684edo

← 683edo

684edo

685edo →

Prime factorization	22 × 32 × 19
Step size	1.75439 ¢
Fifth	400\684 (701.754 ¢) (→ 100\171)
Semitones (A1:m2)	64:52 (112.3 ¢ : 91.23 ¢)
Consistency limit	17
Distinct consistency limit	17

684 equal divisions of the octave (abbreviated 684edo or 684ed2), also called 684-tone equal temperament (684tet) or 684 equal temperament (684et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 684 equal parts of about 1.75 ¢ each. Each step represents a frequency ratio of 2^1/684, or the 684th root of 2.

Theory

684edo divides the steps of 171edo into four. It is consistent to the 17-odd-limit, tempering out 2401/2400, 3025/3024, 4225/4224, 4375/4374, 9801/9800, 10648/10647, 32805/32768, 41503/41472, 43923/43904, 50193/50176, 59535/59488, and 65625/65536 in the 13-limit; 1089/1088, 1225/1224, 1701/1700, 2025/2023, 2058/2057, 2500/2499, 8624/8619, and 14875/14872 in the 17-limit.

Prime harmonics

Approximation of prime harmonics in 684edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.201	-0.349	-0.405	-0.441	-0.177	+0.308	+0.733	-0.204	+0.247	+0.578
	Relative (%)	+0.0	-11.4	-19.9	-23.1	-25.1	-10.1	+17.5	+41.8	-11.6	+14.1	+33.0
Steps (reduced)		684 (0)	1084 (400)	1588 (220)	1920 (552)	2366 (314)	2531 (479)	2796 (60)	2906 (170)	3094 (358)	3323 (587)	3389 (653)

Subsets and supersets

Since 684 factors into 2² × 3² × 19, 684edo has subset edos 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342.

Approximation to JI

Zeta peak index

Tuning			Strength				Octave (cents)		Integer limit
ZPI	Steps per 8ve	Step size (cents)	Height		Integral	Gap	Size	Stretch	Consistent	Distinct
			Tempered	Pure
5818zpi	683.938935	1.754543	14.267321	7.268914	1.773752	20.109967	1200.107141	0.107141	18	18

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5.7.11.13	2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768	[⟨684 1084 1588 1920 2366 2531]]	+0.0994	0.0558	3.18
2.3.5.7.11.13.17	1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005	[⟨684 1084 1588 1920 2366 2531 2796]]	+0.0744	0.0800	4.56

• 684et is the first equal temperament past 494 with a lower 13-limit absolute error. The next equal temperament that is better tuned is 764.

Rank-2 temperaments

Note: 11-limit temperaments supported by 342et are not shown.

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
18	271\684 (5\684)	475.44 (8.77)	1053/800 (1287/1280)	Semihemiennealimmal
38	151\684 (7\684)	264.91 (12.28)	500/429 (144/143)	Semihemienneadecal

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

Scales

• Diamond15: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64

Commas of the 2.3.5.7.11.13 subgroup tempered out in 684edo patent val

Commas with numerator <= 2^32:

