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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|84}}
{{ED intro}}


== Theory ==
== Theory ==
In the [[13-limit]] it is the [[optimal patent val]] for the rank five temperament tempering out [[144/143]].
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]]. In the [[13-limit]] it is the [[optimal patent val]] for the rank-5 temperament tempering out [[144/143]].  


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]]. From a regular temperament perspective, 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''' }} representing [[newspeak]]. 84edo orwell offers mosses 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.
 
Being a small multiple of 12, 84et tempers out the [[Pythagorean comma]], thus supporting the period-12 temperament [[compton]]. Being a small multiple of 28, it tempers out the [[Oquatonic|oquatonic comma]], which maps 5/4 to 9\28.


=== Prime harmonics ===
=== Prime harmonics ===
Line 14: Line 12:


=== Subsets and supersets ===
=== Subsets and supersets ===
84edo is a [[largely composite]] number. Since 84 factors as {{factorization|84}}, 84edo has subset edos {{EDOs|1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42}}.
84 is a [[largely composite]] number. Since 84 factors as {{factorization|84}}, 84edo has subset edos {{EDOs| 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 }}. Being a small multiple of 28, it tempers out the [[oquatonic|oquatonic comma]], which maps 5/4 to 9\28.


== Table of intervals ==
== Intervals ==
For this table, the notation of Orwell[9] from the [[4L 5s]] page is taken. 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 a "at" (the symbol "@" unlike in the 4L 5s page cannot be used because of technical details).
{| class="wikitable center-1 right-2"
{| class="wikitable"
|-
|+Table of 84edo intervals
! #
! Degree
! Cents
! Size (Cents)
! Approximate ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[4L 5s|4L 5s Notation]]
! Associated ratio
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| Perfect 1sn
| Perfect 1sn
| P1
| P1
| D
| D
| Perfect 1sn
| P1
| J
| 1/1 exact
|-
|-
| 1
| 1
| 14.286
| 14.3
| ''[[81/80]]'', [[105/104]], [[126/125]], [[169/168]], [[196/195]]
| Up 1sn
| Up 1sn
| ^1
| ^1
| ^D
| ^D
| Up 1sn
| ^1
| J^
|
|-
|-
| 2
| 2
| 28.571
| 28.6
| [[50/49]], [[64/63]], [[65/64]], ''[[91/90]]''
| Dup 1sn
| Dup 1sn
| ^^1
| ^^1
| ^^D
| ^^D
| Downaug 1sn
| vA1
| Jv&
|
|-
|-
| 3
| 3
| 42.857
| 42.9
