84edo: Difference between revisions

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== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
|-
! #
! #
Line 26: Line 26:
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| 1/1
| Perfect 1sn
| Perfect 1sn
Line 33: Line 33:
|-
|-
| 1
| 1
| 14.286
| 14.3
| ''81/80'', 105/104, 126/125, 169/168, 196/195
| ''81/80'', 105/104, 126/125, 169/168, 196/195
| Up 1sn
| Up 1sn
Line 40: Line 40:
|-
|-
| 2
| 2
| 28.571
| 28.6
| 50/49, 64/63, 65/64, ''91/90''
| 50/49, 64/63, 65/64, ''91/90''
| Dup 1sn
| Dup 1sn
Line 47: Line 47:
|-
|-
| 3
| 3
| 42.857
| 42.9
| 36/35, 40/39, 46/45, 49/48
| 36/35, 40/39, 46/45, 49/48
| Trup 1sn
| Trup 1sn
Line 54: Line 54:
|-
|-
| 4
| 4
| 57.143
| 57.1
| ''27/26''
| ''27/26''
| Trudminor 2nd
| Trudminor 2nd
Line 61: Line 61:
|-
|-
| 5
| 5
| 71.429
| 71.4
| 24/23, 25/24, 26/25, ''28/27''
| 24/23, 25/24, 26/25, ''28/27''
| Dudminor 2nd
| Dudminor 2nd
Line 68: Line 68:
|-
|-
| 6
| 6
| 85.714
| 85.7
| 20/19, 21/20
| 20/19, 21/20
| Downminor 2nd
| Downminor 2nd
Line 75: Line 75:
|-
|-
| 7
| 7
| 100.000
| 100.0
| 19/18
| 19/18
| Minor 2nd
| Minor 2nd
Line 82: Line 82:
|-
|-
| 8
| 8
| 114.286
| 114.3
| 15/14, 16/15
| 15/14, 16/15
| Upminor 2nd
| Upminor 2nd
Line 89: Line 89:
|-
|-
| 9
| 9
| 128.571
| 128.6
| 14/13
| 14/13
| Dupminor 2nd
| Dupminor 2nd
Line 96: Line 96:
|-
|-
| 10
| 10
| 142.857
| 142.9
| 13/12
| 13/12
| Trupminor 2nd
| Trupminor 2nd
Line 103: Line 103:
|-
|-
| 11
| 11
| 157.143
| 157.1
| 23/21
| 23/21
| Trudmajor 2nd
| Trudmajor 2nd
Line 110: Line 110:
|-
|-
| 12
| 12
| 171.429
| 171.4
| 21/19
| 21/19
| Dudmajor 2nd
| Dudmajor 2nd
Line 117: Line 117:
|-
|-
| 13
| 13
| 185.714
| 185.7
| 10/9
| 10/9
| Downmajor 2nd
| Downmajor 2nd
Line 124: Line 124:
|-
|-
| 14
| 14
| 200.000
| 200.0
| 9/8
| 9/8
| Major 2nd
| Major 2nd
Line 131: Line 131:
|-
|-
| 15
| 15
| 214.286
| 214.3
| 26/23
| 26/23
| Upmajor 2nd
| Upmajor 2nd
Line 138: Line 138:
|-
|-
| 16
| 16
| 228.571
| 228.6
| 8/7
| 8/7
| Dupmajor 2nd
| Dupmajor 2nd
Line 145: Line 145:
|-
|-
| 17
| 17
| 242.857
| 242.9
| 15/13, 23/20
| 15/13, 23/20
| Trupmajor 2nd
| Trupmajor 2nd
Line 152: Line 152:
|-
|-
| 18
| 18
| 257.143
| 257.1
| 52/45
| 52/45
| Trudminor 3rd
| Trudminor 3rd
Line 159: Line 159:
|-
|-
| 19
| 19
| 271.429
| 271.4
| 7/6
| 7/6
| Dudminor 3rd
| Dudminor 3rd
Line 166: Line 166:
|-
|-
| 20
| 20
| 285.714
| 285.7
| 45/38, 46/39
| 45/38, 46/39
| Downminor 3rd
| Downminor 3rd
Line 173: Line 173:
|-
|-
| 21
| 21
| 300.000
| 300.0
| 19/16, 25/21, 32/27
| 19/16, 25/21, 32/27
| Minor 3rd
| Minor 3rd
Line 180: Line 180:
|-
|-
| 22
| 22
| 314.286
| 314.3
| 6/5
| 6/5
| Upminor 3rd
| Upminor 3rd
Line 187: Line 187:
|-
|-
| 23
| 23
| 328.571
| 328.6
| 23/19
| 23/19
| Dupminor 3rd
| Dupminor 3rd
Line 194: Line 194:
|-
|-
| 24
| 24
| 342.857
| 342.9
| 28/23, 39/32
| 28/23, 39/32
| Trupminor 3rd
| Trupminor 3rd
Line 201: Line 201:
|-
|-
| 25
| 25
| 357.143
| 357.1
| 16/13
| 16/13
| Trudmajor 3rd
| Trudmajor 3rd
Line 208: Line 208:
|-
|-
| 26
| 26
| 371.429
| 371.4
| 26/21
| 26/21
| Dudmajor 3rd
