84edo: Difference between revisions

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Line 20: Line 20:
! #
! #
! Cents
! Cents
! Approximate Ratios
! Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[4L 5s|4L 5s Notation]]
! colspan="3" | [[4L 5s|4L 5s Notation]]
Line 36: Line 36:
| 1
| 1
| 14.286
| 14.286
|  
| ''81/80'', 126/125
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 46: Line 46:
| 2
| 2
| 28.571
| 28.571
|  
| 50/49, 64/63
| Dup 1sn
| Dup 1sn
| ^^1
| ^^1
Line 56: Line 56:
| 3
| 3
| 42.857
| 42.857
|  
| 36/35, 40/39, 49/48
| Trup 1sn
| Trup 1sn
| ^^^1
| ^^^1
Line 66: Line 66:
| 4
| 4
| 57.143
| 57.143
|  
| ''27/26''
| Trudminor 2nd
| Trudminor 2nd
| vvvm2
| vvvm2
Line 76: Line 76:
| 5
| 5
| 71.429
| 71.429
|  
| 25/24, 26/25, ''28/27''
| Dudminor 2nd
| Dudminor 2nd
| vvm2
| vvm2
Line 86: Line 86:
| 6
| 6
| 85.714
| 85.714
|  
| 21/20
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
Line 106: Line 106:
| 8
| 8
| 114.286
| 114.286
|  
| 15/14, 16/15
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 116: Line 116:
| 9
| 9
| 128.571
| 128.571
|  
| 14/13
| Dupminor 2nd
| Dupminor 2nd
| ^^m2
| ^^m2
Line 126: Line 126:
| 10
| 10
| 142.857
| 142.857
|  
| 13/12
| Trupminor 2nd
| Trupminor 2nd
| ^^^m2
| ^^^m2
Line 156: Line 156:
| 13
| 13
| 185.714
| 185.714
|  
| 10/9
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 166: Line 166:
| 14
| 14
| 200.000
| 200.000
|  
| 9/8
| Major 2nd
| Major 2nd
| M2
| M2
Line 186: Line 186:
| 16
| 16
| 228.571
| 228.571
|  
| 8/7
| Dupmajor 2nd
| Dupmajor 2nd
| ^^M2
| ^^M2
Line 196: Line 196:
| 17
| 17
| 242.857
| 242.857
|  
| 15/13
| Trupmajor 2nd
| Trupmajor 2nd
| ^^^M2
| ^^^M2
Line 216: Line 216:
| 19
| 19
| 271.429
| 271.429
|[[7/6]]
| 7/6
| Dudminor 3rd
| Dudminor 3rd
| vvm2
| vvm2
Line 236: Line 236:
| 21
| 21
| 300.000
| 300.000
|  
| 32/27
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 246: Line 246:
| 22
| 22
| 314.286
| 314.286
|  
| 6/5
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 266: Line 266:
| 24
| 24
| 342.857
| 342.857
|  
| 39/32
| Trupminor 3rd
| Trupminor 3rd
| ^^^m3
| ^^^m3
Line 276: Line 276:
| 25
| 25
| 357.143
| 357.143
|  
| 16/13
| Trudmajor 3rd
| Trudmajor 3rd
| vvvM3
| vvvM3
Line 286: Line 286:
| 26
| 26
| 371.429
| 371.429
|  
| 26/21
| Dudmajor 3rd
| Dudmajor 3rd
| vvM3
| vvM3
Line 296: Line 296:
| 27
| 27
| 385.714
| 385.714
|  
| 5/4
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 326: Line 326:
| 30
| 30
| 428.571
| 428.571
|  
| 9/7
| Dupmajor 3rd
| Dupmajor 3rd
| ^^M3
| ^^M3
Line 346: Line 346:
| 32
| 32
| 457.143
| 457.143
|  
| 13/10
| Trud 4th
| Trud 4th
| vvv4
| vvv4
Line 356: Line 356:
| 33
| 33
| 471.429
| 471.429
|  
| 21/16
| Dud 4th
| Dud 4th
| vv4
| vv4
Line 376: Line 376:
| 35
| 35
| 500.000
| 500.000
|  
| 4/3
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 386: Line 386:
| 36
| 36
| 514.286
| 514.286
|  
| 27/20
| Up 4th
| Up 4th
| ^4
| ^4
Line 416: Line 416:
| 39
| 39
| 557.143
| 557.143
|  
| 18/13
| Trudaug 4th
| Trudaug 4th
| vvvA4
| vvvA4
Line 436: Line 436:
| 41
| 41
| 585.714
| 585.714
|  
| 7/5
| Downaug 4th
| Downaug 4th
| vA4
