84edo
84edo divides the octave into 84 equal parts of size 14.286 cents each.
Theory
Its patent val ⟨84 133 195 236 291] makes it an excellent orwell tuning and also a good one for compton, and the 84e val, ⟨84 133 195 236 290], is almost identical to the 11-limit POTE tuning for orwell. 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. The maximum evenness orwell in this temperament is a 31 note scale, and other MOS are of size 9, 13, and 22. Orwell in 84edo comes in two varieties - 31e & 84, being the proper orwell, and 31 & 84, being newspeak.
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 | |
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).