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84edo: Difference between revisions

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→Rank-2 temperaments: While both 162/125 and 125/81 could be generators for Sensei, 31\84 is the former, which is also what is listed in the Sensi temperament entry.
 
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''84edo'' divides the [[Octave|octave]] into 84 equal parts of size 14.286 [[cent|cent]]s each and it is the highest edo where the size of 3 has a common factor with its cardinality. It makes for an excellent orwell tuning and also a good one for compton, and the 84e val, <84 133 195 236 290|, is almost identical to the 11-limit POTE tuning for orwell. In the [[13-limit|13-limit]] it is the [[Optimal_patent_val|optimal patent val]] for the rank five temperament tempering out 144/143.
{{Infobox ET}}
{{ED intro}}


[[5-limit|5-limit]] commas: 78732/78125, 531441/524288, 2109375/2097152
== Theory ==
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]].


[[7-limit|7-limit]] commas: 225/224, 1728/1715, 2430/2401, 6144/6125
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.


[[11-limit|11-limit]] commas: 441/440, 1344/1331, 1375/1372
=== Prime harmonics ===
{{Harmonics in equal|84|columns=12}}
{{Harmonics in equal|84|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 84edo (continued)}}


84e: 99/98, 121/120, 176/175, 385/384, 540/539, 5632/5625
=== Subsets and supersets ===
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.


