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
1610510000/1610497161	[4 -4 4 -6 5 -2⟩	0.014
123201/123200	[-6 6 -2 -1 -1 2⟩	0.014
781258401/781250000	[-4 2 -11 2 6⟩	0.019
3294225/3294172	[-2 2 2 -7 4⟩	0.028
1475824779/1475789056	[-8 8 0 -8 3 2⟩	0.042
1433600000/1433562273	[16 -3 5 1 -1 -6⟩	0.046
3456000/3455881	[10 3 3 0 -2 -4⟩	0.060
199297406/199290375	[1 -13 -3 7 2⟩	0.061
78125000/78121827	[3 -13 10 -2⟩	0.070
1574640/1574573	[4 9 1 -1 -3 -2⟩	0.074
218709491/218700000	[-5 -7 -5 6 1 2⟩	0.075
66015625/66012408	[-3 -7 8 -3 -1 2⟩	0.084
14348907/14348180	[-2 15 -1 -2 -4⟩	0.088
1771561/1771470	[-1 -11 -1 0 6⟩	0.089
480024727/480000000	[-11 -1 -7 5 0 4⟩	0.089
1650478599/1650390625	[0 11 -10 1 3 -2⟩	0.092
446265625/446240256	[-9 -1 6 -4 -2 4⟩	0.098
974308500/974251369	[2 11 3 -8 1 -2⟩	0.102
27217619/27216000	[-7 -5 -3 -1 5 2⟩	0.103
645700815/645657712	[-4 17 1 -9⟩	0.116
1254484803/1254400000	[-13 1 -5 -2 4 4⟩	0.117
1686616064/1686498489	[11 -10 0 7 0 -4⟩	0.121
214375000/214358881	[3 0 7 3 -8⟩	0.130
3764768/3764475	[5 -4 -2 6 -1 -2⟩	0.135
187404800/187388721	[9 -8 2 0 4 -4⟩	0.149
151263/151250	[-1 2 -4 5 -2⟩	0.149
35156250/35153041	[1 2 9 -4 -4⟩	0.158
1224552875/1224440064	[-8 -14 3 3 0 4⟩	0.159
10648/10647	[3 -2 0 -1 3 -2⟩	0.163
2662250409/2662000000	[-7 8 -6 4 -3 2⟩	0.163
1182568205/1182449664	[-14 -8 1 2 -1 6⟩	0.174
9801/9800	[-3 4 -2 -2 2⟩	0.177
3955078125/3954653486	[-1 4 11 -11⟩	0.186
974358550/974251369	[1 0 2 -8 7 -2⟩	0.190
109772091/109760000	[-9 10 -4 -3 1 2⟩	0.191
1291467969/1291315424	[-5 6 0 -9 6⟩	0.205
2091059712/2090808005	[9 5 -1 5 -4 -4⟩	0.208
21437500/21434787	[2 -11 6 3 -2⟩	0.219
33792000/33787663	[13 1 3 -1 1 -6⟩	0.222
3402639576/3402202375	[3 11 -3 4 -5 -2⟩	0.222
5175625/5174928	[-4 -5 4 2 -3 2⟩	0.233
1399680/1399489	[7 7 1 -2 0 -4⟩	0.236
246071287/246037500	[-2 -9 -5 5 4⟩	0.238
47265625/47258883	[0 -9 8 -4 2⟩	0.247
14994525/14992384	[-10 1 2 1 -4 4⟩	0.247
3188646/3188185	[1 13 -1 -3 -1 -2⟩	0.250
540078539/540000000	[-8 -3 -7 4 3 2⟩	0.252
29046875/29042496	[-6 -3 6 -5 1 2⟩	0.261
1162261467/1162084000	[-5 19 -3 -4 -2⟩	0.264
214358881/214326000	[-4 -7 -3 -2 8⟩	0.266
