282edo: Difference between revisions

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== Theory ==

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]] (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.

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), 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]].

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.

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=== Subsets and supersets ===

Since 282 factors into {{~~factorization~~|~~282~~}}, 282edo has subset edos {{EDOs| 2, 3, 47, 94, and 141 }}.

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

== Regular temperament properties ==

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

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

| {{~~mapping~~| 282 447 655 }}

| {{Mapping| 282 447 655 }}

| −0.1684

| 0.1671

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| 2.3.5.7

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

| {{~~mapping~~| 282 447 655 792 }}

| {{Mapping| 282 447 655 792 }}

| −0.2498

| 0.2020

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| 2.3.5.7.11

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

| {{~~mapping~~| 282 447 655 792 976 }}

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

| −0.3081

| 0.2151

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| 2.3.5.7.11.13

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

| {{~~mapping~~| 282 447 655 792 976 1044 }}

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

| −0.3480

| 0.2156

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| 2.3.5.7.11.13.17

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

| {{~~mapping~~| 282 447 655 792 976 1044 1153 }}

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

| −0.3481

| 0.1996

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| 2.3.5.7.11.13.17.19

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

| {{~~mapping~~| 282 447 655 792 976 1044 1153 1198 }}

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

| −0.3152

| 0.2061

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| 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

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

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

| −0.3173

| 0.1944

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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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| 565.96

| 4096/2835

| [[~~Trident~~]] (~~282ef~~)

| [[Alphatrident]] (7-limit)

|-

| 2

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|-

| 6

| 51\282 (4\282)

| 51\282 (4\282)

| 217.02 (17.02)

| 217.02 (17.02)

| 17/15 (105/104)

| 17/15 (105/104)

| [[Stearnscape]]

|-

| 6

| 80\282 (14\282)

| 80\282 (14\282)

| 340.43 (59.57)

| 340.43 (59.57)

| 162/133 (88/85)

| 162/133 (88/85)

| [[Semiseptichrome]]

|-

| 6

| 117\282 (23\282)

| 117\282 (23\282)

| 497.87 (97.87)

| 497.87 (97.87)

| 4/3 (128/121)

| 4/3 (128/121)

| [[Sextile]]

|}

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

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

[[Category:Jupiter]]

[[Category:Septisuperfourth]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|282}}
+{{ED intro}}
 == Theory ==
-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]] (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.
+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), 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]].
+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.
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-Since 282 factors into {{factorization|282}}, 282edo has subset edos {{EDOs| 2, 3, 47, 94, and 141 }}.
+Since 282 factors into primes as {{nowrap| 2 × 3 × 47 }}, 282edo has subset edos {{EDOs| 2, 3, 47, 94, and 141 }}.
 == Regular temperament properties ==
@@ Line 21: / Line 21: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 28: / Line 28: @@
 |-
 | 2.3.5
-| {{monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}
+| {{Monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}
-| {{mapping| 282 447 655 }}
+| {{Mapping| 282 447 655 }}
 | −0.1684
 | 0.1671
@@ Line 36: / Line 36: @@
 | 2.3.5.7
 | 6144/6125, 118098/117649, 250047/250000
-| {{mapping| 282 447 655 792 }}
+| {{Mapping| 282 447 655 792 }}
 | −0.2498
 | 0.2020
@@ Line 43: / Line 43: @@
 | 2.3.5.7.11
 | 540/539, 4000/3993, 5632/5625, 137781/137500
-| {{mapping| 282 447 655 792 976 }}
+| {{Mapping| 282 447 655 792 976 }}
 | −0.3081
 | 0.2151
@@ Line 50: / Line 50: @@
 | 2.3.5.7.11.13
 | 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575
-| {{mapping| 282 447 655 792 976 1044 }}
+| {{Mapping| 282 447 655 792 976 1044 }}
 | −0.3480
 | 0.2156
@@ Line 57: / Line 57: @@
 | 2.3.5.7.11.13.17
 | 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197
-| {{mapping| 282 447 655 792 976 1044 1153 }}
+| {{Mapping| 282 447 655 792 976 1044 1153 }}
 | −0.3481
 | 0.1996
@@ Line 64: / Line 64: @@
 | 2.3.5.7.11.13.17.19
 | 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573
-| {{mapping| 282 447 655 792 976 1044 1153 1198 }}
+| {{Mapping| 282 447 655 792 976 1044 1153 1198 }}
 | −0.3152
 | 0.2061
@@ Line 71: / 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
-| {{mapping| 282 447 655 792 976 1044 1153 1198 1276 }}
+| {{Mapping| 282 447 655 792 976 1044 1153 1198 1276 }}
 | −0.3173
 | 0.1944
@@ Line 82: / Line 82: @@
 |+ 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 98: / Line 98: @@
 | 565.96
 | 4096/2835
-| [[Trident]] (282ef)
+| [[Alphatrident]] (7-limit)
 |-
 | 2
@@ Line 125: / Line 125: @@
 |-
 | 6
-| 51\282<br />(4\282)
+| 51\282<br>(4\282)
-| 217.02<br />(17.02)
+| 217.02<br>(17.02)
-| 17/15<br />(105/104)
+| 17/15<br>(105/104)
 | [[Stearnscape]]
 |-
 | 6
-| 80\282<br />(14\282)
+| 80\282<br>(14\282)
-| 340.43<br />(59.57)
+| 340.43<br>(59.57)
-| 162/133<br />(88/85)
+| 162/133<br>(88/85)
 | [[Semiseptichrome]]
 |-
 | 6
-| 117\282<br />(23\282)
+| 117\282<br>(23\282)
-| 497.87<br />(97.87)
+| 497.87<br>(97.87)
-| 4/3<br />(128/121)
+| 4/3<br>(128/121)
 | [[Sextile]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 [[Category:Jupiter]]
 [[Category:Septisuperfourth]]