[[13-limit|13-limit]] commas: 144/143, 351/350, 364/363, 625/625
== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios*
! colspan="3" | [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| Perfect 1sn
| P1
| D
|-
| 1
| 14.3
| ''[[81/80]]'', [[105/104]], [[126/125]], [[169/168]], [[196/195]]
| Up 1sn
| ^1
| ^D
|-
| 2
| 28.6
| [[50/49]], [[64/63]], [[65/64]], ''[[91/90]]''
| Dup 1sn
| ^^1
| ^^D
|-
| 3
| 42.9
| [[36/35]], [[40/39]], [[46/45]], [[49/48]]
| Trup 1sn
| ^^^1
| ^^^D
|-
| 4
| 57.1
| ''[[27/26]]''
| Trudminor 2nd
| vvvm2
| vvvEb
|-
| 5
| 71.4
| [[24/23]], [[25/24]], [[26/25]], ''[[28/27]]''
| Dudminor 2nd
| vvm2
| vvEb
|-
| 6
| 85.7
| [[20/19]], [[21/20]]
| Downminor 2nd
| vm2
| vEb
|-
| 7
| 100.0
| [[19/18]]
| Minor 2nd
| m2
| Eb
|-
| 8
| 114.3
| [[15/14]], [[16/15]]
| Upminor 2nd
| ^m2
| ^Eb
|-
| 9
| 128.6
| [[14/13]]
| Dupminor 2nd
| ^^m2
| ^^Eb
|-
| 10
| 142.9
| [[13/12]]
| Trupminor 2nd
| ^^^m2
| ^^^Eb
|-
| 11
| 157.1
| [[23/21]]
| Trudmajor 2nd
| vvvM2
| vvvE
|-
| 12
| 171.4
| [[21/19]]
| Dudmajor 2nd
| vvM2
| vvE
|-
| 13
| 185.7
| [[10/9]]
| Downmajor 2nd
| vM2
| vE
|-
| 14
| 200.0
| [[9/8]]
| Major 2nd
| M2
| E
|-
| 15
| 214.3
| [[26/23]]
| Upmajor 2nd
| ^M2
| ^E
|-
| 16
| 228.6
| [[8/7]]
| Dupmajor 2nd
| ^^M2
| ^^E
|-
| 17
| 242.9
| [[15/13]], [[23/20]]
| Trupmajor 2nd
| ^^^M2
| ^^^E
|-
| 18
| 257.1
| [[52/45]]
| Trudminor 3rd
| vvvm3
| vvvF
|-
| 19
| 271.4
| [[7/6]]
| Dudminor 3rd
| vvm2
| vvF
|-
| 20
| 285.7
| [[45/38]], [[46/39]]
| Downminor 3rd
| vm3
| vF
|-
| 21
| 300.0
| [[19/16]], [[25/21]], [[32/27]]
| Minor 3rd
| m3
| F
|-
| 22
| 314.3
| [[6/5]]
| Upminor 3rd
| ^m3
| ^F
|-
| 23
| 328.6
| [[23/19]]
| Dupminor 3rd
| ^^m3
| ^^F
|-
| 24
| 342.9
| [[28/23]], [[39/32]]
| Trupminor 3rd
| ^^^m3
| ^^^F
|-
| 25
| 357.1
| [[16/13]]
| Trudmajor 3rd
| vvvM3
| vvvF#
|-
| 26
| 371.4
| [[26/21]]
| Dudmajor 3rd
| vvM3
| vvF#
|-
| 27
| 385.7
| [[5/4]]
| Downmajor 3rd
| vM3
| vF#
|-
| 28
| 400.0
| [[24/19]]
| Major 3rd
| M3
| F#
|-
| 29
| 414.3
| [[19/15]]
| Upmajor 3rd
| ^M3
| ^F#
|-
| 30
| 428.6
| [[9/7]], [[23/18]], [[32/25]]
| Dupmajor 3rd
| ^^M3
| ^^F#
|-
| 31
| 442.9
| [[84/65]]
| Trupmajor 3rd
| ^^^M3
| ^^^F#
|-
| 32
| 457.1
| [[13/10]], [[30/23]]
| Trud 4th
| vvv4
| vvvG
|-
| 33
| 471.4
| [[21/16]]
| Dud 4th
| vv4
| vvG
|-
| 34
| 485.7
| [[65/49]]
| Down 4th
| v4
| vG
|-
| 35
| 500.0
| [[4/3]]
| Perfect 4th
| P4
| G
|-
| 36
| 514.3
| [[27/20]]
| Up 4th
| ^4
| ^G
|-
| 37
| 528.6
| [[19/14]]
| Dup 4th
| ^^4
| ^^G
|-
| 38
| 542.9
| [[26/19]]
| Trup 4th
| ^^^4
| ^^^G
|-
| 39
| 557.1
| [[18/13]]
| Trudaug 4th
| vvvA4
| vvvG#
|-
| 40
| 571.4
| [[25/18]], [[32/23]]
| Dudaug 4th
| vvA4
| vvG#
|-
| 41
| 585.7
| [[7/5]]
| Downaug 4th
| vA4