481966875/481890304	[-12 3 4 -6 0 4⟩	0.275
3293331899/3292800000	[-10 -1 -5 -3 7 2⟩	0.280
245000000/244958571	[6 -2 7 2 -5 -2⟩	0.293
520613632/520524225	[8 -6 -2 5 2 -4⟩	0.297
1771875/1771561	[0 4 5 1 -6⟩	0.307
105644/105625	[2 0 -4 4 1 -2⟩	0.311
31250000/31244213	[4 0 9 -5 -1 -2⟩	0.321
1247410125/1247178944	[-6 10 3 0 -7 2⟩	0.321
1377751375/1377495072	[-5 -16 3 2 3 2⟩	0.322
113379904/113358609	[6 -4 0 -2 6 -4⟩	0.325
250047/250000	[-4 6 -6 3⟩	0.325
56953125/56942116	[-2 6 7 -6 -2⟩	0.335
120955835/120932352	[-11 -10 1 1 2 4⟩	0.336
1449459/1449175	[0 2 -2 -3 5 -2⟩	0.339
53094899/53084160	[-17 -4 -1 0 1 6⟩	0.350
96059601/96040000	[-6 8 -4 -4 4⟩	0.353
288240050/288178803	[1 -9 2 8 -4⟩	0.368
265531392/265474495	[12 3 -1 4 -1 -6⟩	0.371
269500000/269440587	[5 -13 6 2 1 -2⟩	0.382
139178767/139148064	[-5 -3 0 7 -5 2⟩	0.382
432081216/431985125	[6 9 -3 3 -2 -4⟩	0.385
4375/4374	[-1 -7 4 1⟩	0.396
1655976960/1655595487	[10 5 1 -3 3 -6⟩	0.399
703096443/702934375	[0 15 -5 2 -3 -2⟩	0.399
4225/4224	[-7 -1 2 0 -1 2⟩	0.410
342956592/342874805	[4 11 -1 -4 2 -4⟩	0.413
607645423/607500000	[-5 -5 -7 3 6⟩	0.414
228765625/228709656	[-3 -5 6 -6 4⟩	0.424
6940323/6938624	[-13 5 0 -1 -2 4⟩	0.424
1420541793/1420191500	[-2 17 -3 -5 1 -2⟩	0.427
4000989993/4000000000	[-11 1 -9 2 5 2⟩	0.428
421760625/421654016	[-9 1 4 -7 3 2⟩	0.438
3294172000/3293331899	[5 0 3 7 -7 -2⟩	0.442
280000000/279926361	[9 -4 7 1 -2 -4⟩	0.455
428830605/428717762	[-1 6 1 6 -8⟩	0.456
3487704605/3486784401	[0 -20 1 8 2⟩	0.457
726364926/726171875	[1 2 -8 9 -1 -2⟩	0.460
225000/224939	[3 2 5 0 -3 -2⟩	0.469
1530966437/1530550080	[-6 -14 -1 7 1 2⟩	0.471
160699616/160655625	[5 -2 -4 3 4 -4⟩	0.474
820125/819896	[-3 8 3 -1 -4⟩	0.483
1550115875/1549681956	[-2 -18 3 1 6⟩	0.485
3360173089/3359232000	[-12 -8 -3 6 0 4⟩	0.485
5282739/5281250	[-1 4 -6 2 3 -2⟩	0.488
139218750/139178767	[1 4 7 -7 1 -2⟩	0.497
4041608805/4040447488	[-9 14 1 -2 -5 2⟩	0.498
136088095/136048896	[-8 -12 1 0 5 2⟩	0.499
1714871048/1714374025	[3 0 -2 -4 8 -4⟩	0.502
50014503/50000000	[-7 10 -8 1 2⟩	0.502
184528125/184473632	[-5 10 5 -8⟩	0.511
418161601/418037760	[-14 -6 -1 -1 4 4⟩	0.513
329417200/329316273	[4 -11 2 7 -1 -2⟩	0.530
