84edo
← 83edo | 84edo | 85edo → |
84 equal divisions of the octave (abbreviated 84edo or 84ed2), also called 84-tone equal temperament (84tet) or 84 equal temperament (84et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 84 equal parts of about 14.3 ¢ each. Each step represents a frequency ratio of 21/84, or the 84th root of 2.
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
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) |
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 | 0.000 | 1/1 | Perfect 1sn | P1 | D |
1 | 14.286 | 81/80, 105/104, 126/125, 169/168, 196/195 | Up 1sn | ^1 | ^D |
2 | 28.571 | 50/49, 64/63, 65/64, 91/90 | Dup 1sn | ^^1 | ^^D |
3 | 42.857 | 36/35, 40/39, 46/45, 49/48 | Trup 1sn | ^^^1 | ^^^D |
4 | 57.143 | 27/26 | Trudminor 2nd | vvvm2 | vvvEb |
5 | 71.429 | 24/23, 25/24, 26/25, 28/27 | Dudminor 2nd | vvm2 | vvEb |
6 | 85.714 | 20/19, 21/20 | Downminor 2nd | vm2 | vEb |
7 | 100.000 | 19/18 | Minor 2nd | m2 | Eb |
8 | 114.286 | 15/14, 16/15 | Upminor 2nd | ^m2 | ^Eb |
9 | 128.571 | 14/13 | Dupminor 2nd | ^^m2 | ^^Eb |
10 | 142.857 | 13/12 | Trupminor 2nd | ^^^m2 | ^^^Eb |
11 | 157.143 | 23/21 | Trudmajor 2nd | vvvM2 | vvvE |
12 | 171.429 | 21/19 | Dudmajor 2nd | vvM2 | vvE |
13 | 185.714 | 10/9 | Downmajor 2nd | vM2 | vE |
14 | 200.000 | 9/8 | Major 2nd | M2 | E |
15 | 214.286 | 26/23 | Upmajor 2nd | ^M2 | ^E |
16 | 228.571 | 8/7 | Dupmajor 2nd | ^^M2 | ^^E |
17 | 242.857 | 15/13, 23/20 | Trupmajor 2nd | ^^^M2 | ^^^E |
18 | 257.143 | 52/45 | Trudminor 3rd | vvvm3 | vvvF |
19 | 271.429 | 7/6 | Dudminor 3rd | vvm2 | vvF |
20 | 285.714 | 45/38, 46/39 | Downminor 3rd | vm3 | vF |
21 | 300.000 | 19/16, 25/21, 32/27 | Minor 3rd | m3 | F |
22 | 314.286 | 6/5 | Upminor 3rd | ^m3 | ^F |
23 | 328.571 | 23/19 | Dupminor 3rd | ^^m3 | ^^F |
24 | 342.857 | 28/23, 39/32 | Trupminor 3rd | ^^^m3 | ^^^F |
25 | 357.143 | 16/13 | Trudmajor 3rd | vvvM3 | vvvF# |
26 | 371.429 | 26/21 | Dudmajor 3rd | vvM3 | vvF# |
27 | 385.714 | 5/4 | Downmajor 3rd | vM3 | vF# |
28 | 400.000 | 24/19 | Major 3rd | M3 | F# |
29 | 414.286 | 19/15 | Upmajor 3rd | ^M3 | ^F# |
30 | 428.571 | 9/7, 23/18, 32/25 | Dupmajor 3rd | ^^M3 | ^^F# |
31 | 442.857 | 84/65 | Trupmajor 3rd | ^^^M3 | ^^^F# |
32 | 457.143 | 13/10, 30/23 | Trud 4th | vvv4 | vvvG |
33 | 471.429 | 21/16 | Dud 4th | vv4 | vvG |
34 | 485.714 | 65/49 | Down 4th | v4 | vG |
35 | 500.000 | 4/3 | Perfect 4th | P4 | G |
36 | 514.286 | 27/20 | Up 4th | ^4 | ^G |
37 | 528.571 | 19/14 | Dup 4th | ^^4 | ^^G |
38 | 542.857 | 26/19 | Trup 4th | ^^^4 | ^^^G |
39 | 557.143 | 18/13 | Trudaug 4th | vvvA4 | vvvG# |
40 | 571.429 | 25/18, 32/23 | Dudaug 4th | vvA4 | vvG# |
41 | 585.714 | 7/5 | Downaug 4th | vA4 | vG# |