| Dudmajor 3rd
Line 215: Line 215:
|-
|-
| 27
| 27
| 385.714
| 385.7
| 5/4
| 5/4
| Downmajor 3rd
| Downmajor 3rd
Line 222: Line 222:
|-
|-
| 28
| 28
| 400.000
| 400.0
| 24/19
| 24/19
| Major 3rd
| Major 3rd
Line 229: Line 229:
|-
|-
| 29
| 29
| 414.286
| 414.3
| 19/15
| 19/15
| Upmajor 3rd
| Upmajor 3rd
Line 236: Line 236:
|-
|-
| 30
| 30
| 428.571
| 428.6
| 9/7, 23/18, 32/25
| 9/7, 23/18, 32/25
| Dupmajor 3rd
| Dupmajor 3rd
Line 243: Line 243:
|-
|-
| 31
| 31
| 442.857
| 442.9
| 84/65
| 84/65
| Trupmajor 3rd
| Trupmajor 3rd
Line 250: Line 250:
|-
|-
| 32
| 32
| 457.143
| 457.1
| 13/10, 30/23
| 13/10, 30/23
| Trud 4th
| Trud 4th
Line 257: Line 257:
|-
|-
| 33
| 33
| 471.429
| 471.4
| 21/16
| 21/16
| Dud 4th
| Dud 4th
Line 264: Line 264:
|-
|-
| 34
| 34
| 485.714
| 485.7
| 65/49
| 65/49
| Down 4th
| Down 4th
Line 271: Line 271:
|-
|-
| 35
| 35
| 500.000
| 500.0
| 4/3
| 4/3
| Perfect 4th
| Perfect 4th
Line 278: Line 278:
|-
|-
| 36
| 36
| 514.286
| 514.3
| 27/20
| 27/20
| Up 4th
| Up 4th
Line 285: Line 285:
|-
|-
| 37
| 37
| 528.571
| 528.6
| 19/14
| 19/14
| Dup 4th
| Dup 4th
Line 292: Line 292:
|-
|-
| 38
| 38
| 542.857
| 542.9
| 26/19
| 26/19
| Trup 4th
| Trup 4th
Line 299: Line 299:
|-
|-
| 39
| 39
| 557.143
| 557.1
| 18/13
| 18/13
| Trudaug 4th
| Trudaug 4th
Line 306: Line 306:
|-
|-
| 40
| 40
| 571.429
| 571.4
| 25/18, 32/23
| 25/18, 32/23
| Dudaug 4th
| Dudaug 4th
Line 313: Line 313:
|-
|-
| 41
| 41
| 585.714
| 585.7
| 7/5
| 7/5
| Downaug 4th
| Downaug 4th
Line 320: Line 320:
|-
|-
| 42
| 42
| 600.000
| 600.0
| 27/19, 38/27
| 27/19, 38/27
| Aug 4th, Dim 5th
| Aug 4th, Dim 5th
Line 327: Line 327:
|-
|-
| 43
| 43
| 614.286
| 614.3
| 10/7
| 10/7
| Updim 5th
| Updim 5th
Line 334: Line 334:
|-
|-
| 44
| 44
| 628.571
| 628.6
| 23/16, 36/25
| 23/16, 36/25
| Dupdim 5th
| Dupdim 5th
Line 341: Line 341:
|-
|-
| 45
| 45
| 642.857
| 642.9
| 13/9
| 13/9
| Trupdim 5th
| Trupdim 5th
Line 348: Line 348:
|-
|-
| 46
| 46
| 657.143
| 657.1
| 19/13
| 19/13
| Trud 5th
| Trud 5th
Line 355: Line 355:
|-
|-
| 47
| 47
| 671.429
| 671.4
| 28/19
| 28/19
| Dud 5th
| Dud 5th
Line 362: Line 362:
|-
|-
| 48
| 48
| 685.714
| 685.7
| 40/27
| 40/27
| Down 5th
| Down 5th
Line 369: Line 369:
|-
|-
| 49
| 49
| 700.000
| 700.0
| 3/2
| 3/2
| Perfect 5th
| Perfect 5th
Line 376: Line 376:
|-
|-
| 50
| 50
| 714.286
| 714.3
| 98/65
| 98/65
| Up 5th
| Up 5th
Line 383: Line 383:
|-
|-
| 51
| 51
| 728.571
| 728.6
| 32/21
| 32/21
| Dup 5th
| Dup 5th
Line 390: Line 390:
|-
|-
| 52
| 52
| 742.857
| 742.9
| 20/13, 23/15
| 20/13, 23/15
| Trup 5th
| Trup 5th
Line 397: Line 397:
|-
|-
| 53
| 53
| 757.143
| 757.1
| 65/42
| 65/42
| Trudminor 6th
| Trudminor 6th
Line 404: Line 404:
|-
|-
| 54
| 54
| 771.429
| 771.4
| 14/9, 25/16, 36/23
| 14/9, 25/16, 36/23
| Dudminor 6th
| Dudminor 6th
Line 411: Line 411:
|-
|-
| 55
| 55
| 785.714
| 785.7
| 30/19
| 30/19
| Downminor 6th
| Downminor 6th
Line 418: Line 418:
|-
|-
| 56
| 56
| 800.000
| 800.0
| 19/12
| 19/12
| Minor 6th
| Minor 6th
Line 425: Line 425:
|-
|-
| 57
| 57
| 814.286
| 814.3
| 8/5
| 8/5
| Upminor 6th
| Upminor 6th
Line 432: Line 432:
