84edo: Difference between revisions

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== Theory ==

~~84 = 7 × 12, and~~ 84edo shares the [[3/2|perfect fifth]] with [[12edo]], [[tempering out]] the [[Pythagorean comma]] in its [[patent val]]. In the [[5-limit]] it tempers out the [[sensipent comma]]; in the [[7-limit]] [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[support]]ing [[orwell]], [[compton]], and [[sensei]].

84edo shares the [[3/2|perfect fifth]] with [[12edo]], [[tempering out]] the [[Pythagorean comma]] in its [[patent val]]. In the [[5-limit]] it tempers out the [[sensipent comma]]; in the [[7-limit]] [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[support]]ing [[orwell]], [[compton]], and [[sensei]].

84edo is where the orwell temperament takes its name from, since the generator of [[7/6]] is equal to 19 steps of the edo, referencing the [[Wikipedia: Nineteen Eighty-Four|book 1984]]. Orwell in 84edo comes in two ~~varieties – the~~ 84e val {{val| 84 133 195 236 '''290''' }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 '''291''' }} supporting [[newspeak]]. 84edo orwell offers [[mos scale]]s of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale.

84edo is where the orwell temperament takes its name from, since the generator of [[7/6]] is equal to 19 steps of the edo, referencing the [[Wikipedia: Nineteen Eighty-Four|book 1984]]. Orwell in 84edo comes in two varieties—the 84e val {{val| 84 133 195 236 '''290''' }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 '''291''' }} supporting [[newspeak]]. 84edo orwell offers [[mos scale]]s of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale.

It has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], and [[31/1|31]]. In fact, it is [[consistent]] to the no-11 no-17 [[25-odd-limit]]. In the [[13-limit]] it is the [[optimal patent val]] for the rank-5 temperament tempering out [[144/143]].

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== Intervals ==

{| class="wikitable"

|-

! #

! Cents

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| D

|}

<nowiki>*~~</nowiki> as~~ a 2.3.5.7.13.19.23-subgroup temperament

<nowiki />* As a 2.3.5.7.13.19.23-subgroup temperament

== Notation ==

=== 4L 5s (gramitonic) notation ===

~~The~~ notation of Orwell[9]. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & "amp" and @ "at".

This notation is based on Orwell[9]. Notes are denoted as {{nowrap|LsLsLsLss {{=}} JKLMNOPQRJ}}, and raising and lowering by a chroma ({{nowrap|L − s}}), 3 steps in this instance, is denoted by & ("amp") and @ ("at").

{| class="wikitable center-1 right-2 center-3"

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== Regular temperament properties ==

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

|-

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

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

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

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

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

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

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 225/224, 351/350, 640/637, 1701/1690

| {{mapping| 84 133 195 236 311 }}

| -0.013

| −0.013

| 0.754

| 5.28

|-style="border-top: double;"

|- style="border-top: double;"

| 2.3.5.7.11

| 225/224, 441/440, 1344/1331, 1728/1715

| {{mapping| 84 133 195 236 291 }} (84)

| -0.225

| −0.225

| 1.003

| 7.02

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| 144/143, 225/224, 351/350, 441/440, 975/968

| {{mapping| 84 133 195 236 291 311 }} (84)

| -0.292

| −0.292

| 0.928

| 6.50

|-style="border-top: double;"

|- style="border-top: double;"

| 2.3.5.7.11

| 99/98, 121/120, 176/175, 78732/78125

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=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

! Periods per 8ve

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! ~~Temperaments~~

! Temperament

|-

| 1

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| [[Oquatonic]]

|}

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

== Scales ==

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* [[Orwell]]