| [[36/35]], [[40/39]], [[46/45]], [[49/48]]
| Trup 1sn
| Trup 1sn
| ^^^1
| ^^^1
| ^^^D
| ^^^D
| Aug 1sn
| A1
| J&
|
|-
|-
| 4
| 4
| 57.143
| 57.1
| ''[[27/26]]''
| Trudminor 2nd
| Trudminor 2nd
| vvvm2
| vvvm2
| vvvEb
| vvvEb
| Upaug 1sn, Downdim 2nd
| ^A1, vd2
| J^&, Kvaa
|
|-
|-
| 5
| 5
| 71.429
| 71.4
| [[24/23]], [[25/24]], [[26/25]], ''[[28/27]]''
| Dudminor 2nd
| Dudminor 2nd
| vvm2
| vvm2
| vvEb
| vvEb
| Dim 2nd
| d2
| Kaa
|
|-
|-
| 6
| 6
| 85.714
| 85.7
| [[20/19]], [[21/20]]
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
| vEb
| vEb
| Updim 2nd
| ^d2
| K^aa
|
|-
|-
| 7
| 7
| 100.000
| 100.0
| [[19/18]]
| Minor 2nd
| Minor 2nd
| m2
| m2
| Eb
| Eb
| Downminor 2nd
| vm2
| Kva
|
|-
|-
| 8
| 8
| 114.286
| 114.3
| [[15/14]], [[16/15]]
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
| ^Eb
| ^Eb
| Minor 2nd
| m2
| Ka
|
|-
|-
| 9
| 9
| 128.571
| 128.6
| [[14/13]]
| Dupminor 2nd
| Dupminor 2nd
| ^^m2
| ^^m2
| ^^Eb
| ^^Eb
| Upminor 2nd
| ^m2
| K^a
|
|-
|-
| 10
| 10
| 142.857
| 142.9
| [[13/12]]
| Trupminor 2nd
| Trupminor 2nd
| ^^^m2
| ^^^m2
| ^^^Eb
| ^^^Eb
| Downmajor 2nd
| vM2
| Kv
|
|-
|-
| 11
| 11
| 157.143
| 157.1
| [[23/21]]
| Trudmajor 2nd
| Trudmajor 2nd
| vvvM2
| vvvM2
| vvvE
| vvvE
| Major 2nd
| M2
| K
|
|-
|-
| 12
| 12
| 171.429
| 171.4
| [[21/19]]
| Dudmajor 2nd
| Dudmajor 2nd
| vvM2
| vvM2
| vvE
| vvE
| Upmajor 2nd
| ^M2
| K^
|
|-
|-
| 13
| 13
| 185.714
| 185.7
| [[10/9]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
| vE
| vE
| Downaug 2nd
| vA2
| Kv&
|
|-
|-
| 14
| 14
| 200.000
| 200.0
| [[9/8]]
| Major 2nd
| Major 2nd
| M2
| M2
| E
| E
| Aug 2nd
| A2
| K&
|
|-
|-
| 15
| 15
| 214.286
| 214.3
| [[26/23]]
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
| ^E
| ^E
| Upaug 2nd, Downdim 3rd
| ^A2, vd3
| K^&, Lva
|
|-
|-
| 16
| 16
| 228.571
| 228.6
| [[8/7]]
| Dupmajor 2nd
| Dupmajor 2nd
| ^^M2
| ^^M2
| ^^E
| ^^E
| Dim 3rd
| d3
| La
|
|-
|-
| 17
| 17
| 242.857
| 242.9
| [[15/13]], [[23/20]]
| Trupmajor 2nd
| Trupmajor 2nd
| ^^^M2
| ^^^M2
| ^^^E
| ^^^E
| Updim 3rd
| ^d3
| L^a
|
|-
|-
| 18
| 18
| 257.143
| 257.1
| [[52/45]]
| Trudminor 3rd
| Trudminor 3rd
| vvvm3
| vvvm3
| vvvF
| vvvF
| Down 3rd
| v3
| Lv
|
|-
|-
| 19
| 19
| 271.429
| 271.4
| [[7/6]]
| Dudminor 3rd
| Dudminor 3rd
| vvm2
| vvm2
| vvF
| vvF
| Perfect 3rd
| P3
| L
| [[7/6]]
|-
|-
| 20
| 20
| 285.714
| 285.7
| [[45/38]], [[46/39]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
| vF
| vF
| Up 3rd
| ^3
| L^
|
|-
|-
| 21
| 21
| 300.000
| 300.0
| [[19/16]], [[25/21]], [[32/27]]