Ratio	Monzo	Cents
2401/2400	[-5 -1 -2 4⟩	0.721
3025/3024	[-4 -3 2 -1 2⟩	0.572
4225/4224	[-7 -1 2 0 -1 2⟩	0.410
4375/4374	[-1 -7 4 1⟩	0.396
9801/9800	[-3 4 -2 -2 2⟩	0.177
10648/10647	[3 -2 0 -1 3 -2⟩	0.163
32805/32768	[-15 8 1⟩	1.954
41503/41472	[-9 -4 0 3 2⟩	1.294
43923/43904	[-7 1 0 -3 4⟩	0.749
50193/50176	[-10 3 0 -2 1 2⟩	0.586
59535/59488	[-5 5 1 2 -1 -2⟩	1.367
65625/65536	[-16 1 5 1⟩	2.349
78125/78078	[-1 -1 7 -1 -1 -2⟩	1.042
85750/85683	[1 -1 3 3 0 -4⟩	1.353
86625/86528	[-9 2 3 1 1 -2⟩	1.940
91125/91091	[0 6 3 -2 -1 -2⟩	0.646
102487/102400	[-12 0 -2 1 4⟩	1.470
105644/105625	[2 0 -4 4 1 -2⟩	0.311
114345/114244	[-2 3 1 1 2 -4⟩	1.530
117649/117612	[-2 -5 0 6 -2⟩	0.545
123201/123200	[-6 6 -2 -1 -1 2⟩	0.014
142884/142805	[2 6 -1 2 0 -4⟩	0.957
151263/151250	[-1 2 -4 5 -2⟩	0.149
160083/160000	[-8 3 -4 2 2⟩	0.898
200000/199927	[6 0 5 -1 0 -4⟩	0.632
216513/216320	[-8 9 -1 0 1 -2⟩	1.544
225000/224939	[3 2 5 0 -3 -2⟩	0.469
234375/234256	[-4 1 7 0 -4⟩	0.879
250047/250000	[-4 6 -6 3⟩	0.325
369754/369603	[1 -7 0 5 1 -2⟩	0.707
405769/405504	[-12 -2 0 4 -1 2⟩	1.131
420175/419904	[-6 -8 2 5⟩	1.117
456533/456300	[-2 -3 -2 3 3 -2⟩	0.884
496125/495616	[-12 4 3 2 -2⟩	1.777
703125/702464	[-11 2 7 -3⟩	1.628
759375/758912	[-7 5 5 -2 -2⟩	1.056
819819/819200	[-15 2 -2 2 1 2⟩	1.308
820125/819896	[-3 8 3 -1 -4⟩	0.483
900375/899756	[-2 1 3 4 -3 -2⟩	1.191
928125/927472	[-4 3 5 -3 1 -2⟩	1.218
1071875/1070784	[-6 -2 5 3 -1 -2⟩	1.763
1161875/1161216	[-11 -4 4 -1 1 2⟩	0.982
1240029/1239040	[-11 11 -1 1 -2⟩	1.381
1399680/1399489	[7 7 1 -2 0 -4⟩	0.236
1449459/1449175	[0 2 -2 -3 5 -2⟩	0.339
1574573/1572864	[-19 -1 0 1 3 2⟩	1.880
1574640/1574573	[4 9 1 -1 -3 -2⟩	0.074
1771561/1769472	[-16 -3 0 0 6⟩	2.043
1771561/1771470	[-1 -11 -1 0 6⟩	0.089
1771875/1771561	[0 4 5 1 -6⟩	0.307
1830125/1827904	[-6 0 3 0 4 -4⟩	2.102
1890625/1889568	[-5 -10 6 0 2⟩	0.968
1953125/1951488	[-8 -2 9 -1 -2⟩	1.452
2100875/2097152	[-21 0 3 5⟩	3.071
2250423/2249390	[-1 8 -1 3 -3 -2⟩	0.795
2460375/2458624	[-10 9 3 -4⟩	1.233
2657205/2656192	[-6 12 1 -3 -2⟩	0.660
2773155/2768896	[-14 1 1 5 1 -2⟩	2.661
3188646/3188185	[1 13 -1 -3 -1 -2⟩	0.250
3247695/3246152	[-3 10 1 -4 1 -2⟩	0.823
3294225/3294172	[-2 2 2 -7 4⟩	0.028
3327500/3326427	[2 -9 4 0 3 -2⟩	0.558
3382071/3380000	[-5 1 -4 1 5 -2⟩	1.060
3456000/3455881	[10 3 3 0 -2 -4⟩	0.060
3457440/3455881	[5 2 1 4 -2 -4⟩	0.781
3675375/3670016	[-19 5 3 -1 2⟩	2.526
3720087/3718000	[-4 12 -3 1 -1 -2⟩	0.972
3750705/3748096	[-8 7 1 3 -4⟩	1.205
3764768/3764475	[5 -4 -2 6 -1 -2⟩	0.135
4026275/4024566	[-1 -5 2 -2 5 -2⟩	0.735
4117715/4112784	[-4 -2 1 7 0 -4⟩	2.074
4159375/4153344	[-13 -1 5 0 3 -2⟩	2.512
4198467/4194304	[-22 1 0 2 0 4⟩	1.717
