282edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 2 × 3 × 47~~

~~| Step size = 4.25532¢~~

~~| Fifth = 165\282 (702.12¢) (→ [[94edo|55\94]])~~

~~| Semitones = 27:21 (114.89¢ : 89.36¢)~~

~~| Consistency = 29~~

}}

The '''282 equal divisions of the octave''' ('''282edo'''), or the '''282(-tone) equal temperament''' ('''282tet''', '''282et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 282 parts of about 4.26 [[cent]]s each.

== Theory ==

282edo is the smallest ~~equal temperament uniquely~~ [[consistent]] through to the [[23-odd-limit]], and also the smallest consistent to the [[29-odd-limit]]. It shares the same 3rd, 7th, and 13th harmonics with [[94edo]] (282 = 3 × 94), as well as [[11/10]] and [[20/17]] (~~supporting~~ the [[Stearnsmic clan #Garistearn|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29. It tempers out [[6144/6125]] (porwell), 118098/117649 (stearnsma), and [[250047/250000]] (landscape comma) in the 7-limit, and [[540/539]] and 5632/5625 in the 11-limit, so that it provides the [[optimal patent val]] for the [[jupiter]] temperament; it also tempers out [[4000/3993]] and 234375/234256, providing the optimal patent val for [[septisuperfourth]] temperament. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[10648/10647]]. It allows [[essentially tempered chord]]s including [[swetismic chords]], [[squbemic chords]], and [[petrmic ~~triad~~]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.

282edo is the smallest edo [[consistency|distinctly consistent]] through to the [[23-odd-limit]], and also the smallest consistent to the [[29-odd-limit]]. It shares the same 3rd, 7th, and 13th harmonics with [[94edo]] ({{nowrap| 282 {{=}} 3 × 94 }}), as well as [[11/10]] and [[20/17]] ([[support]]ing the [[Stearnsmic clan #Garistearn|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29.

The equal temperament [[tempering out|tempers out]] [[6144/6125]] (porwell comma), 118098/117649 (stearnsma), and [[250047/250000]] (landscape comma) in the 7-limit, and [[540/539]] and [[5632/5625]] in the 11-limit, so that it provides the [[optimal patent val]] for the [[jupiter]] temperament; it also tempers out [[4000/3993]] and 234375/234256, providing the optimal patent val for [[septisuperfourth]] temperament. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[10648/10647]].

It allows [[essentially tempered chord]]s including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 282 factors into primes as {{nowrap| 2 × 3 × 47 }}, 282edo has subset edos {{EDOs| 2, 3, 47, 94, and 141 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

|-

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

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

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

Line 26:

Line 28:

|-

| 2.3.5

| {{~~monzo~~| 32 -7 -9 }}, {{monzo| -7 22 -12 }}

| {{Monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}

| [{{~~val~~| 282 447 655 }}]

| {{Mapping| 282 447 655 }}

| -0.1684

| −0.1684

| 0.1671

| 3.93

Line 34:

Line 36:

| 2.3.5.7

| 6144/6125, 118098/117649, 250047/250000

| [{{~~val~~| 282 447 655 792 }}]

| {{Mapping| 282 447 655 792 }}

| -0.2498

| −0.2498

| 0.2020

| 4.75

Line 41:

Line 43:

| 2.3.5.7.11

| 540/539, 4000/3993, 5632/5625, 137781/137500

| [{{~~val~~| 282 447 655 792 976 }}]

| {{Mapping| 282 447 655 792 976 }}

| -0.3081

| −0.3081

| 0.2151

| 5.06

Line 48:

Line 50:

| 2.3.5.7.11.13

| 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575

| [{{~~val~~| 282 447 655 792 976 1044 }}]

| {{Mapping| 282 447 655 792 976 1044 }}

| -0.3480

| −0.3480

| 0.2156

| 5.07

Line 55:

Line 57:

| 2.3.5.7.11.13.17

| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197

| [{{~~val~~| 282 447 655 792 976 1044 1153 }}]

| {{Mapping| 282 447 655 792 976 1044 1153 }}

| -0.3481

| −0.3481

| 0.1996

| 4.69

Line 62:

Line 64:

| 2.3.5.7.11.13.17.19

| 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573

| [{{~~val~~| 282 447 655 792 976 1044 1153 1198 }}]

| {{Mapping| 282 447 655 792 976 1044 1153 1198 }}

| -0.3152

| −0.3152

| 0.2061

| 4.84

Line 69:

Line 71:

| 2.3.5.7.11.13.17.19.23

| 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287

| [{{~~val~~| 282 447 655 792 976 1044 1153 1198 1276 }}]

| {{Mapping| 282 447 655 792 976 1044 1153 1198 1276 }}

| -0.3173

| −0.3173

| 0.1944

| 4.57

|}

* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo|270]] and before [[311edo|311]].

=== Rank-2 temperaments ===

{| 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 per ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

Line 94:

Line 98:

| 565.96

| 4096/2835

| [[~~Tricot]] / [[trident~~]] (~~282ef~~)

| [[Alphatrident]] (7-limit)

|-

| 2

Line 118:

Line 122:

| 157.45

| 35/32

| [[Nessafof]]

| [[Nessafof]] (7-limit)

|-

| 6

| 51\282 (4\282)

| 217.02 (17.02)

| ~~567~~/~~500~~ (~~245~~/~~243~~)

| 17/15 (105/104)

| [[Stearnscape]]

|-

| 6

| 80\282 (14\282)

| 340.43 (59.57)

| 162/133 (88/85)

| [[Semiseptichrome]]

|-

| 6

Line 132:

Line 142:

| [[Sextile]]

|}

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

[[Category:~~Equal divisions of the octave]]~~

[[Category:Jupiter]]

~~[[Category:29-limit~~]]

[[Category:Septisuperfourth]]