| Minor 3rd
| Minor 3rd
| m3
| m3
| F
| F
| Downaug 3rd
| vA3
| Lv&
|
|-
|-
| 22
| 22
| 314.286
| 314.3
| [[6/5]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
| ^F
| ^F
| Aug 3rd
| A3
| L&
|
|-
|-
| 23
| 23
| 328.571
| 328.6
| [[23/19]]
| Dupminor 3rd
| Dupminor 3rd
| ^^m3
| ^^m3
| ^^F
| ^^F
| Upaug 3rd, Downdim 4th
| ^A3, vd4
| L^&, Mvaa
|
|-
|-
| 24
| 24
| 342.857
| 342.9
| [[28/23]], [[39/32]]
| Trupminor 3rd
| Trupminor 3rd
| ^^^m3
| ^^^m3
| ^^^F
| ^^^F
| Dim 4th
| d4
| Maa
|
|-
|-
| 25
| 25
| 357.143
| 357.1
| [[16/13]]
| Trudmajor 3rd
| Trudmajor 3rd
| vvvM3
| vvvM3
| vvvF#
| vvvF#
| Updim 4th
| ^d4
| M^aa
|
|-
|-
| 26
| 26
| 371.429
| 371.4
| [[26/21]]
| Dudmajor 3rd
| Dudmajor 3rd
| vvM3
| vvM3
| vvF#
| vvF#
| Downminor 4th
| vm4
| Mva
|
|-
|-
| 27
| 27
| 385.714
| 385.7
| [[5/4]]
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
| vF#
| vF#
| Minor 4th
| m4
| Ma
|
|-
|-
| 28
| 28
| 400.000
| 400.0
| [[24/19]]
| Major 3rd
| Major 3rd
| M3
| M3
| F#
| F#
| Upminor 4th
| ^m4
| M^a
|
|-
|-
| 29
| 29
| 414.286
| 414.3
| [[19/15]]
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
| ^F#
| ^F#
| Downmajor 4th
| vM4
| Mv
|
|-
|-
| 30
| 30
| 428.571
| 428.6
| [[9/7]], [[23/18]], [[32/25]]
| Dupmajor 3rd
| Dupmajor 3rd
| ^^M3
| ^^M3
| ^^F#
| ^^F#
| Major 4th
| M4
| M
|
|-
|-
| 31
| 31
| 442.857
| 442.9
| [[84/65]]
| Trupmajor 3rd
| Trupmajor 3rd
| ^^^M3
| ^^^M3
| ^^^F#
| ^^^F#
| Upmajor 4th
| ^M4
| M^
|
|-
|-
| 32
| 32
| 457.143
| 457.1
| [[13/10]], [[30/23]]
| Trud 4th
| Trud 4th
| vvv4
| vvv4
| vvvG
| vvvG
| Downaug 4th
| vA4
| Mv&
|
|-
|-
| 33
| 33
| 471.429
| 471.4
| [[21/16]]
| Dud 4th
| Dud 4th
| vv4
| vv4
| vvG
| vvG
| Aug 4th
| A4
| M&
|
|-
|-
| 34
| 34
| 485.714
| 485.7
| [[65/49]]
| Down 4th
| Down 4th
| v4
| v4
| vG
| vG
| Downminor 5th
| vm5
| Nva
|
|-
|-
| 35
| 35
| 500.000
| 500.0
| [[4/3]]
| Perfect 4th
| Perfect 4th
| P4
| P4
| G
| G
| Minor 5th
| m5
| Na
|
|-
|-
| 36
| 36
| 514.286
| 514.3
| [[27/20]]
| Up 4th
| Up 4th
| ^4
| ^4
| ^G
| ^G
| Upminor 5th
| ^m5
| N^a
|
|-
|-
| 37
| 37
| 528.571
| 528.6
| [[19/14]]
| Dup 4th
| Dup 4th
| ^^4
| ^^4
| ^^G
| ^^G
| Downmajor 5th
| vM5
| Nv
|
|-
|-
| 38
| 38
| 542.857
| 542.9
| [[26/19]]
| Trup 4th
| Trup 4th
| ^^^4
| ^^^4
| ^^^G
| ^^^G
| Major 5th
| M5
| N
| [[11/8]] in the 84b val
|-
|-
| 39
| 39
| 557.143
| 557.1
| [[18/13]]
| Trudaug 4th
| Trudaug 4th
| vvvA4
| vvvA4
| vvvG#
| vvvG#
| Upmajor 5th
| ^M5
| N^
|
|-
|-
| 40
| 40
| 571.429
| 571.4
| [[25/18]], [[32/23]]