| vA4
Line 456: Line 456:
| 43
| 43
| 614.286
| 614.286
|  
| 10/7
| Updim 5th
| Updim 5th
| ^d5
| ^d5
Line 476: Line 476:
| 45
| 45
| 642.857
| 642.857
|  
| 13/9
| Trupdim 5th
| Trupdim 5th
| ^^^d5
| ^^^d5
Line 506: Line 506:
| 48
| 48
| 685.714
| 685.714
|  
| 40/27
| Down 5th
| Down 5th
| v5
| v5
Line 516: Line 516:
| 49
| 49
| 700.000
| 700.000
|[[3/2]]
| 3/2
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 536: Line 536:
| 51
| 51
| 728.571
| 728.571
|  
| 32/21
| Dup 5th
| Dup 5th
| ^^5
| ^^5
Line 546: Line 546:
| 52
| 52
| 742.857
| 742.857
|  
| 20/13
| Trup 5th
| Trup 5th
| ^^^5
| ^^^5
Line 566: Line 566:
| 54
| 54
| 771.429
| 771.429
|  
| 14/9
| Dudminor 6th
| Dudminor 6th
| vvm6
| vvm6
Line 596: Line 596:
| 57
| 57
| 814.286
| 814.286
|[[5/3]]
| 8/5
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 606: Line 606:
| 58
| 58
| 828.571
| 828.571
|  
| 21/13
| Dupminor 6th
| Dupminor 6th
| ^^m6
| ^^m6
Line 616: Line 616:
| 59
| 59
| 842.857
| 842.857
|  
| 13/8
| Trupminor 6th
| Trupminor 6th
| ^^^m6
| ^^^m6
Line 626: Line 626:
| 60
| 60
| 857.143
| 857.143
|[[105/64]]
| 64/39
| Trudmajor 6th
| Trudmajor 6th
| vvvM6
| vvvM6
Line 646: Line 646:
| 62
| 62
| 885.714
| 885.714
|  
| 5/3
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 656: Line 656:
| 63
| 63
| 900.000
| 900.000
|  
| 27/16
| Major 6th
| Major 6th
| M6
| M6
Line 676: Line 676:
| 65
| 65
| 928.571
| 928.571
|  
| 12/7
| Dupmajor 6th
| Dupmajor 6th
| ^^M6
| ^^M6
Line 696: Line 696:
| 67
| 67
| 957.143
| 957.143
|  
| 26/15
| Trudminor 7th
| Trudminor 7th
| vvvm7
| vvvm7
Line 706: Line 706:
| 68
| 68
| 971.429
| 971.429
|  
| 7/4
| Dudminor 7th
| Dudminor 7th
| vvm7
| vvm7
Line 726: Line 726:
| 70
| 70
| 1000.000
| 1000.000
|  
| 16/9
| Minor 7th
| Minor 7th
| m7
| m7
Line 736: Line 736:
| 71
| 71
| 1014.286
| 1014.286
|  
| 9/5
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 766: Line 766:
| 74
| 74
| 1057.143
| 1057.143
|  
| 24/13
| Trudmajor 7th
| Trudmajor 7th
| vvvM7
| vvvM7
Line 776: Line 776:
| 75
| 75
| 1071.429
| 1071.429
|  
| 13/7
| Dudmajor 7th
| Dudmajor 7th
| vvM7
| vvM7
Line 786: Line 786:
| 76
| 76
| 1085.714
| 1085.714
|  
| 15/8, 28/15
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 803: Line 803:
| ^M9
| ^M9
| R^
| R^
|-
|-
|-
| 78
| 78
| 1114.286
| 1114.286
|  
| 40/21
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 817: Line 816:
| 79
| 79
| 1128.571
| 1128.571
|  
| 25/13, ''27/14'', 48/25
| Dupmajor 7th
| Dupmajor 7th
| ^^M7
| ^^M7
Line 827: Line 826:
| 80
| 80
| 1142.857
| 1142.857
|  
| ''52/27''
| Trupmajor 7th
| Trupmajor 7th
| ^^^M7
| ^^^M7
Line 837: Line 836:
| 81
| 81
| 1157.143
| 1157.143
|  
| 35/18, 39/20, 96/49
| Trud 8ve
| Trud 8ve
| vvv8
| vvv8
Line 847: Line 846:
| 82
| 82
| 1171.429
| 1171.429
|  
| 49/25, 63/32
| Dud 8ve
| Dud 8ve
| vv8
| vv8
Line 857: Line 856:
| 83
| 83
| 1185.714
| 1185.714
|  
| 125/63, ''160/81''
| Down 8ve
| Down 8ve
| v8
| v8
Line 867: Line 866:
| 84
| 84
| 1200.000
| 1200.000
| [[2/1]]
| 2/1
| Perfect 8ve
| Perfect 8ve
| P8
| P8
Line 876: Line 875:
|-
|-
|}
|}
<nowiki>*</nowiki> as a 2.3.5.7.13-subgroup temperament


