84edo
| ← 83edo | 84edo | 85edo → |
Theory
In the 13-limit it is the optimal patent val for the rank five 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. From a regular temperament perspective, 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] representing newspeak. 84edo orwell offers mosses 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 comma, which maps 5/4 to 9\28.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | -0.60 | +2.60 | +5.82 | +2.33 | -4.96 | +2.49 | +0.30 | -1.01 | -2.18 |
| Relative (%) | +0.0 | -13.7 | -4.2 | +18.2 | +40.8 | +16.3 | -34.7 | +17.4 | +2.1 | -7.0 | -15.2 | |
| Steps (reduced) |
84 (0) |
133 (49) |
195 (27) |
236 (68) |
291 (39) |
311 (59) |
343 (7) |
357 (21) |
380 (44) |
408 (72) |
416 (80) | |
Miscellaneous properties
84edo is a significantly composite number, with divisors of 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
Table of 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 @ "at".
| Degree | Size (Cents) | Ups and Downs Notation | 4L 5s Notation | Associated ratio | ||||
|---|---|---|---|---|---|---|---|---|
| 0 | 0.000 | Perfect 1sn | P1 | D | Perfect 1sn | P1 | J | 1/1 exact |
| 1 | 14.286 | Up 1sn | ^1 | ^D | Up 1sn | ^1 | J^ | |
| 2 | 28.571 | Dup 1sn | ^^1 | ^^D | Downaug 1sn | vA1 | Jv& | |
| 3 | 42.857 | Trup 1sn | ^^^1 | ^^^D | Aug 1sn | A1 | J& | |
| 4 | 57.143 | Trudminor 2nd | vvvm2 | vvvEb | Upaug 1sn, Downdim 2nd | ^A1, vd2 | J^&, Kv@@ | |
| 5 | 71.429 | Dudminor 2nd | vvm2 | vvEb | Dim 2nd | d2 | K@@ | |
| 6 | 85.714 | Downminor 2nd | vm2 | vEb | Updim 2nd | ^d2 | K^@@ | |
| 7 | 100.000 | Minor 2nd | m2 | Eb | Downminor 2nd | vm2 | Kv@ | |
| 8 | 114.286 | Upminor 2nd | ^m2 | ^Eb | Minor 2nd | m2 | K@ | |
| 9 | 128.571 | Dupminor 2nd | ^^m2 | ^^Eb | Upminor 2nd | ^m2 | K^@ | |
| 10 | 142.857 | 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 | Downmajor 2nd | vM2 | vE | Downaug 2nd | vA2 | Kv& | |
| 14 | 200.000 | Major 2nd | M2 | E | Aug 2nd | A2 | K& | |
| 15 | 214.286 | Upmajor 2nd | ^M2 | ^E | Upaug 2nd, Downdim 3rd | ^A2, vd3 | K^&, Lv@ | |
| 16 | 228.571 | Dupmajor 2nd | ^^M2 | ^^E | Dim 3rd | d3 | L@ | |
| 17 | 242.857 | Trupmajor 2nd | ^^^M2 | ^^^E | Updim 3rd | ^d3 | L^@ | |
| 18 | 257.143 | Trudminor 3rd | vvvm3 | vvvF | Down 3rd | v3 | Lv | |
| 19 | 271.429 | Dudminor 3rd | vvm2 | vvF | Perfect 3rd | P3 | L | 7/6 |
| 20 | 285.714 | Downminor 3rd | vm3 | vF | Up 3rd | ^3 | L^ | |
| 21 | 300.000 | Minor 3rd | m3 | F | Downaug 3rd | vA3 | Lv& | |
| 22 | 314.286 | Upminor 3rd | ^m3 | ^F | Aug 3rd | A3 | L& | |