| Dudaug 4th
| Dudaug 4th
| vvA4
| vvA4
| vvG#
| vvG#
| Downaug 5th
| vA5
| Nv&
|
|-
|-
| 41
| 41
| 585.714
| 585.7
| [[7/5]]
| Downaug 4th
| Downaug 4th
| vA4
| vA4
| vG#
| vG#
| Aug 5th
| A5
| N&
|
|-
|-
| 42
| 42
| 600.000
| 600.0
| [[27/19]], [[38/27]]
| Aug 4th, Dim 5th
| Aug 4th, Dim 5th
| A4, d5
| A4, d5
| G#, Ab
| G#, Ab
| Upaug 5th, Downdim 6th
| ^A5, vd6
| N^&, Ovaa
|
|-
|-
| 43
| 43
| 614.286
| 614.3
| [[10/7]]
| Updim 5th
| Updim 5th
| ^d5
| ^d5
| ^Ab
| ^Ab
| Dim 6th
| d6
| Oaa
|
|-
|-
| 44
| 44
| 628.571
| 628.6
| [[23/16]], [[36/25]]
| Dupdim 5th
| Dupdim 5th
| ^^d5
| ^^d5
| ^^Ab
| ^^Ab
| Updim 6th
| ^d6
| O^aa
|
|-
|-
| 45
| 45
| 642.857
| 642.9
| [[13/9]]
| Trupdim 5th
| Trupdim 5th
| ^^^d5
| ^^^d5
| ^^^Ab
| ^^^Ab
| Downminor 6th
| vm6
| Ova
|
|-
|-
| 46
| 46
| 657.143
| 657.1
| [[19/13]]
| Trud 5th
| Trud 5th
| vvv5
| vvv5
| vvvA
| vvvA
| Minor 6th
| m6
| Oa
|
|-
|-
| 47
| 47
| 671.429
| 671.4
| [[28/19]]
| Dud 5th
| Dud 5th
| vv5
| vv5
| vvA
| vvA
| Upminor 6th
| ^m6
| O^a
|
|-
|-
| 48
| 48
| 685.714
| 685.7
| [[40/27]]
| Down 5th
| Down 5th
| v5
| v5
| vA
| vA
| Downmajor 6th
| vM6
| Ov
|
|-
|-
| 49
| 49
| 700.000
| 700.0
| [[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
| A
| A
| Major 6th
| M6
| O
| [[3/2]]
|-
|-
| 50
| 50
| 714.286
| 714.3
| [[98/65]]
| Up 5th
| Up 5th
| ^5
| ^5
| ^A
| ^A
| Upmajor 6th
| ^M6
| O^
|
|-
|-
| 51
| 51
| 728.571
| 728.6
| [[32/21]]
| Dup 5th
| Dup 5th
| ^^5
| ^^5
| ^^A
| ^^A
| Dim 7th
| d7
| Paa
|
|-
|-
| 52
| 52
| 742.857
| 742.9
| [[20/13]], [[23/15]]
| Trup 5th
| Trup 5th
| ^^^5
| ^^^5
| ^^^A
| ^^^A
| Aug 6th
| A6
| O&
|
|-
|-
| 53
| 53
| 757.143
| 757.1
| [[65/42]]
| Trudminor 6th
| Trudminor 6th
| vvvm6
| vvvm6
| vvvBb
| vvvBb
| Downminor 7th
| vm7
| Pva
|
|-
|-
| 54
| 54
| 771.429
| 771.4
| [[14/9]], [[25/16]], [[36/23]]
| Dudminor 6th
| Dudminor 6th
| vvm6
| vvm6
| vvBb
| vvBb
| Minor 7th
| m7
| Pa
|
|-
|-
| 55
| 55
| 785.714
| 785.7
| [[30/19]]
| Downminor 6th
| Downminor 6th
| vm6
| vm6
| vBb
| vBb
| Upminor 7th
| ^m7
| P^a
|
|-
|-
| 56
| 56
| 800.000
| 800.0
| [[19/12]]
| Minor 6th
| Minor 6th
| m6
| m6
| Bb
| Bb
| Downmajor 7th
| vM7
| Pv
|
|-
|-
| 57
| 57
| 814.286
| 814.3
| [[8/5]]
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
| ^Bb
| ^Bb
| Major 7th
| M7
| P
| [[5/3]]
|-
|-
| 58
| 58
| 828.571
| 828.6
| [[21/13]]
| Dupminor 6th
| Dupminor 6th
| ^^m6
| ^^m6
| ^^Bb
| ^^Bb
| Upmajor 7th
| ^M7
| P^
|
|-
|-
| 59
| 59
| 842.857
| 842.9
| [[13/8]]
| Trupminor 6th
| Trupminor 6th
