282edo: Difference between revisions

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== Theory ==

282edo is the smallest edo ~~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. 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 chords]] 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]] (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.

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]].

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 ===

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=== Subsets and supersets ===

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

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

== Regular temperament properties ==

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| 2.3.5

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

| [{{~~val~~| 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

| [{{~~val~~| 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

| [{{~~val~~| 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

| [{{~~val~~| 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

| [{{~~val~~| 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

| [{{~~val~~| 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

| [{{~~val~~| 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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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| 565.96

| 4096/2835

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

| [[Trident]] (282ef)

|-

| 2

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| 157.45

| 35/32

| [[Nessafof]]

| [[Nessafof]] (7-limit)

|-

| 6

| 51\282 (4\282)

| 217.02 (17.02)

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

| 17/15 (245/243)

| [[Stearnscape]]

|-

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| [[Sextile]]

|}

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

[[Category:Septisuperfourth]]

[[Category:Jupiter]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is the smallest edo 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. 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 chords]] 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]] (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.
+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]].
+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 ===
@@ Line 9: / Line 13: @@
 === Subsets and supersets ===
-Since 282 factors into 2 × 3 × 47, it has subset edos {{EDOs| 2, 3, 47, 94, and 141 }}.
+Since 282 factors into {{factorization|282}}, 282edo has subset edos {{EDOs| 2, 3, 47, 94, and 141 }}.
 == Regular temperament properties ==
@@ Line 24: / Line 28: @@
 | 2.3.5
 | {{monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}
-| [{{val| 282 447 655 }}]
+| {{mapping| 282 447 655 }}
 | -0.1684
 | 0.1671
@@ Line 31: / Line 35: @@
 | 2.3.5.7
 | 6144/6125, 118098/117649, 250047/250000
-| [{{val| 282 447 655 792 }}]
+| {{mapping| 282 447 655 792 }}
 | -0.2498
 | 0.2020
@@ Line 38: / Line 42: @@
 | 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
 | 0.2151
@@ Line 45: / Line 49: @@
 | 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
 | 0.2156
@@ Line 52: / Line 56: @@
 | 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
 | 0.1996
@@ Line 59: / Line 63: @@
 | 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
 | 0.2061
@@ Line 66: / Line 70: @@
 | 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
 | 0.1944
@@ Line 76: / Line 80: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
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 | 565.96
 | 4096/2835
-| [[Tricot]] / [[trident]] (282ef)
+| [[Trident]] (282ef)
 |-
 | 2
@@ Line 115: / Line 119: @@
 | 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>(245/243)
 | [[Stearnscape]]
 |-
@@ Line 129: / Line 133: @@
 | [[Sextile]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Septisuperfourth]]
 [[Category:Jupiter]]