444edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|444}} == Theory == 444et is only consistent to the 5-limit. Using the patent val, it tempers out 67108864/66976875, 29360128/29296875 and 250047/..."
 
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== Theory ==
== Theory ==
444et is only consistent to the 5-limit. Using the patent val, it tempers out 67108864/66976875, 29360128/29296875 and [[250047/250000]] in the 7-limit; 100663296/100656875, 2097152/2096325, [[131072/130977]], 172032/171875, [[5632/5625]], 47265625/47258883, [[3025/3024]], 160083/160000, 42592/42525, 391314/390625, 102487/102400, 322102/321489 and [[1771561/1769472]] in the 11-limit. It provides the [[optimal patent val]] for the [[magnesium]] temperament.
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.


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
444 factors into 2<sub>2</sub> × 3 × 37, with subset edos {{EDOs|2, 3, 4, 6, 12, 37, 74, 111, 148, and 222}}. [[1332edo]], which triples it, gives a good correction to the harmonic 7.  
Since 444 factors into {{factorization|444}}, 444edo has subset edos {{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 ==
== 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]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|176 -111}}
| {{monzo| 176 -111 }}
|{{mapping|444 704}}
| {{mapping| 444 704 }}
| -0.2359
| -0.2359
| 0.2358
| 0.2358
| 8.72
| 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
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)*
! Generator*
! Cents<br>(reduced)*
! Cents*
! Associated<br>Ratio*
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|13\444
| 13\444
|35.14
| 35.14
|1990656/1953125
| 1990656/1953125
|[[Gammic]]
| [[Gammic]] (5-limit)
|-
|-
|4
| 4
|184\444<br>(38\444)
| 184\444<br>(38\444)
|497.30<br>(102.70)
| 497.30<br>(102.70)
|4/3<br>(35/33)
| 4/3<br>(35/33)
|[[Undim]]
| [[Undim]]
|}
|}
<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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 15:22, 13 December 2023

← 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

444et is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. 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 [176 -111⟩ [⟨444 704]] -0.2359 0.2358 8.72
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

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

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