84edo

← 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	7
Distinct consistency limit	7

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.

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".

Table of 84edo intervals

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

• Orwell[9] - 4L 5s
• Orwell[13] - 9L 4s
• 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).