445edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|445}} == Theory == 445et is consistent to the 7-odd-limit. Using the patent val, it tempers out 1220703125/1219784832, 48828125/48771072, 95703..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|445}}
{{ED intro}}


== Theory ==
== Theory ==
445et is consistent to the [[7-odd-limit]]. Using the patent val, it tempers out 1220703125/1219784832, 48828125/48771072, 95703125/95551488 and [[2401/2400]] in the 7-limit; 78121827/77948684, 56723625/56689952, 10333575/10307264, 35156250/35153041, 234375/234256, 1366875/1362944, 104162436/103984375, 1953125/1951488, 151263/151250, 472392/471625, 137781/137500, 24057/24010, [[8019/8000]], [[3025/3024]], [[41503/41472]], 539055/537824, 35937/35840, 244515348/244140625, 805255/802816, 39135393/39062500 in the 11-limit. It [[support]]s [[misneb]].
445edo is [[enfactoring|enfactored]] in the [[3-limit]] with the same tuning as [[89edo]], but the approximation to some of the higher harmonics are improved. It is [[consistent]] to the [[7-odd-limit]] with [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]] all tuned flat, and it allows an extension to the [[11-limit]]. The equal temperament [[tempering out|tempers out]] [[2401/2400]], 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, 35937/35840, [[41503/41472]], 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably [[support]]s [[neptune]].


=== Odd harmonics ===
=== Odd harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
445 factors into 5 × 89, with [[5edo]] and [[89edo]] as its subset edos.
Since 445 factors into {{factorisation|445}}, 445edo has [[5edo]] and 89edo as its subsets.


== 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]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|-141 89}}
! rowspan="2" | [[Comma list]]
|{{mapping|445 705}}
! rowspan="2" | [[Mapping]]
| 0.2623
! rowspan="2" | Optimal<br />8ve stretch (¢)
| 0.2623
! colspan="2" | Tuning error
| 9.73
|-
|-
|2.3.5
! [[TE error|Absolute]] (¢)
|{{monzo|-28 25 -5}}, {{monzo|-29 -11 20}}
! [[TE simple badness|Relative]] (%)
|{{mapping|445 705 1033}}
|-
| 0.2748
| 2.3.5
| {{monzo| -28 25 -5 }}, {{monzo| -29 -11 20 }}
| {{mapping| 445 705 1033 }}
| +0.2748
| 0.2149
| 0.2149
| 7.97
| 7.97
|-
|-
|2.3.5.7
| 2.3.5.7
|2401/2400, 48828125/48771072, 43046721/42875000
| 2401/2400, 7381125/7340032, 43046721/42875000
|{{mapping|445 705 1033 1249}}
| {{mapping| 445 705 1033 1249 }}
| 0.2716
| +0.2716
| 0.1862
| 0.1862
| 6.90
| 6.90
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 2401/2400, 8019/8000, 234375/234256
| 2401/2400, 3025/3024, 8019/8000, 234375/234256
|{{mapping|445 705 1033 1249 1539}}
| {{mapping| 445 705 1033 1249 1539 }}
| 0.2870
| +0.2870
| 0.1694
| 0.1694
| 6.28
| 6.28
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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
|-
|-
|1
| 1
|13\445
| 13\445
|35.06
| 35.06
|1990656/1953125
| 1990656/1953125
|[[Gammic]]
| [[Gammic]] (5-limit)
|-
|-
|1
| 1
|42\445
| 42\445
|113.26
| 113.26
|16/15
| 16/15
|[[Misneb]]
| [[Misneb]]
|-
|-
|1
| 1
|216\445
| 216\445
|582.47
| 582.47
|7/5
| 7/5
|[[Neptune]]
| [[Neptune]] (7-limit)
|-
|-
|5
| 5
|185\445<br>(7\445)
| 185\445<br>(7\445)
|498.88<br>(18.88)
| 498.88<br>(18.88)
|4/3<br>(81/80)
| 4/3<br>(81/80)
|[[Pental]]
| [[Pental (temperament)|Pental]] (5-limit)
|}
|}
 
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if 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

Latest revision as of 13:33, 13 March 2026

← 444edo 445edo 446edo →
Prime factorization 5 × 89
Step size 2.69663 ¢ 
Fifth 260\445 (701.124 ¢) (→ 52\89)
Semitones (A1:m2) 40:35 (107.9 ¢ : 94.38 ¢)
Consistency limit 7
Distinct consistency limit 7

445 equal divisions of the octave (abbreviated 445edo or 445ed2), also called 445-tone equal temperament (445tet) or 445 equal temperament (445et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 445 equal parts of about 2.7 ¢ each. Each step represents a frequency ratio of 21/445, or the 445th root of 2.

Theory

445edo is enfactored in the 3-limit with the same tuning as 89edo, but the approximation to some of the higher harmonics are improved. It is consistent to the 7-odd-limit with harmonics 3, 5, 7 all tuned flat, and it allows an extension to the 11-limit. The equal temperament tempers out 2401/2400, 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; 3025/3024, 8019/8000, 24057/24010, 35937/35840, 41503/41472, 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably supports neptune.

Odd harmonics

Approximation of odd harmonics in 445edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.83 -0.70 -0.74 +1.03 -1.21 +0.82 +1.17 +0.21 -0.88 +1.13 +0.04
Relative (%) -30.8 -25.8 -27.3 +38.3 -44.7 +30.4 +43.4 +7.9 -32.8 +41.9 +1.5
Steps
(reduced) 		705
(260) 		1033
(143) 		1249
(359) 		1411
(76) 		1539
(204) 		1647
(312) 		1739
(404) 		1819
(39) 		1890
(110) 		1955
(175) 		2013
(233)

Subsets and supersets

Since 445 factors into 5 × 89, 445edo has 5edo and 89edo as its subsets.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [-28 25 -5⟩, [-29 -11 20⟩ [⟨445 705 1033]] +0.2748 0.2149 7.97
2.3.5.7 2401/2400, 7381125/7340032, 43046721/42875000 [⟨445 705 1033 1249]] +0.2716 0.1862 6.90
2.3.5.7.11 2401/2400, 3025/3024, 8019/8000, 234375/234256 [⟨445 705 1033 1249 1539]] +0.2870 0.1694 6.28

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 13\445 35.06 1990656/1953125 Gammic (5-limit)
1 42\445 113.26 16/15 Misneb
1 216\445 582.47 7/5 Neptune (7-limit)
5 185\445
(7\445) 		498.88
(18.88) 		4/3
(81/80) 		Pental (5-limit)

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

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