| 23 | 328.571 | Dupminor 3rd | ^^m3 | ^^F | Upaug 3rd, Downdim 4th | ^A3, vd4 | L^&, Mv@@ | |
| 24 | 342.857 | Trupminor 3rd | ^^^m3 | ^^^F | Dim 4th | d4 | M@@ | |
| 25 | 357.143 | Trudmajor 3rd | vvvM3 | vvvF# | Updim 4th | ^d4 | M^@@ | |
| 26 | 371.429 | Dudmajor 3rd | vvM3 | vvF# | Downminor 4th | vm4 | Mv@ | |
| 27 | 385.714 | Downmajor 3rd | vM3 | vF# | Minor 4th | m4 | M@ | |
| 28 | 400.000 | Major 3rd | M3 | F# | Upminor 4th | ^m4 | M^@ | |
| 29 | 414.286 | Upmajor 3rd | ^M3 | ^F# | Downmajor 4th | vM4 | Mv | |
| 30 | 428.571 | Dupmajor 3rd | ^^M3 | ^^F# | Major 4th | M4 | M | |
| 31 | 442.857 | Trupmajor 3rd | ^^^M3 | ^^^F# | Upmajor 4th | ^M4 | M^ | |
| 32 | 457.143 | Trud 4th | vvv4 | vvvG | Downaug 4th | vA4 | Mv& | |
| 33 | 471.429 | Dud 4th | vv4 | vvG | Aug 4th | A4 | M& | |
| 34 | 485.714 | Down 4th | v4 | vG | Downminor 5th | vm5 | Nv@ | |
| 35 | 500.000 | Perfect 4th | P4 | G | Minor 5th | m5 | N@ | |
| 36 | 514.286 | Up 4th | ^4 | ^G | Upminor 5th | ^m5 | N^@ | |
| 37 | 528.571 | Dup 4th | ^^4 | ^^G | Downmajor 5th | vM5 | Nv | |
| 38 | 542.857 | Trup 4th | ^^^4 | ^^^G | Major 5th | M5 | N | 11/8 in the 84b val |
| 39 | 557.143 | Trudaug 4th | vvvA4 | vvvG# | Upmajor 5th | ^M5 | N^ | |
| 40 | 571.429 | Dudaug 4th | vvA4 | vvG# | Downaug 5th | vA5 | Nv& | |
| 41 | 585.714 | 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^&, Ov@@ | |
| 43 | 614.286 | Updim 5th | ^d5 | ^Ab | Dim 6th | d6 | O@@ | |
| 44 | 628.571 | Dupdim 5th | ^^d5 | ^^Ab | Updim 6th | ^d6 | O^@@ | |
| 45 | 642.857 | Trupdim 5th | ^^^d5 | ^^^Ab | Downminor 6th | vm6 | Ov@ | |
| 46 | 657.143 | Trud 5th | vvv5 | vvvA | Minor 6th | m6 | O@ | |
| 47 | 671.429 | Dud 5th | vv5 | vvA | Upminor 6th | ^m6 | O^@ | |
| 48 | 685.714 | Down 5th | v5 | vA | Downmajor 6th | vM6 | Ov | |
| 49 | 700.000 | Perfect 5th | P5 | A | Major 6th | M6 | O | 3/2 |
| 50 | 714.286 | Up 5th | ^5 | ^A | Upmajor 6th | ^M6 | O^ | |
| 51 | 728.571 | Dup 5th | ^^5 | ^^A | Dim 7th | d7 | P@@ | |
| 52 | 742.857 | Trup 5th | ^^^5 | ^^^A | Aug 6th | A6 | O& | |
| 53 | 757.143 | Trudminor 6th | vvvm6 | vvvBb | Downminor 7th | vm7 | Pv@ | |
| 54 | 771.429 | Dudminor 6th | vvm6 | vvBb | Minor 7th | m7 | P@ | |
| 55 | 785.714 | Downminor 6th | vm6 | vBb | Upminor 7th | ^m7 | P^@ | |
| 56 | 800.000 | Minor 6th | m6 | Bb | Downmajor 7th | vM7 | Pv | |
| 57 | 814.286 | Upminor 6th | ^m6 | ^Bb | Major 7th | M7 | P | 5/3 |
| 58 | 828.571 | Dupminor 6th | ^^m6 | ^^Bb | Upmajor 7th | ^M7 | P^ | |
| 59 | 842.857 | Trupminor 6th | ^^^m6 | ^^^Bb | Downaug 7th | vA7 | Pv& | |