| vG#
|-
| 42
| 600.0
| [[27/19]], [[38/27]]
| Aug 4th, Dim 5th
| A4, d5
| G#, Ab
|-
| 43
| 614.3
| [[10/7]]
| Updim 5th
| ^d5
| ^Ab
|-
| 44
| 628.6
| [[23/16]], [[36/25]]
| Dupdim 5th
| ^^d5
| ^^Ab
|-
| 45
| 642.9
| [[13/9]]
| Trupdim 5th
| ^^^d5
| ^^^Ab
|-
| 46
| 657.1
| [[19/13]]
| Trud 5th
| vvv5
| vvvA
|-
| 47
| 671.4
| [[28/19]]
| Dud 5th
| vv5
| vvA
|-
| 48
| 685.7
| [[40/27]]
| Down 5th
| v5
| vA
|-
| 49
| 700.0
| [[3/2]]
| Perfect 5th
| P5
| A
|-
| 50
| 714.3
| [[98/65]]
| Up 5th
| ^5
| ^A
|-
| 51
| 728.6
| [[32/21]]
| Dup 5th
| ^^5
| ^^A
|-
| 52
| 742.9
| [[20/13]], [[23/15]]
| Trup 5th
| ^^^5
| ^^^A
|-
| 53
| 757.1
| [[65/42]]
| Trudminor 6th
| vvvm6
| vvvBb
|-
| 54
| 771.4
| [[14/9]], [[25/16]], [[36/23]]
| Dudminor 6th
| vvm6
| vvBb
|-
| 55
| 785.7
| [[30/19]]
| Downminor 6th
| vm6
| vBb
|-
| 56
| 800.0
| [[19/12]]
| Minor 6th
| m6
| Bb
|-
| 57
| 814.3
| [[8/5]]
| Upminor 6th
| ^m6
| ^Bb
|-
| 58
| 828.6
| [[21/13]]
| Dupminor 6th
| ^^m6
| ^^Bb
|-
| 59
| 842.9
| [[13/8]]
| Trupminor 6th
| ^^^m6
| ^^^Bb
|-
| 60
| 857.1
| [[23/14]], [[64/39]]
| Trudmajor 6th
| vvvM6
| vvvB
|-
| 61
| 871.4
| [[38/23]]
| Dudmajor 6th
| vvM6
| vvB
|-
| 62
| 885.7
| [[5/3]]
| Downmajor 6th
| vM6
| vB
|-
| 63
| 900.0
| [[32/19]], [[27/16]], [[42/25]]
| Major 6th
| M6
| B
|-
| 64
| 914.3
| [[39/23]], [[76/45]]
| Upmajor 6th
| ^M6
| ^B
|-
| 65
| 928.6
| [[12/7]]
| Dupmajor 6th
| ^^M6
| ^^B
|-
| 66
| 942.9
| [[45/26]]
| Trupmajor 6th
| ^^^M6
| ^^^B
|-
| 67
| 957.1
| [[26/15]], [[40/23]]
| Trudminor 7th
| vvvm7
| vvvC
|-
| 68
| 971.4
| [[7/4]]
| Dudminor 7th
| vvm7
| vvC
|-
| 69
| 985.7
| [[23/13]]
| Downminor 7th
| vm7
| vC
|-
| 70
| 1000.0
| [[16/9]]
| Minor 7th
| m7
| C
|-
| 71
| 1014.3
| [[9/5]]
| Upminor 7th
| ^m7
| ^C
|-
| 72
| 1028.6
| [[38/21]]
| Dupminor 7th
| ^^m7
| ^^C
|-
| 73
| 1042.9
| [[42/23]]
| Trupminor 7th
| ^^^m7
| ^^^C
|-
| 74
| 1057.1
| [[24/13]]
| Trudmajor 7th
| vvvM7
| vvvC#
|-
| 75
| 1071.4
| [[13/7]]
| Dudmajor 7th
| vvM7
| vvC#
|-
| 76
| 1085.7
| [[15/8]], [[28/15]]
| Downmajor 7th
| vM7
| vC#
|-
| 77
| 1100.0
| [[36/19]]
| Major 7th
| M7
| C#
|-
| 78
| 1114.3
| [[19/10]], [[40/21]]
| Upmajor 7th
| ^M7
| ^C#
|-
| 79
| 1128.6
| [[23/12]], [[25/13]], ''[[27/14]]'', [[48/25]]
| Dupmajor 7th
| ^^M7
| ^^C#
|-
| 80
| 1142.9
| ''[[52/27]]''
| Trupmajor 7th
| ^^^M7
| ^^^C#
|-
| 81
| 1157.1
| [[35/18]], [[39/20]], [[96/49]]
| Trud 8ve
| vvv8
| vvvD
|-
| 82
| 1171.4
| [[45/23]], [[49/25]], [[63/32]], [[128/65]], ''[[180/91]]''
| Dud 8ve
| vv8
| vvD
|-
| 83
| 1185.7
| [[125/63]], ''[[160/81]]'', [[195/98]], [[336/169]]
| Down 8ve
| v8
| vD
|-
| 84
| 1200.0
| [[2/1]]
| Perfect 8ve
| P8
| D
|}
<nowiki/>* As a 2.3.5.7.13.19.23-subgroup temperament


