77edo

← 76edo

77edo

78edo →

Prime factorization	7 × 11
Step size	15.5844 ¢
Fifth	45\77 (701.299 ¢)
Semitones (A1:m2)	7:6 (109.1 ¢ : 93.51 ¢)
Consistency limit	9
Distinct consistency limit	9

The 77 equal divisions of the octave (77edo), or the 77(-tone) equal temperament (77et, 77tet) when viewed from a regular temperament perspective, divides the octave into 77 steps of size about 15.6 cents each.

Theory

With fifths less than a cent flat, major thirds a bit over three cents sharp and 7/4's less flat than that, 77edo represents an excellent tuning choice for both valentine, the 31&46 temperament, and starling, the 126/125 planar temperament, giving the optimal patent val for 11-limit valentine and its 13-limit extensions dwynwen and valentino, as well as 11-limit starling and oxpecker temperaments. It also gives the optimal patent val for grackle and various members of the unicorn family, with a generator of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit alicorn and 11- and 13-limit camahueto.

77et tempers out 32805/32768 in the 5-limit, 126/125, 1029/1024 and 6144/6125 in the 7-limit, 121/120, 176/175, 385/384 and 441/440 in the 11-limit, and 196/195, 351/350, 352/351, 676/675 and 729/728 in the 13-limit.

77edo is an excellent EDO for Carlos Alpha scale, since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only -0.042912 cents.

Prime harmonics

Approximation of prime harmonics in 77edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.66	+3.30	-2.59	-5.86	+1.03	+4.14	-1.41	-4.90	-1.01	-7.37
	Relative (%)	+0.0	-4.2	+21.2	-16.6	-37.6	+6.6	+26.5	-9.0	-31.4	-6.5	-47.3
Steps (reduced)		77 (0)	122 (45)	179 (25)	216 (62)	266 (35)	285 (54)	315 (7)	327 (19)	348 (40)	374 (66)	381 (73)

Intervals

Degree	Cents	Approximate Ratios in the 13-limit
0	0.000	1/1
1	15.584	81/80, 99/98
2	31.169	64/63, 49/48
3	46.753	33/32, 36/35
4	62.338	28/27, 26/25
5	77.922	21/20, 25/24
6	93.506	135/128
7	109.091	16/15
8	124.675	15/14
9	140.260	13/12
10	155.844	12/11, 11/10
11	171.429	72/65
12	187.013	10/9
13	202.597	9/8
14	218.182	256/225
15	233.766	8/7
16	249.351	15/13
17	264.935	7/6
18	280.519	33/28
19	296.104	32/27, 13/11
20	311.688	6/5
21	327.273	98/81
22	342.857	11/9, 39/32
23	358.442	16/13
24	374.026	56/45, 26/21
25	389.610	5/4
26	405.195	33/26, 81/64
27	420.779	14/11, 32/25
28	436.364	9/7
29	451.948	13/10
30	467.532	21/16
31	483.117	120/91
32	498.701	4/3
33	514.286	27/20
34	529.870	49/36
35	545.455	11/8, 15/11
36	561.039	18/13
37	576.623	7/5
38	592.208	45/32
39	607.792	64/45
40	623.377	10/7
41	638.961	13/9
42	654.545	16/11, 22/15
43	670.130	72/49
44	685.714	40/27
45	701.299	3/2
46	716.883	91/60
47	732.468	32/21
48	748.052	20/13
49	763.636	14/9
50	779.221	11/7, 25/16
51	794.805	52/33, 128/81
52	810.390	8/5
53	825.974	45/28, 21/13
54	841.558	13/8
55	857.143	18/11, 64/39
56	872.727	81/49
57	888.312	5/3
58	903.896	27/16, 22/13
59	919.481	56/33
60	935.065	12/7
61	950.649	26/15
62	966.234	7/4
63	981.818	225/128
64	997.403	16/9
65	1012.987	9/5
66	1028.571	65/36
67	1044.156	11/6, 20/11
68	1059.740	24/13
69	1075.325	28/15
70	1090.909	15/8
71	1106.494	256/135
72	1122.078	40/21, 48/25
73	1137.662	27/14, 25/13
74	1153.247	64/33, 35/18
75	1168.831	63/32, 96/49
76	1184.416	160/81, 196/99
77	1200.000	2/1

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-122 77⟩	[⟨77 122]]	+0.207	0.207	1.33
2.3.5	32805/32768, 1594323/1562500	[⟨77 122 179]]	-0.336	0.785	5.04
2.3.5.7	126/125, 1029/1024, 10976/10935	[⟨77 122 179 216]]	-0.021	0.872	5.59
2.3.5.7.11	121/120, 126/125, 176/175, 10976/10935	[⟨77 122 179 216 266]]	+0.322	1.039	6.66
2.3.5.7.11.13	121/120, 126/125, 176/175, 196/195, 676/675	[⟨77 122 179 216 266 285]]	+0.222	0.974	6.25

Rank-2 temperaments

Periods per Octave	Generator	Temperaments
1	1\77
1	2\77
1	3\77
1	4\77	Unicorn / alicorn / camahueto / qilin
1	5\77	Valentine
1	6\77
1	8\77
1	9\77	Tsaharuk
1	10\77
1	12\77
1	13\77
1	15\77	Guiron
1	16\77	Hemischis
1	17\77
1	18\77
1	19\77
1	20\77	Oolong
1	23\77	Restles
1	24\77
1	25\77
1	26\77
1	27\77
1	29\77
1	30\77
1	31\77	Hemiseven
1	32\77	Helmholtz / grackle
1	34\77	Mabila
1	36\77
1	37\77
1	38\77
7	1\77	Absurdity
7	2\77
7	3\77
7	4\77
7	5\77
11	1\77	Hendecatonic
11	2\77
11	3\77

Music

Jake Freivald

• A Seed Planted, in an organ version of Claudi Meneghin.

Joel Grant Taylor

• Star 1-GrimaldiA+Bmod

Chris Vaisvil

• 77et Star