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	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

84edo shares the perfect fifth with 12edo, tempering out the Pythagorean comma in its patent val. In the 5-limit it tempers out the sensipent comma; in the 7-limit 225/224, 1728/1715, 2430/2401, 6144/6125, supporting orwell, compton, and sensei. In the 13-limit it is the optimal patent val for the rank-5 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. 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] supporting newspeak. 84edo orwell offers mos scales of size 9, 13, 22, and 31, of which the 31-note scale is the maximal evenness scale.

High limit consistency and coverage

It has fairly good approximation to higher prime harmonics such as 13, 19, 23, 29, 31, 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit (with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics). In fact, if we avoid all intervals of 11 and 17 as well as the complex compound prime powers 27 and 49, it is completely consistent in the no-37's no-47's 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~1.3 ¢ (off by ~7.3 ¢), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too (when supported by context). Except 17, the only missing primes are thus 37, 47, 67, 71, 79 and 83, which coincidentally are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).

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
	Relative (%)	+0.0	-13.7	-4.2	+18.2	+40.8	+16.3	-34.7	+17.4	+2.1	-7.0	-15.2	+40.6
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
	Relative (%)	-3.4	+19.4	+41.5	-14.5	-14.2	-18.2	+44.9	+42.1	+5.5	+48.2	+49.7	+3.8
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

84 is a largely composite number. Since 84 factors as 2² × 3 × 7, 84edo has subset edos 2, 3, 4, 6, 7, 12, 14, 21, 28, 42. Being a small multiple of 28, it tempers out the oquatonic comma, which maps 5/4 to 9\28.

Intervals

#	Cents	Approximate ratios*	Ups and downs notation
0	0.0	1/1	Perfect 1sn	P1	D
1	14.3	81/80, 105/104, 126/125, 169/168, 196/195	Up 1sn	^1	^D
2	28.6	50/49, 64/63, 65/64, 91/90	Dup 1sn	^^1	^^D
3	42.9	36/35, 40/39, 46/45, 49/48	Trup 1sn	^^^1	^^^D
4	57.1	27/26	Trudminor 2nd	vvvm2	vvvEb
5	71.4	24/23, 25/24, 26/25, 28/27	Dudminor 2nd	vvm2	vvEb
6	85.7	20/19, 21/20	Downminor 2nd	vm2	vEb
7	100.0	19/18	Minor 2nd	m2	Eb
8	114.3	15/14, 16/15	Upminor 2nd	^m2	^Eb
9	128.6	14/13	Dupminor 2nd	^^m2	^^Eb
10	142.9	13/12	Trupminor 2nd	^^^m2	^^^Eb
11	157.1	23/21	Trudmajor 2nd	vvvM2	vvvE
12	171.4	21/19	Dudmajor 2nd	vvM2	vvE
13	185.7	10/9	Downmajor 2nd	vM2	vE
14	200.0	9/8	Major 2nd	M2	E
15	214.3	26/23	Upmajor 2nd	^M2	^E
16	228.6	8/7	Dupmajor 2nd	^^M2	^^E
17	242.9	15/13, 23/20	Trupmajor 2nd	^^^M2	^^^E
18	257.1	52/45	Trudminor 3rd	vvvm3	vvvF
19	271.4	7/6	Dudminor 3rd	vvm2	vvF
20	285.7	45/38, 46/39	Downminor 3rd	vm3	vF
21	300.0	19/16, 25/21, 32/27	Minor 3rd	m3	F
22	314.3	6/5	Upminor 3rd	^m3	^F
23	328.6	23/19	Dupminor 3rd	^^m3	^^F
24	342.9	28/23, 39/32	Trupminor 3rd	^^^m3	^^^F
25	357.1	16/13	Trudmajor 3rd	vvvM3	vvvF#
26	371.4	26/21	Dudmajor 3rd	vvM3	vvF#
27	385.7	5/4	Downmajor 3rd	vM3	vF#
28	400.0	24/19	Major 3rd	M3	F#
29	414.3	19/15	Upmajor 3rd	^M3	^F#
30	428.6	9/7, 23/18, 32/25	Dupmajor 3rd	^^M3	^^F#
31	442.9	84/65	Trupmajor 3rd	^^^M3	^^^F#
32	457.1	13/10, 30/23	Trud 4th	vvv4	vvvG
33	471.4	21/16	Dud 4th	vv4	vvG
34	485.7	65/49	Down 4th	v4	vG
35	500.0	4/3	Perfect 4th	P4	G
36	514.3	27/20	Up 4th	^4	^G
37	528.6	19/14	Dup 4th	^^4	^^G
38	542.9	26/19	Trup 4th	^^^4	^^^G
39	557.1	18/13	Trudaug 4th	vvvA4	vvvG#
40	571.4	25/18, 32/23	Dudaug 4th	vvA4	vvG#
41	585.7	7/5	Downaug 4th	vA4	vG#
42	600.0	27/19, 38/27	Aug 4th, Dim 5th	A4, d5	G#, Ab
43	614.3	10/7	Updim 5th	^d5	^Ab
44	628.6	23/16, 36/25	Dupdim 5th	^^d5	^^Ab
45	642.9	13/9	Trupdim 5th	^^^d5	^^^Ab
46	657.1	19/13	Trud 5th	vvv5	vvvA
47	671.4	28/19	Dud 5th	vv5	vvA
48	685.7	40/27	Down 5th	v5	vA
49	700.0	3/2	Perfect 5th	P5	A
50	714.3	98/65	Up 5th	^5	^A
51	728.6	32/21	Dup 5th	^^5	^^A
52	742.9	20/13, 23/15	Trup 5th	^^^5	^^^A
53	757.1	65/42	Trudminor 6th	vvvm6	vvvBb
54	771.4	14/9, 25/16, 36/23	Dudminor 6th	vvm6	vvBb
55	785.7	30/19	Downminor 6th	vm6	vBb
56	800.0	19/12	Minor 6th	m6	Bb
57	814.3	8/5	Upminor 6th	^m6	^Bb
58	828.6	21/13	Dupminor 6th	^^m6	^^Bb
59	842.9	13/8	Trupminor 6th	^^^m6	^^^Bb
60	857.1	23/14, 64/39	Trudmajor 6th	vvvM6	vvvB
61	871.4	38/23	Dudmajor 6th	vvM6	vvB
62	885.7	5/3	Downmajor 6th	vM6	vB
63	900.0	32/19, 27/16, 42/25	Major 6th	M6	B
64	914.3	39/23, 76/45	Upmajor 6th	^M6	^B
65	928.6	12/7	Dupmajor 6th	^^M6	^^B
66	942.9	45/26	Trupmajor 6th	^^^M6	^^^B
67	957.1	26/15, 40/23	Trudminor 7th	vvvm7	vvvC
68	971.4	7/4	Dudminor 7th	vvm7	vvC
69	985.7	23/13	Downminor 7th	vm7	vC
70	1000.0	16/9	Minor 7th	m7	C
71	1014.3	9/5	Upminor 7th	^m7	^C
72	1028.6	38/21	Dupminor 7th	^^m7	^^C
73	1042.9	42/23	Trupminor 7th	^^^m7	^^^C
74	1057.1	24/13	Trudmajor 7th	vvvM7	vvvC#
75	1071.4	13/7	Dudmajor 7th	vvM7	vvC#
76	1085.7	15/8, 28/15	Downmajor 7th	vM7	vC#
77	1100.0	36/19	Major 7th	M7	C#
78	1114.3	19/10, 40/21	Upmajor 7th	^M7	^C#
79	1128.6	23/12, 25/13, 27/14, 48/25	Dupmajor 7th	^^M7	^^C#
80	1142.9	52/27	Trupmajor 7th	^^^M7	^^^C#
81	1157.1	35/18, 39/20, 96/49	Trud 8ve	vvv8	vvvD
82	1171.4	45/23, 49/25, 63/32, 128/65, 180/91	Dud 8ve	vv8	vvD
83	1185.7	125/63, 160/81, 195/98, 336/169	Down 8ve	v8	vD
84	1200.0	2/1	Perfect 8ve	P8	D

