445edo: Difference between revisions

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== Theory ==

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]].

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 (temperament)|neptune]].

=== Odd harmonics ===

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! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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|-

| 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

| 0.2149

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| 2.3.5.7

| 2401/2400, 7381125/7340032, 43046721/42875000

| {{~~mapping~~| 445 705 1033 1249 }}

| {{Mapping| 445 705 1033 1249 }}

| +0.2716

| 0.1862

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| 2.3.5.7.11

| 2401/2400, 3025/3024, 8019/8000, 234375/234256

| {{~~mapping~~| 445 705 1033 1249 1539 }}

| {{Mapping| 445 705 1033 1249 1539 }}

| +0.2870

| 0.1694

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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

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| 582.47

| 7/5

| [[Neptune]] (7-limit)

| [[Neptune (temperament)|Neptune]] (7-limit)

|-

| 5

| ~~185\445 (~~7\445)

| 7\445

| ~~498.88 (~~18.88)

| 18.88

| ~~4/3 (~~81/80)

| 81/80

| [[~~Pental (temperament)|Pental~~]] ~~(5-limit)~~

| [[Quintile]]

|}

<nowiki />* [[~~Normal lists|Octave~~-~~reduced~~ form~~]], reduced to the first half-octave, and [[Normal lists~~|minimal form]] ~~in parentheses if distinct~~

<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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]].
+edo 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 (temperament)|neptune]].
 === Odd harmonics ===
@@ Line 17: / Line 17: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 24: / Line 24: @@
 |-
 | 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
 | 0.2149
@@ Line 32: / Line 32: @@
 | 2.3.5.7
 | 2401/2400, 7381125/7340032, 43046721/42875000
-| {{mapping| 445 705 1033 1249 }}
+| {{Mapping| 445 705 1033 1249 }}
 | +0.2716
 | 0.1862
@@ Line 39: / Line 39: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 8019/8000, 234375/234256
-| {{mapping| 445 705 1033 1249 1539 }}
+| {{Mapping| 445 705 1033 1249 1539 }}
 | +0.2870
 | 0.1694
@@ Line 49: / Line 49: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 71: / Line 71: @@
 | 582.47
 | 7/5
-| [[Neptune]] (7-limit)
+| [[Neptune (temperament)|Neptune]] (7-limit)
 |-
 | 5
-| 185\445<br>(7\445)
+| 7\445
-| 498.88<br>(18.88)
+| 18.88
-| 4/3<br>(81/80)
+| 81/80
-| [[Pental (temperament)|Pental]] (5-limit)
+| [[Quintile]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]