84e: 275/273, 640/637, 351/350, 352/351, 625/624, 1001/1000
== Notation ==
=== Ups and downs notation ===


==Music==
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}}


''<span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 14px;">'''Ten'''</span>''<span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 14px;"> by John Cage, 1991, for chamber ensemble. [http://youtu.be/PE_2Ds_6qGk Ives Ensemble] </span>recording (YouTube).
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7|84}}


''<span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 14px;">'''Two4'''</span>''<span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 14px;"> by John Cage, 1991, for violin and piano or shō. [http://youtu.be/sLOrpd5onCs Harr &amp; Miyata] recording (YouTube).</span>
=== 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").  


''<span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 14px;">'''Two5'''</span>''<span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 14px;"> by John Cage, 1991, for tenor trombone and piano. [http://youtu.be/YOtQZIqfY1w Fulkerson &amp; Denyer] recording (YouTube).</span>
{| class="wikitable center-1 right-2 center-3"
|-
! #
! Cents
! Note
! Name
! Associated Ratio
|-
| 0
| 0.0
| J
| 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
|}


''<span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 14px;">'''Two6'''</span>''<span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 14px;"> by John Cage, 1992, for violin and piano. </span>[http://youtu.be/XkX37zH6AbU Haar &amp; Snijders] recording (YouTube).
== Approximation to JI ==
[[Category:compton]]
=== 15-odd-limit intervals ===
[[Category:Equal divisions of the octave]]
{{Q-odd-limit intervals|84}}
[[Category:john_cage]]
 
[[Category:listen]]
=== Higher-limit JI ===
[[Category:orwell]]
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 ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 78732/78125, 531441/524288
| {{Mapping| 84 133 195 }}
| +0.498
| 0.531
| 3.72
|-
| 2.3.5.7
| 225/224, 1728/1715, 78732/78125
| {{Mapping| 84 133 195 236 }}
| +0.141
| 0.769
| 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
| 225/224, 441/440, 1344/1331, 1728/1715
| {{Mapping| 84 133 195 236 291 }} (84)
| −0.225
| 1.003
| 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
| 99/98, 121/120, 176/175, 78732/78125
| {{Mapping| 84 133 195 236 290 }} (84e)
| +0.601
| 1.151
| 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 ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 19\84
| 271.43
| 7/6
| [[Orwell]] (84e) / [[newspeak]] (84)
|-
| 1
| 25\84
| 357.14
| 768/625
| [[Dodifo]]
|-
| 1
| 27\84
| 385.71
| 5/4
| [[Mutt]]
|-
| 1
| 31\84
| 442.86
| 162/125
| [[Sensei]]
|-
| 1
| 41\84
| 585.71
| 7/5
| [[Merman]]
|-
| 2
| 5\84
| 71.43
| 25/24
| [[Narayana]]
|-
| 2
| 11\84
| 157.14
| 35/32
| [[Bison]]
|-
| 2
| 13\84
| 185.71
| 10/9
| [[Secant]]
|-
| 3
| 11\84
| 157.14
| 35/32
| [[Nessafof]]
|-
| 7
| 5\84
| 500.00<br>(14.29)
| 4/3<br>(81/80)
| [[Absurdity]]
|-
| 12
| 27\84<br>(1\84)
| 385.71<br>(14.29)
| 5/4<br>(126/125)
| [[Compton]]
|-
| 12
| 16\84<br>(2\84)
| 228.6<br>(28.57)
| 8/7<br>(64/63)
| [[Catler]] (84c)
|-
| 21
| 41\84<br>(1\84)
| 585.71<br>(14.29)
| 91875/65536<br>(126/125)
| [[Akjayland]]
|-
| 28
| 49\84<br>(1\84)
| 500.00<br>(14.29)
| 4/3<br>(105/104)
| [[Oquatonic]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
== Scales ==
=== MOS ===
Brightest mode is listed.
 
* [[Orwell]]
** 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[22] ([[13L 9s]])
** Orwell[31] ([[22L 9s]])
 
=== Other ===
* [[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 ==
; {{W|John Cage}}
* ''Ten'' for chamber ensemble (1991) [https://youtu.be/PE_2Ds_6qGk Ives Ensemble recording (YouTube)] {{dead link}}
* ''Two4'' for violin and piano or shō (1991) [https://youtu.be/sLOrpd5onCs Harr &amp; Miyata 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)]
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/Sqkxrmwggr0 ''microtonal improvisation in 84edo''] (2025)
* [https://www.youtube.com/shorts/Qu6UIA2NmmQ ''84edo groove''] (2026)
 
; [[Eliora]]
* [https://www.youtube.com/watch?v=wTgOV_gVzuQ ''Requiem in Gb 1/7 Orwell''] (2023)
 
; [[JUMBLE]]
* [https://www.youtube.com/watch?v=blFYmTtFUWY ''Undiminished''] (2023)
 
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[[Category:John Cage]]
[[Category:Listen]]
[[Category:Compton]]
[[Category:Grossmic]]
[[Category:Orwell]]
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