== Regular temperament properties ==
== Regular temperament properties ==

Revision as of 09:22, 21 June 2024

← 83edo 84edo 85edo →
Prime factorization 22 × 3 × 7
Step size 14.2857 ¢ 
Fifth 49\84 (700 ¢) (→ 7\12)
Semitones (A1:m2) 7:7 (100 ¢ : 100 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

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 book 1984. Orwell in 84edo comes in two varieties – the 84e val ⟨84 133 195 236 290], supporting the original orwell, and its patent val ⟨84 133 195 236 291] supporting newspeak. 84edo orwell offers mosses of size 9, 13, 22, and 31, of which the 31-note scale is the maximal evenness scale.

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

Prime harmonics

Approximation of prime harmonics in 84edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.00 -1.96 -0.60 +2.60 +5.82 +2.33 -4.96 +2.49 +0.30 -1.01 -2.18 +5.80
Relative (%) +0.0 -13.7 -4.2 +18.2 +40.8 +16.3 -34.7 +17.4 +2.1 -7.0 -15.2 +40.6
Steps
(reduced) 		84
(0) 		133
(49) 		195
(27) 		236
(68) 		291
(39) 		311
(59) 		343
(7) 		357
(21) 		380
(44) 		408
(72) 		416
(80) 		438
(18)
Approximation of prime harmonics in 84edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) -0.49 +2.77 +5.92 -2.08 -2.03 -2.60 +6.41 +6.02 +0.78 +6.89 +7.10 +0.55
Relative (%) -3.4 +19.4 +41.5 -14.5 -14.2 -18.2 +44.9 +42.1 +5.5 +48.2 +49.7 +3.8
Steps
(reduced) 		450
(30) 		456
(36) 		467
(47) 		481
(61) 		494
(74) 		498
(78) 		510
(6) 		517
(13) 		520
(16) 		530
(26) 		536
(32) 		544
(40)

Subsets and supersets

84edo is a largely composite number. Since 84 factors as 22 × 3 × 7, 84edo has subset edos 2, 3, 4, 6, 7, 12, 14, 21, 28, 42. Being a small multiple of 12, 84et tempers out the Pythagorean comma, thus supporting the 1/12-octave temperament compton. Being a small multiple of 28, it tempers out the oquatonic comma, which maps 5/4 to 9\28.

Interval table

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).