4079910912/4078653605	[15 1 -1 3 2 -8⟩	0.534
117649/117612	[-2 -5 0 6 -2⟩	0.545
603542016/603351125	[9 7 -3 2 1 -6⟩	0.548
1220703125/1220312709	[0 -5 13 -3 -4⟩	0.554
3327500/3326427	[2 -9 4 0 3 -2⟩	0.558
8521149/8518400	[-8 1 -2 5 -3 2⟩	0.559
89282088/89253125	[3 13 -5 1 0 -4⟩	0.562
3025/3024	[-4 -3 2 -1 2⟩	0.572
1162261467/1161875000	[-3 19 -7 0 -1 -2⟩	0.576
50193/50176	[-10 3 0 -2 1 2⟩	0.586
1500512167/1500000000	[-8 -1 -9 1 8⟩	0.591
1107225625/1106841792	[-6 -1 4 -8 6⟩	0.600
562166163/561971200	[-16 9 -2 -3 0 4⟩	0.601
3764768000/3763454409	[8 -2 3 6 -4 -4⟩	0.604
3520000000/3518743761	[12 -6 7 0 1 -6⟩	0.618
27227340/27217619	[2 4 1 5 -5 -2⟩	0.618
200000/199927	[6 0 5 -1 0 -4⟩	0.632
141776649/141724880	[-4 10 -1 4 -6⟩	0.632
1722499009/1721868840	[-3 -16 -1 6 4⟩	0.633
91125/91091	[0 6 3 -2 -1 -2⟩	0.646
3780549773/3779136000	[-9 -10 -3 5 3 2⟩	0.648
446433372/446265625	[2 2 -6 1 6 -4⟩	0.651
2657205/2656192	[-6 12 1 -3 -2⟩	0.660
1071794405/1071385056	[-5 -14 1 -1 8⟩	0.661
1056655611/1056250000	[-4 8 -8 0 5 -2⟩	0.665
3293331899/3292047360	[-11 -8 -1 -2 7 2⟩	0.675
1977326743/1976535000	[-3 -3 -4 11 -4⟩	0.693
95703125/95664294	[-1 -3 9 2 -6⟩	0.703
369754/369603	[1 -7 0 5 1 -2⟩	0.707
2495390625/2494357888	[-7 3 7 1 -7 2⟩	0.717
2401/2400	[-5 -1 -2 4⟩	0.721
48828125/48807528	[-3 -1 11 -5 -2⟩	0.730
4026275/4024566	[-1 -5 2 -2 5 -2⟩	0.735
14085981/14080000	[-11 5 -4 3 -1 2⟩	0.735
43923/43904	[-7 1 0 -3 4⟩	0.749
4078653605/4076863488	[-24 -5 1 0 0 8⟩	0.760
491941593/491724800	[-13 7 -2 -4 3 2⟩	0.763
3457440/3455881	[5 2 1 4 -2 -4⟩	0.781
19140625/19131876	[-2 -14 8 2⟩	0.792
2129600000/2128622769	[9 -2 5 -2 3 -6⟩	0.795
2250423/2249390	[-1 8 -1 3 -3 -2⟩	0.795
18484375/18475776	[-8 -8 6 1 -1 2⟩	0.806
9801000/9796423	[3 4 3 -3 2 -4⟩	0.809
234365481/234256000	[-7 14 -3 2 -4⟩	0.809
4253517961/4251528000	[-6 -12 -3 4 6⟩	0.810
17850625/17842176	[-14 -2 4 0 -2 4⟩	0.820
3247695/3246152	[-3 10 1 -4 1 -2⟩	0.823
43046721/43025920	[-9 16 -1 -5⟩	0.837
2573571875/2572306572	[-2 -1 5 7 -8⟩	0.851
282475249/282335625	[0 -5 -4 10 -1 -2⟩	0.856
54687500/54660177	[2 -5 9 1 -3 -2⟩	0.865
562448656/562166163	[4 -9 0 4 4 -4⟩	0.870