* As a 2.3.5.7.13.19.23-subgroup temperament

Notation

Ups and downs notation

Using Helmholtz–Ellis accidentals, 84edo can be notated using ups and downs notation:

Semitones	0	1⁄7	2⁄7	3⁄7	4⁄7	5⁄7	6⁄7	1	1+1⁄7	1+2⁄7	1+3⁄7	1+4⁄7	1+5⁄7	1+6⁄7	2	1+1⁄7	1+2⁄7	1+3⁄7
Sharp symbol
Flat symbol

4L 5s (gramitonic) notation

This notation is based on Orwell[9]. 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").

#	Cents	Note	Name	Associated Ratio
0	0.0	J	Perfect 0-gramstep	1/1
8	114.3	K@	Minor 1-gramstep	15/14~16/15
11	157.1	K	Major 1-gramstep	11/10~12/11
16	228.6	L@	Diminished 2-gramstep	8/7
19	271.4	L	Perfect 2-gramstep	7/6
27	385.7	M@	Minor 3-gramstep	5/4
30	428.6	M	Major 3-gramstep	9/7
35	500.0	N@	Minor 4-gramstep	4/3
38	542.9	N	Major 4-gramstep	11/8~15/11
46	657.1	O@	Minor 5-gramstep	16/11~22/15
49	700.0	O	Major 5-gramstep	3/2
54	771.4	P@	Minor 6-gramstep	14/9
57	814.3	P	Major 6-gramstep	8/5
65	928.6	Q@	Perfect 7-gramstep	12/7
68	971.4	Q	Augmented 7-gramstep	7/4
73	1042.9	R@	Minor 8-gramstep	11/6~20/11
76	1085.7	R	Major 8-gramstep	15/8~28/15
84	1200.0	J	Perfect 9-gramstep	2/1

Approximation to JI

Zeta peak index

Tuning			Strength			Closest edo		Integer limit
ZPI	Steps per octave	Step size (cents)	Height	Integral	Gap	Edo	Octave (cents)	Consistent	Distinct
462zpi	83.9972142607288	14.2861880666087	8.020965	1.241945	16.733121	84edo	1200.03979759513	10	10

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
2.3.5.7.13	225/224, 351/350, 640/637, 1701/1690	[⟨84 133 195 236 311]]	−0.013	0.754	5.28
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
2.3.5.7.11.13	144/143, 225/224, 351/350, 441/440, 975/968	[⟨84 133 195 236 291 311]] (84)	−0.292	0.928	6.50
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
2.3.5.7.11.13	99/98, 121/120, 176/175, 275/273, 1701/1690	[⟨84 133 195 236 290 311]] (84e)	+0.396	1.146	8.02

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	19\84	271.43	7/6	Orwell (84e) / newspeak (84)
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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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

Instruments

If you have a precise enough tuner and stable enough instruments, 84edo can be played using 7 instruments tuned a 14th of a tone apart.

You could also try the Lumatone mapping for 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)