|-
|-
| 58
| 58
| 828.571
| 828.6
| 21/13
| 21/13
| Dupminor 6th
| Dupminor 6th
Line 439: Line 439:
|-
|-
| 59
| 59
| 842.857
| 842.9
| 13/8
| 13/8
| Trupminor 6th
| Trupminor 6th
Line 446: Line 446:
|-
|-
| 60
| 60
| 857.143
| 857.1
| 23/14, 64/39
| 23/14, 64/39
| Trudmajor 6th
| Trudmajor 6th
Line 453: Line 453:
|-
|-
| 61
| 61
| 871.429
| 871.4
| 38/23
| 38/23
| Dudmajor 6th
| Dudmajor 6th
Line 460: Line 460:
|-
|-
| 62
| 62
| 885.714
| 885.7
| 5/3
| 5/3
| Downmajor 6th
| Downmajor 6th
Line 467: Line 467:
|-
|-
| 63
| 63
| 900.000
| 900.0
| 32/19, 27/16, 42/25
| 32/19, 27/16, 42/25
| Major 6th
| Major 6th
Line 474: Line 474:
|-
|-
| 64
| 64
| 914.286
| 914.3
| 39/23, 76/45
| 39/23, 76/45
| Upmajor 6th
| Upmajor 6th
Line 481: Line 481:
|-
|-
| 65
| 65
| 928.571
| 928.6
| 12/7
| 12/7
| Dupmajor 6th
| Dupmajor 6th
Line 488: Line 488:
|-
|-
| 66
| 66
| 942.857
| 942.9
| 45/26
| 45/26
| Trupmajor 6th
| Trupmajor 6th
Line 495: Line 495:
|-
|-
| 67
| 67
| 957.143
| 957.1
| 26/15, 40/23
| 26/15, 40/23
| Trudminor 7th
| Trudminor 7th
Line 502: Line 502:
|-
|-
| 68
| 68
| 971.429
| 971.4
| 7/4
| 7/4
| Dudminor 7th
| Dudminor 7th
Line 509: Line 509:
|-
|-
| 69
| 69
| 985.714
| 985.7
| 23/13
| 23/13
| Downminor 7th
| Downminor 7th
Line 516: Line 516:
|-
|-
| 70
| 70
| 1000.000
| 1000.0
| 16/9
| 16/9
| Minor 7th
| Minor 7th
Line 523: Line 523:
|-
|-
| 71
| 71
| 1014.286
| 1014.3
| 9/5
| 9/5
| Upminor 7th
| Upminor 7th
Line 530: Line 530:
|-
|-
| 72
| 72
| 1028.571
| 1028.6
| 38/21
| 38/21
| Dupminor 7th
| Dupminor 7th
Line 537: Line 537:
|-
|-
| 73
| 73
| 1042.857
| 1042.9
| 42/23
| 42/23
| Trupminor 7th
| Trupminor 7th
Line 544: Line 544:
|-
|-
| 74
| 74
| 1057.143
| 1057.1
| 24/13
| 24/13
| Trudmajor 7th
| Trudmajor 7th
Line 551: Line 551:
|-
|-
| 75
| 75
| 1071.429
| 1071.4
| 13/7
| 13/7
| Dudmajor 7th
| Dudmajor 7th
Line 558: Line 558:
|-
|-
| 76
| 76
| 1085.714
| 1085.7
| 15/8, 28/15
| 15/8, 28/15
| Downmajor 7th
| Downmajor 7th
Line 565: Line 565:
|-
|-
| 77
| 77
| 1100.000
| 1100.0
| 36/19
| 36/19
| Major 7th
| Major 7th
Line 572: Line 572:
|-
|-
| 78
| 78
| 1114.286
| 1114.3
| 19/10, 40/21
| 19/10, 40/21
| Upmajor 7th
| Upmajor 7th
Line 579: Line 579:
|-
|-
| 79
| 79
| 1128.571
| 1128.6
| 23/12, 25/13, ''27/14'', 48/25
| 23/12, 25/13, ''27/14'', 48/25
| Dupmajor 7th
| Dupmajor 7th
Line 586: Line 586:
|-
|-
| 80
| 80
| 1142.857
| 1142.9
| ''52/27''
| ''52/27''
| Trupmajor 7th
| Trupmajor 7th
Line 593: Line 593:
|-
|-
| 81
| 81
| 1157.143
| 1157.1
| 35/18, 39/20, 96/49
| 35/18, 39/20, 96/49
| Trud 8ve
| Trud 8ve
Line 600: Line 600:
|-
|-
| 82
| 82
| 1171.429
| 1171.4
| 45/23, 49/25, 63/32, 128/65, ''180/91''
| 45/23, 49/25, 63/32, 128/65, ''180/91''
| Dud 8ve
| Dud 8ve
Line 607: Line 607:
|-
|-
| 83
| 83
| 1185.714
| 1185.7
| 125/63, ''160/81'', 195/98, 336/169
| 125/63, ''160/81'', 195/98, 336/169
| Down 8ve
| Down 8ve
Line 614: Line 614:
|-
|-
| 84
| 84
| 1200.000
| 1200.0
| 2/1
| 2/1
| Perfect 8ve
| Perfect 8ve
Line 620: Line 620:
| D
| D
|}
|}
<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 ==
== Notation ==

