)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

Latest revision as of 06:23, 21 February 2025

← 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

444 equal divisions of the octave (abbreviated 444edo or 444ed2), also called 444-tone equal temperament (444tet) or 444 equal temperament (444et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 444 equal parts of about 2.7 ¢ each. Each step represents a frequency ratio of 21/444, or the 444th root of 2.

Theory

444edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. Since 444 = 4 × 111, 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [41 -20 -4⟩, [-29 -11 20⟩ [⟨444 704 1031]] −0.1821 0.2071 7.66

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
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)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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