42 | 600.000 | 27/19, 38/27 | Aug 4th, Dim 5th | A4, d5 | G#, Ab |
43 | 614.286 | 10/7 | Updim 5th | ^d5 | ^Ab |
44 | 628.571 | 23/16, 36/25 | Dupdim 5th | ^^d5 | ^^Ab |
45 | 642.857 | 13/9 | Trupdim 5th | ^^^d5 | ^^^Ab |
46 | 657.143 | 19/13 | Trud 5th | vvv5 | vvvA |
47 | 671.429 | 28/19 | Dud 5th | vv5 | vvA |
48 | 685.714 | 40/27 | Down 5th | v5 | vA |
49 | 700.000 | 3/2 | Perfect 5th | P5 | A |
50 | 714.286 | 98/65 | Up 5th | ^5 | ^A |
51 | 728.571 | 32/21 | Dup 5th | ^^5 | ^^A |
52 | 742.857 | 20/13, 23/15 | Trup 5th | ^^^5 | ^^^A |
53 | 757.143 | 65/42 | Trudminor 6th | vvvm6 | vvvBb |
54 | 771.429 | 14/9, 25/16, 36/23 | Dudminor 6th | vvm6 | vvBb |
55 | 785.714 | 30/19 | Downminor 6th | vm6 | vBb |
56 | 800.000 | 19/12 | Minor 6th | m6 | Bb |
57 | 814.286 | 8/5 | Upminor 6th | ^m6 | ^Bb |
58 | 828.571 | 21/13 | Dupminor 6th | ^^m6 | ^^Bb |
59 | 842.857 | 13/8 | Trupminor 6th | ^^^m6 | ^^^Bb |
60 | 857.143 | 23/14, 64/39 | Trudmajor 6th | vvvM6 | vvvB |
61 | 871.429 | 38/23 | Dudmajor 6th | vvM6 | vvB |
62 | 885.714 | 5/3 | Downmajor 6th | vM6 | vB |
63 | 900.000 | 32/19, 27/16, 42/25 | Major 6th | M6 | B |
64 | 914.286 | 39/23, 76/45 | Upmajor 6th | ^M6 | ^B |
65 | 928.571 | 12/7 | Dupmajor 6th | ^^M6 | ^^B |
66 | 942.857 | 45/26 | Trupmajor 6th | ^^^M6 | ^^^B |
67 | 957.143 | 26/15, 40/23 | Trudminor 7th | vvvm7 | vvvC |
68 | 971.429 | 7/4 | Dudminor 7th | vvm7 | vvC |
69 | 985.714 | 23/13 | Downminor 7th | vm7 | vC |
70 | 1000.000 | 16/9 | Minor 7th | m7 | C |
71 | 1014.286 | 9/5 | Upminor 7th | ^m7 | ^C |
72 | 1028.571 | 38/21 | Dupminor 7th | ^^m7 | ^^C |
73 | 1042.857 | 42/23 | Trupminor 7th | ^^^m7 | ^^^C |
74 | 1057.143 | 24/13 | Trudmajor 7th | vvvM7 | vvvC# |
75 | 1071.429 | 13/7 | Dudmajor 7th | vvM7 | vvC# |
76 | 1085.714 | 15/8, 28/15 | Downmajor 7th | vM7 | vC# |
77 | 1100.000 | 36/19 | Major 7th | M7 | C# |
78 | 1114.286 | 19/10, 40/21 | Upmajor 7th | ^M7 | ^C# |
79 | 1128.571 | 23/12, 25/13, 27/14, 48/25 | Dupmajor 7th | ^^M7 | ^^C# |
80 | 1142.857 | 52/27 | Trupmajor 7th | ^^^M7 | ^^^C# |
81 | 1157.143 | 35/18, 39/20, 96/49 | Trud 8ve | vvv8 | vvvD |
82 | 1171.429 | 45/23, 49/25, 63/32, 128/65, 180/91 | Dud 8ve | vv8 | vvD |
83 | 1185.714 | 125/63, 160/81, 195/98, 336/169 | Down 8ve | v8 | vD |
84 | 1200.000 | 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
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
- Ten for chamber ensemble (1991) Ives Ensemble recording (YouTube) [dead link]
- Two4 for violin and piano or shō (1991) Harr & Miyata recording (YouTube)
- Two5 for tenor trombone and piano (1991) Fulkerson & Denyer recording (YouTube)
- Two6 for violin and piano (1992) Haar & Snijders recording (YouTube)
- Requiem in Gb 1/7 Orwell (2023)
- Undiminished (2023)