4829440/4826809	[8 0 1 3 1 -6⟩	0.943
4831530/4826809	[1 1 1 0 5 -6⟩	1.692
4851495/4845568	[-12 6 1 -1 3 -2⟩	2.116
5175625/5174928	[-4 -5 4 2 -3 2⟩	0.233
5250987/5242880	[-20 7 -1 4⟩	2.675
5282739/5281250	[-1 4 -6 2 3 -2⟩	0.488
5623475/5619712	[-14 0 2 -3 3 2⟩	1.159
5764801/5760000	[-10 -2 -4 8⟩	1.442
6426225/6422528	[-17 2 2 -2 0 4⟩	0.996
6940323/6938624	[-13 5 0 -1 -2 4⟩	0.424
7144929/7140250	[-1 10 -3 0 2 -4⟩	1.134
7503125/7496192	[-9 0 5 4 -4⟩	1.600
7891499/7884864	[-6 -6 0 2 5 -2⟩	1.456
8521149/8518400	[-8 1 -2 5 -3 2⟩	0.559
9150625/9144576	[-8 -6 4 -2 4⟩	1.145
9453125/9437184	[-20 -2 7 0 2⟩	2.922
9529569/9518080	[-10 4 -1 6 -1 -2⟩	2.088
9801000/9796423	[3 4 3 -3 2 -4⟩	0.809
14085981/14080000	[-11 5 -4 3 -1 2⟩	0.735
14348907/14336000	[-14 15 -3 -1⟩	1.558
14348907/14348180	[-2 15 -1 -2 -4⟩	0.088
14994525/14992384	[-10 1 2 1 -4 4⟩	0.247
15882615/15859712	[-17 3 1 6 -2⟩	2.498
15940925/15925248	[-16 -5 2 3 1 2⟩	1.703
16808715/16777216	[-24 4 1 3 2⟩	3.247
17294403/17279405	[0 1 -1 8 -2 -4⟩	1.502
17850625/17842176	[-14 -2 4 0 -2 4⟩	0.820
17935225/17915904	[-13 -7 2 2 4⟩	1.866
18302823/18279040	[-7 2 -1 5 2 -4⟩	2.251
18484375/18475776	[-8 -8 6 1 -1 2⟩	0.806
18528125/18507528	[-3 -4 5 2 2 -4⟩	1.926
19140625/19131876	[-2 -14 8 2⟩	0.792
19487171/19468800	[-9 -2 -2 0 7 -2⟩	1.633
19487171/19476912	[-4 -1 0 -4 7 -2⟩	0.912
21437500/21434787	[2 -11 6 3 -2⟩	0.219
21484375/21464352	[-5 -4 9 -2 1 -2⟩	1.614
24149664/24134045	[5 4 -1 1 3 -6⟩	1.120
26796875/26763264	[-13 -3 7 3 -2⟩	2.173
27217619/27216000	[-7 -5 -3 -1 5 2⟩	0.103
27227340/27217619	[2 4 1 5 -5 -2⟩	0.618
28588707/28561000	[-3 5 -3 6 0 -4⟩	1.679
28824005/28792192	[-7 0 1 8 -3 -2⟩	1.912
29046875/29042496	[-6 -3 6 -5 1 2⟩	0.261
29296875/29246464	[-10 1 10 0 0 -4⟩	2.981
31250000/31244213	[4 0 9 -5 -1 -2⟩	0.321
32019867/31988320	[-5 7 -1 -1 4 -4⟩	1.707
32019867/32000000	[-11 7 -6 0 4⟩	1.074
33792000/33787663	[13 1 3 -1 1 -6⟩	0.222
35156250/35153041	[1 2 9 -4 -4⟩	0.158
37812500/37786203	[2 -3 7 -2 2 -4⟩	1.204
38671875/38614472	[-3 2 8 0 1 -6⟩	2.572
40353607/40310784	[-11 -9 0 9⟩	1.838
42109375/42052608	[-10 -5 7 2 1 -2⟩	2.335
43046721/43025920	[-9 16 -1 -5⟩	0.837
44289025/44255232	[-11 -2 2 -4 6⟩	1.321
46200375/46137344	[-22 7 3 0 -1 2⟩	2.364
47265625/47258883	[0 -9 8 -4 2⟩	0.247
48828125/48771072	[-12 -5 11 -2⟩	2.024
48828125/48787596	[-2 -8 11 0 -1 -2⟩	1.438
48828125/48807528	[-3 -1 11 -5 -2⟩	0.730
50014503/50000000	[-7 10 -8 1 2⟩	0.502
52734375/52706752	[-6 3 9 -7⟩	0.907
53094899/53084160	[-17 -4 -1 0 1 6⟩	0.350
53144100/53094899	[2 12 2 0 -1 -6⟩	1.604
53156250/53094899	[1 5 6 1 -1 -6⟩	1.999
54687500/54660177	[2 -5 9 1 -3 -2⟩	0.865