121060821/121000000	[-6 1 -6 9 -2⟩	0.870
234375/234256	[-4 1 7 0 -4⟩	0.879
456533/456300	[-2 -3 -2 3 3 -2⟩	0.884
537109375/536832387	[0 -3 11 -6 1 -2⟩	0.893
1155009375/1154413568	[-10 7 5 -1 -5 2⟩	0.893
160083/160000	[-8 3 -4 2 2⟩	0.898
52734375/52706752	[-6 3 9 -7⟩	0.907
19487171/19476912	[-4 -1 0 -4 7 -2⟩	0.912
256134879/256000000	[-14 9 -6 1 1 2⟩	0.912
430489323/430259200	[-10 5 -2 -5 6⟩	0.926
514714375/514434888	[-3 -12 4 7 -2⟩	0.940
4829440/4826809	[8 0 1 3 1 -6⟩	0.943
497067025/496793088	[-9 -6 2 6 -3 2⟩	0.954
142884/142805	[2 6 -1 2 0 -4⟩	0.957
1890625/1889568	[-5 -10 6 0 2⟩	0.968
480024727/479756288	[-15 0 0 5 -4 4⟩	0.968
3720087/3718000	[-4 12 -3 1 -1 -2⟩	0.972
1161875/1161216	[-11 -4 4 -1 1 2⟩	0.982
480298005/480024727	[0 8 1 -5 4 -4⟩	0.985
387420489/387200000	[-10 18 -5 0 -2⟩	0.986
6426225/6422528	[-17 2 2 -2 0 4⟩	0.996
735306250/734875713	[1 -3 5 6 -5 -2⟩	1.014
62500000/62462907	[5 -7 9 0 0 -4⟩	1.028
56723625/56689952	[-5 3 3 5 -6⟩	1.028
63412811/63375000	[-3 -1 -6 8 1 -2⟩	1.033
78125/78078	[-1 -1 7 -1 -1 -2⟩	1.042
173612978/173508075	[1 -5 -2 2 6 -4⟩	1.046
200120949/200000000	[-9 5 -8 7⟩	1.047
759375/758912	[-7 5 5 -2 -2⟩	1.056
3382071/3380000	[-5 1 -4 1 5 -2⟩	1.060
3742230375/3739918336	[-13 11 3 -3 -3 2⟩	1.070
32019867/32000000	[-11 7 -6 0 4⟩	1.074
808836875/808321761	[0 -14 4 6 1 -2⟩	1.103
420175/419904	[-6 -8 2 5⟩	1.117
24149664/24134045	[5 4 -1 1 3 -6⟩	1.120
2516421875/2514778812	[-2 -12 6 -1 5 -2⟩	1.131
405769/405504	[-12 -2 0 4 -1 2⟩	1.131
7144929/7140250	[-1 10 -3 0 2 -4⟩	1.134
9150625/9144576	[-8 -6 4 -2 4⟩	1.145
793510263/792985600	[-18 4 -2 3 -2 4⟩	1.145
473513931/473200000	[-7 16 -5 -1 1 -2⟩	1.148
5623475/5619712	[-14 0 2 -3 3 2⟩	1.159
2519337249/2517630976	[-20 6 0 -4 2 4⟩	1.173
840350000/839779083	[4 -5 5 5 -2 -4⟩	1.177
900375/899756	[-2 1 3 4 -3 -2⟩	1.191
816293376/815730721	[9 13 0 0 0 -8⟩	1.194
37812500/37786203	[2 -3 7 -2 2 -4⟩	1.204
3750705/3748096	[-8 7 1 3 -4⟩	1.205
4227952113/4225000000	[-6 3 -8 6 3 -2⟩	1.209
928125/927472	[-4 3 5 -3 1 -2⟩	1.218
214358881/214207500	[-2 -1 -4 0 8 -4⟩	1.223
2460375/2458624	[-10 9 3 -4⟩	1.233
1977326743/1975897638	[-1 -12 0 11 -1 -2⟩	1.252
