84edo
Theory
In the 13-limit it is the optimal patent val for the rank five temperament tempering out 144/143.
Orwell
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 MOS of size 9, 13, 22, and 31, of which the 31 note scale is the maximum evenness scale.
Other
84edo is a significantly composite number, with divisors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Being a small multiple of 12, it tempres out the Pythagorean comma, thus supporting period-12 temperament compton. Being a small multiple of 28, it tempers out the oquatonic comma, which maps 5/4 to 9\28.Script error: No such module "primes_in_edo".
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".
| Step | Size (Cents) | Orwell note
(if tonic is J) |
Orwellian Name | Associated ratio |
|---|---|---|---|---|
| 0 | 0.000 | J | unison, prime | 1/1 exact |
| 3 | 42.857 | J& | ||
| 11 | 157.143 | K | second | |
| 19 | 271.429 | L | third | 7/6 |
| 22 | 314.286 | L& | major third | |
| 30 | 428.571 | M | fourth | |
| 38 | 542.857 | N | fifth | 11/8 in the 84b val |
| 41 | 585.714 | N& | ||
| 49 | 700.000 | O | sixth | 3/2 |
| 57 | 814.286 | P | seventh | 5/3 |
| 60 | 857.143 | P& | 105/64 | |
| 68 | 971.429 | Q | eighth | |
| 76 | 1085.714 | R | ninth | |
| 79 | 1128.571 | R& | ||
| 84 | 1200.000 | J (tenth above) | perfect tenth | 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 | |
| 2.3.5.7 | 225/224, 1728/1715, 321489/320000 | ⟨84 133 195 236] | 0.141 | 0.769 | |
| 2.3.5.7.11 | 225/224, 441/440, 1944/1925, 8019/8000 | ⟨84 133 195 236 291] | -0.225 | 1.003 | |
| 2.3.5.7.11 | 99/98, 121/120, 1728/1715, 321489/320000 | ⟨84 133 195 236 290] (84e) | 0.601 | 1.151 | |
Rank-2 temperaments by generator
| Periods
per octave |
Generator | Cents | Associated
ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 19\84 | 271.428 | 7/6 | Orwell (84e val) |
| Newspeak (84p val) | ||||
| 1 | 27\84 | 385.714 | 5/4 | Mutt |
| 12 | 27\84
(6\84) |
385.714
(85.714) |
5/4
(20480/19683) |
Compton |
| 28 | 49\84
(1\84) |
500.000
(14.286) |
4/3
(105/104) |
Oquatonic |
Scales
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).