| ^^^m6
| ^^^m6
| ^^^Bb
| ^^^Bb
| Downaug 7th
| vA7
| Pv&
|
|-
|-
| 60
| 60
| 857.143
| 857.1
| [[23/14]], [[64/39]]
| Trudmajor 6th
| Trudmajor 6th
| vvvM6
| vvvM6
| vvvB
| vvvB
| Aug 7th
| A7
| P&
| [[105/64]]
|-
|-
| 61
| 61
| 871.429
| 871.4
| [[38/23]]
| Dudmajor 6th
| Dudmajor 6th
| vvM6
| vvM6
| vvB
| vvB
| Upaug 7th, Downdim 8th
| ^A7, vd8
| P^&, Qvaa
|
|-
|-
| 62
| 62
| 885.714
| 885.7
| [[5/3]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
| vB
| vB
| Dim 8th
| d8
| Qaa
|
|-
|-
| 63
| 63
| 900.000
| 900.0
| [[32/19]], [[27/16]], [[42/25]]
| Major 6th
| Major 6th
| M6
| M6
| B
| B
| Updim 8th
| ^d8
| Q^aa
|
|-
|-
| 64
| 64
| 914.286
| 914.3
| [[39/23]], [[76/45]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
| ^B
| ^B
| Down 8th
| v8
| Qva
|
|-
|-
| 65
| 65
| 928.571
| 928.6
| [[12/7]]
| Dupmajor 6th
| Dupmajor 6th
| ^^M6
| ^^M6
| ^^B
| ^^B
| Perfect 8th
| P8
| Qa
|
|-
|-
| 66
| 66
| 942.857
| 942.9
| [[45/26]]
| Trupmajor 6th
| Trupmajor 6th
| ^^^M6
| ^^^M6
| ^^^B
| ^^^B
| Up 8th
| ^8
| Q^a
|
|-
|-
| 67
| 67
| 957.143
| 957.1
| [[26/15]], [[40/23]]
| Trudminor 7th
| Trudminor 7th
| vvvm7
| vvvm7
| vvvC
| vvvC
| Downaug 8th
| vA8
| Qv
|
|-
|-
| 68
| 68
| 971.429
| 971.4
| [[7/4]]
| Dudminor 7th
| Dudminor 7th
| vvm7
| vvm7
| vvC
| vvC
| Aug 8th
| A8
| Q
|
|-
|-
| 69
| 69
| 985.714
| 985.7
| [[23/13]]
| Downminor 7th
| Downminor 7th
| vm7
| vm7
| vC
| vC
| Upaug 8th, Downdim 9th
| ^A8, vd9
| Q^, Rvaa
|
|-
|-
| 70
| 70
| 1000.000
| 1000.0
| [[16/9]]
| Minor 7th
| Minor 7th
| m7
| m7
| C
| C
| Dim 9th
| d9
| Raa
|
|-
|-
| 71
| 71
| 1014.286
| 1014.3
| [[9/5]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
| ^C
| ^C
| Updim 9th
| ^d9
| R^aa
|
|-
|-
| 72
| 72
| 1028.571
| 1028.6
| [[38/21]]
| Dupminor 7th
| Dupminor 7th
| ^^m7
| ^^m7
| ^^C
| ^^C
| Downminor 9th
| vm9
| Rva
|
|-
|-
| 73
| 73
| 1042.857
| 1042.9
| [[42/23]]
| Trupminor 7th
| Trupminor 7th
| ^^^m7
| ^^^m7
| ^^^C
| ^^^C
| Minor 9th
| m9
| Ra
|
|-
|-
| 74
| 74
| 1057.143
| 1057.1
| [[24/13]]
| Trudmajor 7th
| Trudmajor 7th
| vvvM7
| vvvM7
| vvvC#
| vvvC#
| Upminor 9th
| ^m9
| R^a
|
|-
|-
| 75
| 75
| 1071.429
| 1071.4
| [[13/7]]
| Dudmajor 7th
| Dudmajor 7th
| vvM7
| vvM7
| vvC#
| vvC#
| Downmajor 9th
| vM9
| Rv
|
|-
|-
| 76
| 76
| 1085.714
| 1085.7
| [[15/8]], [[28/15]]
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
| vC#
| vC#
| Major 9th
| M9
| R
|
|-
|-
| 77
| 77
| 1100.000
| 1100.0
| [[36/19]]
| Major 7th
| Major 7th
| M7
| M7
| C#
| C#
| Upmajor 9th
| ^M9
| R^