Revision as of 06:54, 16 February 2025

← 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 shares the 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, supporting 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 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 mos scales of size 9, 13, 22, and 31, of which the 31-note scale is the maximal evenness scale.

High limit consistency and coverage

It has fairly good approximation to higher prime harmonics such as 13, 19, 23, 29, 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, if we avoid all intervals of 11 and 17 as well as the complex compound prime powers 27 and 49, it is completely consistent in the no-37's no-47's 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~1.3 ¢ (off by ~7.3 ¢), 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). Except 17, the only missing primes are thus 37, 47, 67, 71, 79 and 83, which coincidentally 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).

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

84 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 28, it tempers out the oquatonic comma, which maps 5/4 to 9\28.

Intervals

# Cents Approximate ratios* 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

* As a 2.3.5.7.13.19.23-subgroup temperament

Notation

4L 5s (gramitonic) notation

This notation is based on 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").

# 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

Ups and downs notation

Using Helmholtz–Ellis accidentals, 84edo can be notated using ups and downs notation:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp symbol  
Flat symbol

Approximation to JI

Zeta peak index

Tuning Strength Closest edo Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap Edo Octave (cents) Consistent Distinct
462zpi 83.9972142607288 14.2861880666087 8.020965 1.241945 16.733121 84edo 1200.03979759513 10 10

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
2.3.5.7.13 225/224, 351/350, 640/637, 1701/1690 [⟨84 133 195 236 311]] −0.013 0.754 5.28
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
2.3.5.7.11.13 144/143, 225/224, 351/350, 441/440, 975/968 [⟨84 133 195 236 291 311]] (84) −0.292 0.928 6.50
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
2.3.5.7.11.13 99/98, 121/120, 176/175, 275/273, 1701/1690 [⟨84 133 195 236 290 311]] (84e) +0.396 1.146 8.02

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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\84 442.86 125/81 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 distinct

Scales

MOS

Brightest mode is listed.

Other

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

John Cage
Eliora
JUMBLE
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