282475249/282268800	[-7 -6 -2 10 -2⟩	1.266
341796875/341545248	[-5 -6 11 1 -4⟩	1.275
79893275/79834248	[-3 -10 2 4 3 -2⟩	1.280
4081676544/4078653605	[8 2 -1 0 6 -8⟩	1.283
330078125/329832448	[-11 0 9 0 -5 2⟩	1.289
41503/41472	[-9 -4 0 3 2⟩	1.294
4226622444/4223457875	[2 8 -3 -1 5 -6⟩	1.297
1220703125/1219784832	[-7 -4 13 -6⟩	1.303
819819/819200	[-15 2 -2 2 1 2⟩	1.308
44289025/44255232	[-11 -2 2 -4 6⟩	1.321
1311721047/1310720000	[-21 8 -4 1 0 4⟩	1.322
2204627139/2202927104	[-17 4 0 -5 5 2⟩	1.336
1089547389/1088704760	[-3 3 -1 9 -5 -2⟩	1.339
85750/85683	[1 -1 3 3 0 -4⟩	1.353
59535/59488	[-5 5 1 2 -1 -2⟩	1.367
726171875/725594112	[-12 -11 8 0 1 2⟩	1.378
137259375/137149922	[-1 1 5 -4 4 -4⟩	1.381
1240029/1239040	[-11 11 -1 1 -2⟩	1.381
446265625/445906944	[-18 -5 6 -1 0 4⟩	1.392
363862125/363569024	[-7 7 3 -5 3 -2⟩	1.395
964565415/963780608	[-13 13 1 -6 2⟩	1.409
443889677/443523600	[-4 -8 -2 9 1 -2⟩	1.428
48828125/48787596	[-2 -8 11 0 -1 -2⟩	1.438
837019575/836323202	[-1 14 2 1 -4 -4⟩	1.441
5764801/5760000	[-10 -2 -4 8⟩	1.442
1953125/1951488	[-8 -2 9 -1 -2⟩	1.452
3486784401/3483855232	[-7 20 0 0 -5 -2⟩	1.455
7891499/7884864	[-6 -6 0 2 5 -2⟩	1.456
1069453125/1068548096	[-14 4 7 -2 -3 2⟩	1.466
102487/102400	[-12 0 -2 1 4⟩	1.470
163980531/163840000	[-18 6 -4 0 3 2⟩	1.484
1929229929/1927561216	[-14 2 0 -6 8⟩	1.498
17294403/17279405	[0 1 -1 8 -2 -4⟩	1.502
2929687500/2927131207	[2 1 12 -1 -4 -4⟩	1.511
1838265625/1836660096	[-7 -15 6 6⟩	1.513
130438000/130323843	[4 -3 3 2 3 -6⟩	1.516
1801088541/1799512000	[-6 7 -3 7 -3 -2⟩	1.516
1775239375/1773674496	[-13 -9 4 5 -1 2⟩	1.527
114345/114244	[-2 3 1 1 2 -4⟩	1.530
1714374025/1712848896	[-19 -3 2 4 -2 4⟩	1.541
216513/216320	[-8 9 -1 0 1 -2⟩	1.544
3514671875/3511517184	[-15 -7 6 -2 3 2⟩	1.555
14348907/14336000	[-14 15 -3 -1⟩	1.558
2159925625/2157969408	[-21 -1 4 -3 2 4⟩	1.569
1313046875/1311844248	[-3 -6 7 5 -3 -2⟩	1.586
816480000/815730721	[8 6 4 1 0 -8⟩	1.589
7503125/7496192	[-9 0 5 4 -4⟩	1.600
53144100/53094899	[2 12 2 0 -1 -6⟩	1.604
1096135733/1095120000	[-7 -4 -4 7 3 -2⟩	1.605
21484375/21464352	[-5 -4 9 -2 1 -2⟩	1.614
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