|
|-
|-
|-
| 78
| 78
| 1114.286
| 1114.3
| [[19/10]], [[40/21]]
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
| ^C#
| ^C#
| Downaug 9th
| vA9
| Rv&
|
|-
|-
| 79
| 79
| 1128.571
| 1128.6
| [[23/12]], [[25/13]], ''[[27/14]]'', [[48/25]]
| Dupmajor 7th
| Dupmajor 7th
| ^^M7
| ^^M7
| ^^C#
| ^^C#
| Aug 9th
| A9
| R&
|
|-
|-
| 80
| 80
| 1142.857
| 1142.9
| ''[[52/27]]''
| Trupmajor 7th
| Trupmajor 7th
| ^^^M7
| ^^^M7
| ^^^C#
| ^^^C#
| Upaug 9th, Downdim 10th
| ^A9, vd10
| R^&, Jva
|
|-
|-
| 81
| 81
| 1157.143
| 1157.1
| [[35/18]], [[39/20]], [[96/49]]
| Trud 8ve
| Trud 8ve
| vvv8
| vvv8
| vvvD
| vvvD
| Dim 10th
| d10
| Ja
|
|-
|-
| 82
| 82
| 1171.429
| 1171.4
| [[45/23]], [[49/25]], [[63/32]], [[128/65]], ''[[180/91]]''
| Dud 8ve
| Dud 8ve
| vv8
| vv8
| vvD
| vvD
| Updim 10th
| ^d10
| J^a
|
|-
|-
| 83
| 83
| 1185.714
| 1185.7
| Down 8ve
| [[125/63]], ''[[160/81]]'', [[195/98]], [[336/169]]
| Down 8ve
| v8
| v8
| vD
| vD
| Down 10th
| v10
| Jv
|
|-
|-
| 84
| 84
| 1200.000
| 1200.0
| [[2/1]]
| Perfect 8ve
| Perfect 8ve
| P8
| P8
| D
| D
| Perfect 10th
|}
| P10
<nowiki/>* As a 2.3.5.7.13.19.23-subgroup temperament
 
== Notation ==
=== Ups and downs notation ===
 
84edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Ups and downs sharpness|84}}
 
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7|84}}
 
=== 4L 5s (gramitonic) notation ===
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"
|-
! #
! Cents
! Note
! Name
! Associated Ratio
|-
| 0
| 0.0
| J
| J
| [[2/1]] exact
| Perfect 0-gramstep
| 1/1
|-
| 8
| 114.3
| K@
| Minor 1-gramstep
| 15/14~16/15
|-
| 11
| 157.1
| K
| Major 1-gramstep
| 11/10~12/11
|-
| 16
| 228.6
| L@
| Diminished 2-gramstep
| 8/7
|-
| 19
| 271.4
| L
| Perfect 2-gramstep
| 7/6
|-
| 27
| 385.7
| M@
| Minor 3-gramstep
| 5/4
|-
| 30
| 428.6
| M
| Major 3-gramstep
| 9/7
|-
| 35
| 500.0
| N@
| Minor 4-gramstep
| 4/3
|-
| 38
| 542.9
| N
| Major 4-gramstep
| 11/8~15/11
|-
| 46
| 657.1
| O@
| Minor 5-gramstep
| 16/11~22/15
|-
| 49
| 700.0
| O
| Major 5-gramstep
| 3/2
|-
| 54
| 771.4
| P@
| Minor 6-gramstep
| 14/9
|-
|-
| 57
| 814.3
| P
| Major 6-gramstep
| 8/5
|-
| 65
| 928.6
| Q@
| Perfect 7-gramstep
| 12/7
|-
| 68
| 971.4
| Q
| Augmented 7-gramstep
| 7/4
|-
| 73
| 1042.9
| R@
| Minor 8-gramstep
| 11/6~20/11
|-
| 76
| 1085.7
| R
| Major 8-gramstep
| 15/8~28/15
|-
| 84
| 1200.0
| J
| Perfect 9-gramstep
| 2/1
|}
|}
== Approximation to JI ==
=== 15-odd-limit intervals ===
{{Q-odd-limit intervals|84}}
=== Higher-limit JI ===