Table of 84edo intervals
# Cents Approximate Ratios* Ups and Downs Notation 4L 5s Notation
0 0.000 1/1 Perfect 1sn P1 D Perfect 1sn P1 J
1 14.286 81/80, 126/125 Up 1sn ^1 ^D Up 1sn ^1 J^
2 28.571 50/49, 64/63 Dup 1sn ^^1 ^^D Downaug 1sn vA1 Jv&
3 42.857 36/35, 40/39, 49/48 Trup 1sn ^^^1 ^^^D Aug 1sn A1 J&
4 57.143 27/26 Trudminor 2nd vvvm2 vvvEb Upaug 1sn, Downdim 2nd ^A1, vd2 J^&, Kvaa
5 71.429 25/24, 26/25, 28/27 Dudminor 2nd vvm2 vvEb Dim 2nd d2 Kaa
6 85.714 21/20 Downminor 2nd vm2 vEb Updim 2nd ^d2 K^aa
7 100.000 Minor 2nd m2 Eb Downminor 2nd vm2 Kva
8 114.286 15/14, 16/15 Upminor 2nd ^m2 ^Eb Minor 2nd m2 Ka
9 128.571 14/13 Dupminor 2nd ^^m2 ^^Eb Upminor 2nd ^m2 K^a
10 142.857 13/12 Trupminor 2nd ^^^m2 ^^^Eb Downmajor 2nd vM2 Kv
11 157.143 Trudmajor 2nd vvvM2 vvvE Major 2nd M2 K
12 171.429 Dudmajor 2nd vvM2 vvE Upmajor 2nd ^M2 K^
13 185.714 10/9 Downmajor 2nd vM2 vE Downaug 2nd vA2 Kv&
14 200.000 9/8 Major 2nd M2 E Aug 2nd A2 K&
15 214.286 Upmajor 2nd ^M2 ^E Upaug 2nd, Downdim 3rd ^A2, vd3 K^&, Lva
16 228.571 8/7 Dupmajor 2nd ^^M2 ^^E Dim 3rd d3 La
17 242.857 15/13 Trupmajor 2nd ^^^M2 ^^^E Updim 3rd ^d3 L^a
18 257.143 Trudminor 3rd vvvm3 vvvF Down 3rd v3 Lv
19 271.429 7/6 Dudminor 3rd vvm2 vvF Perfect 3rd P3 L
20 285.714 Downminor 3rd vm3 vF Up 3rd ^3 L^
21 300.000 32/27 Minor 3rd m3 F Downaug 3rd vA3 Lv&
22 314.286 6/5 Upminor 3rd ^m3 ^F Aug 3rd A3 L&
23 328.571 Dupminor 3rd ^^m3 ^^F Upaug 3rd, Downdim 4th ^A3, vd4 L^&, Mvaa
24 342.857 39/32 Trupminor 3rd ^^^m3 ^^^F Dim 4th d4 Maa
25 357.143 16/13 Trudmajor 3rd vvvM3 vvvF# Updim 4th ^d4 M^aa
26 371.429 26/21 Dudmajor 3rd vvM3 vvF# Downminor 4th vm4 Mva
27 385.714 5/4 Downmajor 3rd vM3 vF# Minor 4th m4 Ma
28 400.000 Major 3rd M3 F# Upminor 4th ^m4 M^a
29 414.286 Upmajor 3rd ^M3 ^F# Downmajor 4th vM4 Mv
30 428.571 9/7 Dupmajor 3rd ^^M3 ^^F# Major 4th M4 M
31 442.857 Trupmajor 3rd ^^^M3 ^^^F# Upmajor 4th ^M4 M^
32 457.143 13/10 Trud 4th vvv4 vvvG Downaug 4th vA4 Mv&
33 471.429 21/16 Dud 4th vv4 vvG Aug 4th A4 M&
34 485.714 Down 4th v4 vG Downminor 5th vm5 Nva
35 500.000 4/3 Perfect 4th P4 G Minor 5th m5 Na
36 514.286 27/20 Up 4th ^4 ^G Upminor 5th ^m5 N^a
37 528.571 Dup 4th ^^4 ^^G Downmajor 5th vM5 Nv
38 542.857 Trup 4th ^^^4 ^^^G Major 5th M5 N
39 557.143 18/13 Trudaug 4th vvvA4 vvvG# Upmajor 5th ^M5 N^
40 571.429 Dudaug 4th vvA4 vvG# Downaug 5th vA5 Nv&
41 585.714 7/5 Downaug 4th vA4 vG# Aug 5th A5 N&
42 600.000 Aug 4th, Dim 5th A4, d5 G#, Ab Upaug 5th, Downdim 6th ^A5, vd6 N^&, Ovaa
43 614.286 10/7 Updim 5th ^d5 ^Ab Dim 6th d6 Oaa
44 628.571 Dupdim 5th ^^d5 ^^Ab Updim 6th ^d6 O^aa
45 642.857 13/9 Trupdim 5th ^^^d5 ^^^Ab Downminor 6th vm6 Ova
46 657.143 Trud 5th vvv5 vvvA Minor 6th m6 Oa
47 671.429 Dud 5th vv5 vvA Upminor 6th ^m6 O^a
48 685.714 40/27 Down 5th v5 vA Downmajor 6th vM6 Ov
49 700.000 3/2 Perfect 5th P5 A Major 6th M6 O
50 714.286 Up 5th ^5 ^A Upmajor 6th ^M6 O^
51 728.571 32/21 Dup 5th ^^5 ^^A Dim 7th d7 Paa