| 60 | 857.143 | Trudmajor 6th | vvvM6 | vvvB | Aug 7th | A7 | P& | 105/64 |
| 61 | 871.429 | Dudmajor 6th | vvM6 | vvB | Upaug 7th, Downdim 8th | ^A7, vd8 | P^&, Qv@@ | |
| 62 | 885.714 | Downmajor 6th | vM6 | vB | Dim 8th | d8 | Q@@ | |
| 63 | 900.000 | Major 6th | M6 | B | Updim 8th | ^d8 | Q^@@ | |
| 64 | 914.286 | Upmajor 6th | ^M6 | ^B | Down 8th | v8 | Qv@ | |
| 65 | 928.571 | Dupmajor 6th | ^^M6 | ^^B | Perfect 8th | P8 | Q@ | |
| 66 | 942.857 | Trupmajor 6th | ^^^M6 | ^^^B | Up 8th | ^8 | Q^@ | |
| 67 | 957.143 | Trudminor 7th | vvvm7 | vvvC | Downaug 8th | vA8 | Qv | |
| 68 | 971.429 | Dudminor 7th | vvm7 | vvC | Aug 8th | A8 | Q | |
| 69 | 985.714 | Downminor 7th | vm7 | vC | Upaug 8th, Downdim 9th | ^A8, vd9 | Q^, Rv@@ | |
| 70 | 1000.000 | Minor 7th | m7 | C | Dim 9th | d9 | R@@ | |
| 71 | 1014.286 | Upminor 7th | ^m7 | ^C | Updim 9th | ^d9 | R^@@ | |
| 72 | 1028.571 | Dupminor 7th | ^^m7 | ^^C | Downminor 9th | vm9 | Rv@ | |
| 73 | 1042.857 | Trupminor 7th | ^^^m7 | ^^^C | Minor 9th | m9 | R@ | |
| 74 | 1057.143 | Trudmajor 7th | vvvM7 | vvvC# | Upminor 9th | ^m9 | R^@ | |
| 75 | 1071.429 | Dudmajor 7th | vvM7 | vvC# | Downmajor 9th | vM9 | Rv | |
| 76 | 1085.714 | Downmajor 7th | vM7 | vC# | Major 9th | M9 | R | |
| 77 | 1100.000 | Major 7th | M7 | C# | Upmajor 9th | ^M9 | R^ | |
| 78 | 1114.286 | Upmajor 7th | ^M7 | ^C# | Downaug 9th | vA9 | Rv& | |
| 79 | 1128.571 | Dupmajor 7th | ^^M7 | ^^C# | Aug 9th | A9 | R& | |
| 80 | 1142.857 | Trupmajor 7th | ^^^M7 | ^^^C# | Upaug 9th, Downdim 10th | ^A9, vd10 | R^&, Jv@ | |
| 81 | 1157.143 | Trud 8ve | vvv8 | vvvD | Dim 10th | d10 | J@ | |
| 82 | 1171.429 | Dud 8ve | vv8 | vvD | Updim 10th | ^d10 | J^@ | |
| 83 | 1185.714 | Down 8ve | v8 | vD | Down 10th | v10 | Jv | |
| 84 | 1200.000 | Perfect 8ve | P8 | D | Perfect 10th | P10 | J | 2/1 exact |
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 (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 19\84 | 271.428 | 7/6 | Orwell (84e) |
| Newspeak (84p) | ||||
| 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 |
| 28 | 49\84 (1\84) |
500.00 (14.29) |
4/3 (105/104) |
Oquatonic |
Scales
Brightest mode is listed.
- 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
Music
- Ten by John Cage, 1991, for chamber ensemble. Ives Ensemble recording (YouTube)
- Two4 by John Cage, 1991, for violin and piano or shō. Harr & Miyata recording (YouTube)
- Two5 by John Cage, 1991, for tenor trombone and piano. Fulkerson & Denyer recording (YouTube).
- Two6 by John Cage, 1992, for violin and piano. Haar & Snijders recording (YouTube).