84edo has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], [[31/1|31]], 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit, with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics. In fact, it is [[consistent]] in the no-11 no-17 no-27 no-37 no-47 no-49 no-51 no-55 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~0.13{{cent}} (off by ~7.3{{cent}}), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too when supported by context. The only missing primes are thus 17, 37, 47, 67, 71, 79 and 83, which except for 17 are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! 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]] (¢)
! [[TE error|Absolute]] (¢)
Line 892: Line 769:
| 2.3.5
| 2.3.5
| 78732/78125, 531441/524288
| 78732/78125, 531441/524288
| {{val| 84 133 195 }}
| {{Mapping| 84 133 195 }}
| +0.498
| +0.498
| 0.531
| 0.531
Line 899: Line 776:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1728/1715, 78732/78125
| 225/224, 1728/1715, 78732/78125
| {{val| 84 133 195 236 }}
| {{Mapping| 84 133 195 236 }}
| +0.141
| +0.141
| 0.769
| 0.769
| 5.39
| 5.39
|-
|-
| 2.3.5.7.13
| 225/224, 351/350, 640/637, 1701/1690
| {{Mapping| 84 133 195 236 311 }}
| −0.013
| 0.754
| 5.28
|- style="border-top: double;"
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 441/440, 1344/1331, 1728/1715
| 225/224, 441/440, 1344/1331, 1728/1715
| {{val| 84 133 195 236 291 }} (84)
| {{Mapping| 84 133 195 236 291 }} (84)
| -0.225
| −0.225
| 1.003
| 1.003
| 7.02
| 7.02
|-
|-
| 2.3.5.7.11.13
| 144/143, 225/224, 351/350, 441/440, 975/968
| {{Mapping| 84 133 195 236 291 311 }} (84)
| −0.292
| 0.928
| 6.50
|- style="border-top: double;"
| 2.3.5.7.11
| 2.3.5.7.11
| 99/98, 121/120, 176/175, 78732/78125
| 99/98, 121/120, 176/175, 78732/78125
| {{val|84 133 195 236 290}} (84e)
| {{Mapping| 84 133 195 236 290 }} (84e)
| +0.601
| +0.601
| 1.151
| 1.151
| 8.05
| 8.05
|-
| 2.3.5.7.11.13
| 99/98, 121/120, 176/175, 275/273, 1701/1690
| {{Mapping| 84 133 195 236 290 311 }} (84e)
| +0.396
| 1.146
| 8.02
|}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| rowspan="2" | 1
| 1
| rowspan="2" | 19\84
| 19\84
| rowspan="2" | 271.43
| 271.4
| rowspan="2" | 7/6
| 7/6
| [[Orwell]] (84e)
| [[Orwell]] (84e) / [[newspeak]] (84)
|-
| [[Newspeak]] (84p)
|-
|-
| 1
| 1
| 25\84
| 25\84
| 357.14
| 357.1
| 768/625
| 768/625
| [[Dodifo]]
| [[Dodifo]]
Line 943: Line 841:
| 1
| 1
| 27\84
| 27\84
| 385.71
| 385.7
| 5/4
| 5/4
| [[Mutt]]
| [[Mutt]]
Line 949: Line 847:
| 1
| 1
| 31\84