52 742.857 20/13 Trup 5th ^^^5 ^^^A Aug 6th A6 O&
53 757.143 Trudminor 6th vvvm6 vvvBb Downminor 7th vm7 Pva
54 771.429 14/9 Dudminor 6th vvm6 vvBb Minor 7th m7 Pa
55 785.714 Downminor 6th vm6 vBb Upminor 7th ^m7 P^a
56 800.000 Minor 6th m6 Bb Downmajor 7th vM7 Pv
57 814.286 8/5 Upminor 6th ^m6 ^Bb Major 7th M7 P
58 828.571 21/13 Dupminor 6th ^^m6 ^^Bb Upmajor 7th ^M7 P^
59 842.857 13/8 Trupminor 6th ^^^m6 ^^^Bb Downaug 7th vA7 Pv&
60 857.143 64/39 Trudmajor 6th vvvM6 vvvB Aug 7th A7 P&
61 871.429 Dudmajor 6th vvM6 vvB Upaug 7th, Downdim 8th ^A7, vd8 P^&, Qvaa
62 885.714 5/3 Downmajor 6th vM6 vB Dim 8th d8 Qaa
63 900.000 27/16 Major 6th M6 B Updim 8th ^d8 Q^aa
64 914.286 Upmajor 6th ^M6 ^B Down 8th v8 Qva
65 928.571 12/7 Dupmajor 6th ^^M6 ^^B Perfect 8th P8 Qa
66 942.857 Trupmajor 6th ^^^M6 ^^^B Up 8th ^8 Q^a
67 957.143 26/15 Trudminor 7th vvvm7 vvvC Downaug 8th vA8 Qv
68 971.429 7/4 Dudminor 7th vvm7 vvC Aug 8th A8 Q
69 985.714 Downminor 7th vm7 vC Upaug 8th, Downdim 9th ^A8, vd9 Q^, Rvaa
70 1000.000 16/9 Minor 7th m7 C Dim 9th d9 Raa
71 1014.286 9/5 Upminor 7th ^m7 ^C Updim 9th ^d9 R^aa
72 1028.571 Dupminor 7th ^^m7 ^^C Downminor 9th vm9 Rva
73 1042.857 Trupminor 7th ^^^m7 ^^^C Minor 9th m9 Ra
74 1057.143 24/13 Trudmajor 7th vvvM7 vvvC# Upminor 9th ^m9 R^a
75 1071.429 13/7 Dudmajor 7th vvM7 vvC# Downmajor 9th vM9 Rv
76 1085.714 15/8, 28/15 Downmajor 7th vM7 vC# Major 9th M9 R
77 1100.000 Major 7th M7 C# Upmajor 9th ^M9 R^
78 1114.286 40/21 Upmajor 7th ^M7 ^C# Downaug 9th vA9 Rv&
79 1128.571 25/13, 27/14, 48/25 Dupmajor 7th ^^M7 ^^C# Aug 9th A9 R&
80 1142.857 52/27 Trupmajor 7th ^^^M7 ^^^C# Upaug 9th, Downdim 10th ^A9, vd10 R^&, Jva
81 1157.143 35/18, 39/20, 96/49 Trud 8ve vvv8 vvvD Dim 10th d10 Ja
82 1171.429 49/25, 63/32 Dud 8ve vv8 vvD Updim 10th ^d10 J^a
83 1185.714 125/63, 160/81 Down 8ve v8 vD Down 10th v10 Jv
84 1200.000 2/1 Perfect 8ve P8 D Perfect 10th P10 J

* as a 2.3.5.7.13-subgroup temperament

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5 78732/78125, 531441/524288 [⟨84 133 195]] +0.498 0.531 3.72
2.3.5.7 225/224, 1728/1715, 78732/78125 [⟨84 133 195 236]] +0.141 0.769 5.39
2.3.5.7.11 225/224, 441/440, 1344/1331, 1728/1715 [⟨84 133 195 236 291]] (84) -0.225 1.003 7.02
2.3.5.7.11 99/98, 121/120, 176/175, 78732/78125 [⟨84 133 195 236 290]] (84e) +0.601 1.151 8.05

Rank-2 temperaments

Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
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\84 442.86 125/81 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
(14.29) 		4/3
(81/80) 		Absurdity
12 27\84
(1\84) 		385.71
(14.29) 		5/4
(126/125) 		Compton
21 41\84
(1\84) 		585.71
(14.29) 		91875/65536
(126/125) 		Akjayland
28 49\84
(1\84) 		500.00
(14.29) 		4/3
(105/104) 		Oquatonic

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

MOS

Brightest mode is listed.

Other

Music

John Cage
Eliora
JUMBLE
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