** Orwell[9], [[4L 5s]] - 11 8 11 8 11 8 11 8 8

** Orwell[9] ([[4L 5s]]) – 11 8 11 8 11 8 11 8 8

** Orwell[13] - [[9L 4s]] - 8 8 8 3 8 8 3 8 8 3 8 8 3

** Orwell[13] ([[9L 4s]]) – 8 8 8 3 8 8 3 8 8 3 8 8 3

** Orwell[22] - [[13L 9s]]

** Orwell[22] ([[13L 9s]])

** Orwell[31] - [[22L 9s]]

** Orwell[31] ([[22L 9s]])

=== Other ===

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|84}}
+{{EDO intro}}
 == Theory ==
-= 7 × 12, and 84edo shares the [[3/2|perfect fifth]] with [[12edo]], [[tempering out]] the [[Pythagorean comma]] in its [[patent val]]. In the [[5-limit]] it tempers out the [[sensipent comma]]; in the [[7-limit]] [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[support]]ing [[orwell]], [[compton]], and [[sensei]].
+edo shares the [[3/2|perfect fifth]] with [[12edo]], [[tempering out]] the [[Pythagorean comma]] in its [[patent val]]. In the [[5-limit]] it tempers out the [[sensipent comma]]; in the [[7-limit]] [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[support]]ing [[orwell]], [[compton]], and [[sensei]].
-edo is where the orwell temperament takes its name from, since the generator of [[7/6]] is equal to 19 steps of the edo, referencing the [[Wikipedia: Nineteen Eighty-Four|book 1984]]. Orwell in 84edo comes in two varieties – the 84e val {{val| 84 133 195 236 '''290''' }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 '''291''' }} supporting [[newspeak]]. 84edo orwell offers [[mos scale]]s of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale.
+edo is where the orwell temperament takes its name from, since the generator of [[7/6]] is equal to 19 steps of the edo, referencing the [[Wikipedia: Nineteen Eighty-Four|book 1984]]. Orwell in 84edo comes in two varieties—the 84e val {{val| 84 133 195 236 '''290''' }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 '''291''' }} supporting [[newspeak]]. 84edo orwell offers [[mos scale]]s of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale.
 It has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], and [[31/1|31]]. In fact, it is [[consistent]] to the no-11 no-17 [[25-odd-limit]]. In the [[13-limit]] it is the [[optimal patent val]] for the rank-5 temperament tempering out [[144/143]].
@@ Line 18: / Line 18: @@
 == Intervals ==
 {| class="wikitable"
+|-
 ! #
 ! Cents
@@ Line 618: / Line 619: @@
 | D
 |}
-<nowiki>*</nowiki> as a 2.3.5.7.13.19.23-subgroup temperament
+<nowiki />* As a 2.3.5.7.13.19.23-subgroup temperament
 == Notation ==
 === 4L 5s (gramitonic) notation ===
-The notation of Orwell[9]. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & "amp" and @ "at".
+This notation is based on Orwell[9]. Notes are denoted as {{nowrap|LsLsLsLss {{=}} JKLMNOPQRJ}}, and raising and lowering by a chroma ({{nowrap|L − s}}), 3 steps in this instance, is denoted by &amp;&nbsp;("amp") and @&nbsp;("at").
 {| class="wikitable center-1 right-2 center-3"
@@ Line 743: / Line 744: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 769: / Line 771: @@
 | 225/224, 351/350, 640/637, 1701/1690
 | {{mapping| 84 133 195 236 311 }}
-| -0.013
+| −0.013
 | 0.754
 | 5.28
-|-style="border-top: double;"
+|- style="border-top: double;"
 | 2.3.5.7.11
 | 225/224, 441/440, 1344/1331, 1728/1715
 | {{mapping| 84 133 195 236 291 }} (84)
-| -0.225
+| −0.225
 | 1.003
 | 7.02
@@ Line 783: / Line 785: @@
 | 144/143, 225/224, 351/350, 441/440, 975/968
 | {{mapping| 84 133 195 236 291 311 }} (84)
-| -0.292
+| −0.292
 | 0.928
 | 6.50
-|-style="border-top: double;"
+|- style="border-top: double;"
 | 2.3.5.7.11
 | 99/98, 121/120, 176/175, 78732/78125
@@ Line 804: / Line 806: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-! Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
-! Temperaments
+! Temperament
 |-
 | 1
@@ Line 888: / Line 892: @@
 | [[Oquatonic]]
 |}
-<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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 896: / Line 900: @@
 * [[Orwell]]
-** Orwell[9], [[4L 5s]] - 11 8 11 8 11 8 11 8 8
+** Orwell[9] ([[4L 5s]]) – 11 8 11 8 11 8 11 8 8
-** Orwell[13] - [[9L 4s]] - 8 8 8 3 8 8 3 8 8 3 8 8 3
+** Orwell[13] ([[9L 4s]]) – 8 8 8 3 8 8 3 8 8 3 8 8 3
-** Orwell[22] - [[13L 9s]]
+** Orwell[22] ([[13L 9s]])
-** Orwell[31] - [[22L 9s]]
+** Orwell[31] ([[22L 9s]])
 === Other ===