~~[[Category:Jupiter]]~~

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2 × 3 × 47
+{{ED intro}}
-| Step size = 4.25532¢
-| Fifth = 165\282 (702.12¢) (→ [[94edo|55\94]])
-| Semitones = 27:21 (114.89¢ : 89.36¢)
-| Consistency = 29
-}}
-The '''282 equal divisions of the octave''' ('''282edo'''), or the '''282(-tone) equal temperament''' ('''282tet''', '''282et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 282 parts of about 4.26 [[cent]]s each.
 == Theory ==
-edo is the smallest equal temperament uniquely [[consistent]] through to the [[23-odd-limit]], and also the smallest consistent to the [[29-odd-limit]]. It shares the same 3rd, 7th, and 13th harmonics with [[94edo]] (282 = 3 × 94), as well as [[11/10]] and [[20/17]] (supporting the [[Stearnsmic clan #Garistearn|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29. It tempers out [[6144/6125]] (porwell), 118098/117649 (stearnsma), and [[250047/250000]] (landscape comma) in the 7-limit, and [[540/539]] and 5632/5625 in the 11-limit, so that it provides the [[optimal patent val]] for the [[jupiter]] temperament; it also tempers out [[4000/3993]] and 234375/234256, providing the optimal patent val for [[septisuperfourth]] temperament. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[10648/10647]]. It allows [[essentially tempered chord]]s including [[swetismic chords]], [[squbemic chords]], and [[petrmic triad]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.
+edo is the smallest edo [[consistency|distinctly consistent]] through to the [[23-odd-limit]], and also the smallest consistent to the [[29-odd-limit]]. It shares the same 3rd, 7th, and 13th harmonics with [[94edo]] ({{nowrap| 282 {{=}} 3 × 94 }}), as well as [[11/10]] and [[20/17]] ([[support]]ing the [[Stearnsmic clan #Garistearn|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29.
+The equal temperament [[tempering out|tempers out]] [[6144/6125]] (porwell comma), 118098/117649 (stearnsma), and [[250047/250000]] (landscape comma) in the 7-limit, and [[540/539]] and [[5632/5625]] in the 11-limit, so that it provides the [[optimal patent val]] for the [[jupiter]] temperament; it also tempers out [[4000/3993]] and 234375/234256, providing the optimal patent val for [[septisuperfourth]] temperament. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[10648/10647]].
+It allows [[essentially tempered chord]]s including [[swetismic chords]], [[squbemic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.
 === Prime harmonics ===
 {{Harmonics in equal|282|columns=11}}
+=== Subsets and supersets ===
+Since 282 factors into primes as {{nowrap| 2 × 3 × 47 }}, 282edo has subset edos {{EDOs| 2, 3, 47, 94, and 141 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
@@ Line 26: / Line 28: @@
 |-
 | 2.3.5
-| {{monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}
+| {{Monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}
-| [{{val| 282 447 655 }}]
+| {{Mapping| 282 447 655 }}
-| -0.1684
+| −0.1684
 | 0.1671
 | 3.93
@@ Line 34: / Line 36: @@
 | 2.3.5.7
 | 6144/6125, 118098/117649, 250047/250000
-| [{{val| 282 447 655 792 }}]
+| {{Mapping| 282 447 655 792 }}
-| -0.2498
+| −0.2498
 | 0.2020
 | 4.75
@@ Line 41: / Line 43: @@
 | 2.3.5.7.11
 | 540/539, 4000/3993, 5632/5625, 137781/137500
-| [{{val| 282 447 655 792 976 }}]
+| {{Mapping| 282 447 655 792 976 }}
-| -0.3081
+| −0.3081
 | 0.2151
 | 5.06
@@ Line 48: / Line 50: @@
 | 2.3.5.7.11.13
 | 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575
-| [{{val| 282 447 655 792 976 1044 }}]
+| {{Mapping| 282 447 655 792 976 1044 }}
-| -0.3480
+| −0.3480
 | 0.2156
 | 5.07
@@ Line 55: / Line 57: @@
 | 2.3.5.7.11.13.17
 | 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197
-| [{{val| 282 447 655 792 976 1044 1153 }}]
+| {{Mapping| 282 447 655 792 976 1044 1153 }}
-| -0.3481
+| −0.3481
 | 0.1996
 | 4.69
@@ Line 62: / Line 64: @@
 | 2.3.5.7.11.13.17.19
 | 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573
-| [{{val| 282 447 655 792 976 1044 1153 1198 }}]
+| {{Mapping| 282 447 655 792 976 1044 1153 1198 }}
-| -0.3152
+| −0.3152
 | 0.2061
 | 4.84
@@ Line 69: / Line 71: @@
 | 2.3.5.7.11.13.17.19.23
 | 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287
-| [{{val| 282 447 655 792 976 1044 1153 1198 1276 }}]
+| {{Mapping| 282 447 655 792 976 1044 1153 1198 1276 }}
-| -0.3173
+| −0.3173
 | 0.1944
 | 4.57
 |}
+* 282et has a lower relative error than any previous equal temperaments in the 23-limit, past [[270edo|270]] and before [[311edo|311]].
 === Rank-2 temperaments ===
 {| 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 octave
+|-
-! Generator<br>(reduced)
+! Periods<br>per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 94: / Line 98: @@
 | 565.96
 | 4096/2835
-| [[Tricot]] / [[trident]] (282ef)
+| [[Alphatrident]] (7-limit)
 |-
 | 2
@@ Line 118: / Line 122: @@
 | 157.45
 | 35/32
-| [[Nessafof]]
+| [[Nessafof]] (7-limit)
 |-
 | 6
 | 51\282<br>(4\282)
 | 217.02<br>(17.02)
-| 567/500<br>(245/243)
+| 17/15<br>(105/104)
 | [[Stearnscape]]
+|-
+| 6
+| 80\282<br>(14\282)
+| 340.43<br>(59.57)
+| 162/133<br>(88/85)
+| [[Semiseptichrome]]
 |-
 | 6
@@ Line 132: / Line 142: @@
 | [[Sextile]]
 |}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
-[[Category:Equal divisions of the octave]]
+[[Category:Jupiter]]
-[[Category:29-limit]]
 [[Category:Septisuperfourth]]
-[[Category:Jupiter]]