56723625/56689952	[-5 3 3 5 -6⟩	1.028
56953125/56942116	[-2 6 7 -6 -2⟩	0.335
62500000/62462907	[5 -7 9 0 0 -4⟩	1.028
63412811/63375000	[-3 -1 -6 8 1 -2⟩	1.033
64304361/64225280	[-18 12 -1 -2 2⟩	2.130
66015625/66012408	[-3 -7 8 -3 -1 2⟩	0.084
72471784/72402135	[3 -1 -1 7 1 -6⟩	1.665
78125000/78121827	[3 -13 10 -2⟩	0.070
79893275/79834248	[-3 -10 2 4 3 -2⟩	1.280
89282088/89253125	[3 13 -5 1 0 -4⟩	0.562
92421875/92274688	[-23 0 7 1 -1 2⟩	2.759
95703125/95551488	[-17 -6 9 2⟩	2.745
95703125/95664294	[-1 -3 9 2 -6⟩	0.703
96059601/96040000	[-6 8 -4 -4 4⟩	0.353
99648703/99532800	[-14 -5 -2 7 2⟩	2.015
102515625/102362624	[-9 8 6 -1 0 -4⟩	2.586
102942875/102795264	[-11 -3 3 7 -1 -2⟩	2.484
103984375/103876864	[-8 0 7 -4 3 -2⟩	1.791
109772091/109760000	[-9 10 -4 -3 1 2⟩	0.191
110716875/110588192	[-5 11 4 0 -2 -4⟩	2.013
110937519/110755840	[-17 5 -1 3 3 -2⟩	2.838
113379904/113358609	[6 -4 0 -2 6 -4⟩	0.325
120955835/120932352	[-11 -10 1 1 2 4⟩	0.336
121060821/121000000	[-6 1 -6 9 -2⟩	0.870
128119761/128000000	[-13 2 -6 6 2⟩	1.619
130438000/130323843	[4 -3 3 2 3 -6⟩	1.516
135320625/135150652	[-2 9 4 -1 1 -6⟩	2.176
136088095/136048896	[-8 -12 1 0 5 2⟩	0.499
137259375/137149922	[-1 1 5 -4 4 -4⟩	1.381
139178767/139148064	[-5 -3 0 7 -5 2⟩	0.382
139218750/139178767	[1 4 7 -7 1 -2⟩	0.497
141776649/141724880	[-4 10 -1 4 -6⟩	0.632
143496441/143360000	[-15 4 -4 -1 6⟩	1.647
160699616/160655625	[5 -2 -4 3 4 -4⟩	0.474
163980531/163840000	[-18 6 -4 0 3 2⟩	1.484
173282571/173056000	[-13 8 -3 4 1 -2⟩	2.265
173612978/173508075	[1 -5 -2 2 6 -4⟩	1.046
175765205/175478784	[-11 -1 1 4 4 -4⟩	2.823
181575625/181398528	[-10 -11 4 4 2⟩	1.689
184528125/184473632	[-5 10 5 -8⟩	0.511
187404800/187388721	[9 -8 2 0 4 -4⟩	0.149
187578125/187388721	[0 -8 7 4 0 -4⟩	1.749
199297406/199290375	[1 -13 -3 7 2⟩	0.061
200120949/200000000	[-9 5 -8 7⟩	1.047
201768035/201326592	[-26 -1 1 9⟩	3.792
214358881/214207500	[-2 -1 -4 0 8 -4⟩	1.223
214358881/214326000	[-4 -7 -3 -2 8⟩	0.266
214375000/214358881	[3 0 7 3 -8⟩	0.130
218709491/218700000	[-5 -7 -5 6 1 2⟩	0.075
221484375/221176384	[-6 4 8 1 -2 -4⟩	2.409
228765625/228709656	[-3 -5 6 -6 4⟩	0.424
234365481/233971712	[-13 14 0 2 0 -4⟩	2.911
234365481/234256000	[-7 14 -3 2 -4⟩	0.809
236328125/236027904	[-15 -1 9 -4 2⟩	2.201
244140625/243662848	[-17 0 12 0 -1 -2⟩	3.391
245000000/244958571	[6 -2 7 2 -5 -2⟩	0.293
246071287/245760000	[-17 -1 -4 5 4⟩	2.191
246071287/246037500	[-2 -9 -5 5 4⟩	0.238
256134879/256000000	[-14 9 -6 1 1 2⟩	0.912
265531392/265474495	[12 3 -1 4 -1 -6⟩	0.371
269028375/268435456	[-28 1 3 2 4⟩	3.820
269500000/269440587	[5 -13 6 2 1 -2⟩	0.382
275653125/275365888	[-14 6 5 -5 2⟩	1.805
280000000/279926361	[9 -4 7 1 -2 -4⟩	0.455
