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..."
 
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== 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 [[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 ===
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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|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|-141 89}}
| {{monzo| -141 89 }}
|{{mapping|445 705}}
| {{mapping| 445 705 }}
| 0.2623
| 0.2623
| 0.2623
| 0.2623
| 9.73
| 9.73
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-28 25 -5}}, {{monzo|-29 -11 20}}
| {{monzo| -28 25 -5 }}, {{monzo| -29 -11 20 }}
|{{mapping|445 705 1033}}
| {{mapping| 445 705 1033 }}
| 0.2748
| 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
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! 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]] (5-limit)
|}
|}
<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 10:17, 21 January 2024

← 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

Template:EDO intro

Theory

445edo 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

445 factors into 5 × 89, with 5edo and 89edo as its subset edos.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-141 89⟩ [⟨445 705]] 0.2623 0.2623 9.73
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 it is distinct

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