| 31\84
| 442.86
| 442.9
| 125/81
| 162/125
| [[Sensei]]
| [[Sensei]]
|-
|-
| 1
| 1
| 41\84
| 41\84
| 585.71
| 585.7
| 7/5
| 7/5
| [[Merman]]
| [[Merman]]
Line 961: Line 859:
| 2
| 2
| 5\84
| 5\84
| 71.43
| 71.4
| 25/24
| 25/24
| [[Narayana]]
| [[Narayana]]
Line 967: Line 865:
| 2
| 2
| 11\84
| 11\84
| 157.14
| 157.1
| 35/32
| 35/32
| [[Bison]]
| [[Bison]]
Line 973: Line 871:
| 2
| 2
| 13\84
| 13\84
| 185.71
| 185.7
| 10/9
| 10/9
| [[Secant]]
| [[Secant]]
Line 979: Line 877:
| 3
| 3
| 11\84
| 11\84
| 157.14
| 157.1
| 35/32
| 35/32
| [[Nessafof]]
| [[Nessafof]]
|-
| 6
| 5\84
| 71.4
| 25/24
| [[Trivish]]
|-
|-
| 7
| 7
| 5\84
| 5\84
| 500.00<br>(14.29)
| 14.3
| 4/3<br>(81/80)
| 81/80
| [[Absurdity]]
| [[Absurdity]]
|-
|-
| 12
| 12
| 27\84<br>(1\84)
| 1\84
| 385.71<br>(14.29)
| 14.3
| 5/4<br>(126/125)
| 126/125
| [[Compton]]
| [[Compton]]
|-
|-
|21
| 12
|41\84<br>(1\84)
| 2\84
|585.71<br>(14.29)
| 28.6
|91875/65536<br>(126/125)
| 64/63
|[[Akjayland]]
| [[Catler]] (84c)
|-
| 21
| 1\84)
| 14.3
| 126/125
| [[Akjayland]]
|-
|-
| 28
| 28
| 49\84<br>(1\84)
| 1\84
| 500.00<br>(14.29)
| 14.3
| 4/3<br>(105/104)
| 105/104
| [[Oquatonic]]
| [[Oquatonic]]
|}
|}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Scales ==
== Scales ==
=== MOS ===
=== MOS ===
Brightest mode is listed.
Brightest mode is listed.


* [[Orwell]]
* [[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 ===
=== Other ===
* [[5- to 10-tone scales in 84edo]]
* [[5- to 10-tone scales in 84edo]]
* [[Maeve Gutierrez|Gutierrez Moonglade scale]]
== Instruments ==
If you have a precise enough tuner and stable enough instruments, 84edo can be played using 7 instruments tuned a 14th of a tone apart.
You could also try the [[Lumatone mapping for 84edo]]


== Music ==
== Music ==
Line 1,028: Line 944:
* ''Two5'' for tenor trombone and piano (1991) [https://youtu.be/YOtQZIqfY1w Fulkerson &amp; Denyer recording (YouTube)]
* ''Two5'' for tenor trombone and piano (1991) [https://youtu.be/YOtQZIqfY1w Fulkerson &amp; Denyer recording (YouTube)]
* ''Two6'' for violin and piano (1992) [https://youtu.be/XkX37zH6AbU Haar &amp; Snijders recording (YouTube)]
* ''Two6'' for violin and piano (1992) [https://youtu.be/XkX37zH6AbU Haar &amp; Snijders recording (YouTube)]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/Sqkxrmwggr0 ''microtonal improvisation in 84edo''] (2025)
* [https://www.youtube.com/shorts/Qu6UIA2NmmQ ''84edo groove''] (2026)


; [[Eliora]]
; [[Eliora]]
Retrieved from "https://en.xen.wiki/w/84edo"