84edo

Prime factorization

2² × 3 × 7

Step size

14.2857¢

Fifth

49\84 (700¢) (→7\12)

Semitones (A1:m2)

7:7 (100¢ : 100¢)

Consistency limit

9

Distinct consistency limit

9

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 2^1/84, or the 84th root of 2.

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

Approximation of prime harmonics in 84edo
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
Error	relative (%)	+0	-14	-4	+18	+41	+16	-35	+17	+2	-7	-15	+41
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)

Approximation of prime harmonics in 84edo (continued)
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
Error	relative (%)	-3	+19	+41	-15	-14	-18	+45	+42	+5	+48	+50	+4
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

84edo is a largely composite number. Since 84 factors as 2² × 3 × 7, 84edo has subset edos 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42.

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 a "at" (the symbol "@" unlike in the 4L 5s page cannot be used because of technical details).

Table of 84edo intervals
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^&, Kvaa
5	71.429	Dudminor 2nd	vvm2	vvEb	Dim 2nd	d2	Kaa
6	85.714	Downminor 2nd	vm2	vEb	Updim 2nd	^d2	K^aa
7	100.000	Minor 2nd	m2	Eb	Downminor 2nd	vm2	Kva
8	114.286	Upminor 2nd	^m2	^Eb	Minor 2nd	m2	Ka
9	128.571	Dupminor 2nd	^^m2	^^Eb	Upminor 2nd	^m2	K^a
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^&, Lva
16	228.571	Dupmajor 2nd	^^M2	^^E	Dim 3rd	d3	La
17	242.857	Trupmajor 2nd	^^^M2	^^^E	Updim 3rd	^d3	L^a
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^&, Mvaa
24	342.857	Trupminor 3rd	^^^m3	^^^F	Dim 4th	d4	Maa
25	357.143	Trudmajor 3rd	vvvM3	vvvF#	Updim 4th	^d4	M^aa
26	371.429	Dudmajor 3rd	vvM3	vvF#	Downminor 4th	vm4	Mva
27	385.714	Downmajor 3rd	vM3	vF#	Minor 4th	m4	Ma
28	400.000	Major 3rd	M3	F#	Upminor 4th	^m4	M^a
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	Nva
35	500.000	Perfect 4th	P4	G	Minor 5th	m5	Na
36	514.286	Up 4th	^4	^G	Upminor 5th	^m5	N^a
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^&, Ovaa
43	614.286	Updim 5th	^d5	^Ab	Dim 6th	d6	Oaa
44	628.571	Dupdim 5th	^^d5	^^Ab	Updim 6th	^d6	O^aa
45	642.857	Trupdim 5th	^^^d5	^^^Ab	Downminor 6th	vm6	Ova
46	657.143	Trud 5th	vvv5	vvvA	Minor 6th	m6	Oa
47	671.429	Dud 5th	vv5	vvA	Upminor 6th	^m6	O^a
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	Paa
52	742.857	Trup 5th	^^^5	^^^A	Aug 6th	A6	O&
53	757.143	Trudminor 6th	vvvm6	vvvBb	Downminor 7th	vm7	Pva
54	771.429	Dudminor 6th	vvm6	vvBb	Minor 7th	m7	Pa
55	785.714	Downminor 6th	vm6	vBb	Upminor 7th	^m7	P^a
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^&, Qvaa
62	885.714	Downmajor 6th	vM6	vB	Dim 8th	d8	Qaa
63	900.000	Major 6th	M6	B	Updim 8th	^d8	Q^aa
64	914.286	Upmajor 6th	^M6	^B	Down 8th	v8	Qva
65	928.571	Dupmajor 6th	^^M6	^^B	Perfect 8th	P8	Qa
66	942.857	Trupmajor 6th	^^^M6	^^^B	Up 8th	^8	Q^a
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^, Rvaa
70	1000.000	Minor 7th	m7	C	Dim 9th	d9	Raa
71	1014.286	Upminor 7th	^m7	^C	Updim 9th	^d9	R^aa
72	1028.571	Dupminor 7th	^^m7	^^C	Downminor 9th	vm9	Rva
73	1042.857	Trupminor 7th	^^^m7	^^^C	Minor 9th	m9	Ra
74	1057.143	Trudmajor 7th	vvvM7	vvvC#	Upminor 9th	^m9	R^a
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^&, Jva
81	1157.143	Trud 8ve	vvv8	vvvD	Dim 10th	d10	Ja
82	1171.429	Dud 8ve	vv8	vvD	Updim 10th	^d10	J^a
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
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	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	271.43	7/6	Orwell (84e)
1	19\84	271.43	7/6	Newspeak (84p)
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

Scales

MOS

Brightest mode is listed.

Orwell
- 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

Other

5- to 10-tone scales in 84edo

Music

John Cage

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)

Eliora

Requiem in Gb 1/7 Orwell (2023)

JUMBLE

Undiminished (2023)