282475249/282268800	[-7 -6 -2 10 -2⟩	1.266
282475249/282335625	[0 -5 -4 10 -1 -2⟩	0.856
288240050/288178803	[1 -9 2 8 -4⟩	0.368
329417200/329316273	[4 -11 2 7 -1 -2⟩	0.530
330078125/329832448	[-11 0 9 0 -5 2⟩	1.289
341796875/341545248	[-5 -6 11 1 -4⟩	1.275
342956592/342874805	[4 11 -1 -4 2 -4⟩	0.413
349274835/348913664	[-18 10 1 1 -3 2⟩	1.791
355117455/354418688	[-21 2 1 2 5 -2⟩	3.410
358722675/358269184	[-8 15 2 -2 0 -4⟩	2.190
363862125/363569024	[-7 7 3 -5 3 -2⟩	1.395
386683451/386144720	[-4 0 -1 4 5 -6⟩	2.414
387420489/387058672	[-4 18 0 -1 -2 -4⟩	1.618
387420489/387200000	[-10 18 -5 0 -2⟩	0.986
399466375/398721024	[-18 -2 3 4 3 -2⟩	3.233
411778125/411041792	[-23 2 5 -2 4⟩	3.099
418161601/418037760	[-14 -6 -1 -1 4 4⟩	0.513
421760625/421654016	[-9 1 4 -7 3 2⟩	0.438
425329947/424759192	[-3 11 0 4 -1 -6⟩	2.325
428830605/428717762	[-1 6 1 6 -8⟩	0.456
430489323/430259200	[-10 5 -2 -5 6⟩	0.926
432081216/431985125	[6 9 -3 3 -2 -4⟩	0.385
443889677/443023360	[-19 0 -1 9 1 -2⟩	3.382
443889677/443523600	[-4 -8 -2 9 1 -2⟩	1.428
446265625/445906944	[-18 -5 6 -1 0 4⟩	1.392
446265625/446240256	[-9 -1 6 -4 -2 4⟩	0.098
446433372/446265625	[2 2 -6 1 6 -4⟩	0.651
468838125/467943424	[-14 7 4 3 0 -4⟩	3.307
470559375/469762048	[-26 4 5 -1 1 2⟩	2.936
473513931/473027282	[-1 16 0 -2 1 -6⟩	1.780
473513931/473200000	[-7 16 -5 -1 1 -2⟩	1.148
478515625/477854091	[0 -2 10 2 -1 -6⟩	2.395
480024727/479756288	[-15 0 0 5 -4 4⟩	0.968
480024727/480000000	[-11 -1 -7 5 0 4⟩	0.089
480298005/480024727	[0 8 1 -5 4 -4⟩	0.985
481966875/481890304	[-12 3 4 -6 0 4⟩	0.275
491941593/491724800	[-13 7 -2 -4 3 2⟩	0.763
497067025/496793088	[-9 -6 2 6 -3 2⟩	0.954
514714375/514434888	[-3 -12 4 7 -2⟩	0.940
520613632/520524225	[8 -6 -2 5 2 -4⟩	0.297
537109375/536832387	[0 -3 11 -6 1 -2⟩	0.893
540078539/540000000	[-8 -3 -7 4 3 2⟩	0.252
543547125/542703616	[-16 3 3 -2 5 -2⟩	2.689
562166163/561971200	[-16 9 -2 -3 0 4⟩	0.601
562448656/562166163	[4 -9 0 4 4 -4⟩	0.870
600362847/599695360	[-13 6 -1 7 -4⟩	1.926
603542016/603351125	[9 7 -3 2 1 -6⟩	0.548
603993159/603351125	[0 3 -3 5 3 -6⟩	1.841
604779175/603979776	[-26 -2 2 1 2 4⟩	2.290
607645423/607500000	[-5 -5 -7 3 6⟩	0.414
618866325/617831552	[-7 8 2 3 1 -6⟩	2.897
643076643/642252800	[-19 1 -2 -2 8⟩	2.219
645700815/645657712	[-4 17 1 -9⟩	0.116
656128473/655360000	[-20 1 -4 6 1 2⟩	2.029
672417207/671088640	[-27 8 -1 1 4⟩	3.424
680440475/679477248	[-23 -4 2 0 5 2⟩	2.452
698709375/697827328	[-19 3 5 2 -3 2⟩	2.187
703096443/702934375	[0 15 -5 2 -3 -2⟩	0.399
717482205/716538368	[-9 4 1 -2 6 -4⟩	2.279
720600125/719634432	[-14 -1 3 8 -4⟩	2.322
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