77edo

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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

Theory

With harmonic 3 less than a cent flat, harmonic 5 a bit over three cents sharp and 7'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, 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

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	15.6		^D, ^^E♭♭
2	31.2		^^D, ^3E♭♭
3	46.8	36/35, 37/36, 38/37, 39/38	^3D, v3E♭
4	62.3	27/26, 28/27, 29/28	v3D♯, vvE♭
5	77.9	22/21, 23/22	vvD♯, vE♭
6	93.5	19/18, 37/35, 39/37	vD♯, E♭
7	109.1	16/15, 33/31	D♯, ^E♭
8	124.7	29/27	^D♯, ^^E♭
9	140.3	13/12, 38/35	^^D♯, ^3E♭
10	155.8	23/21, 35/32	^3D♯, v3E
11	171.4	21/19, 32/29	v3D𝄪, vvE
12	187	29/26, 39/35	vvD𝄪, vE
13	202.6	9/8	E
14	218.2	17/15	^E, ^^F♭
15	233.8	8/7	^^E, ^3F♭
16	249.4	15/13, 37/32	^3E, v3F
17	264.9	7/6	v3E♯, vvF
18	280.5	20/17, 27/23	vvE♯, vF
19	296.1	19/16, 32/27	F
20	311.7		^F, ^^G♭♭
21	327.3	29/24, 35/29	^^F, ^3G♭♭
22	342.9	28/23, 39/32	^3F, v3G♭
23	358.4	16/13	v3F♯, vvG♭
24	374	36/29	vvF♯, vG♭
25	389.6	5/4	vF♯, G♭
26	405.2	24/19	F♯, ^G♭
27	420.8	14/11, 37/29	^F♯, ^^G♭
28	436.4	9/7	^^F♯, ^3G♭
29	451.9	13/10, 35/27	^3F♯, v3G
30	467.5	17/13, 21/16, 38/29	v3F𝄪, vvG
31	483.1	37/28	vvF𝄪, vG
32	498.7	4/3	G
33	514.3	31/23, 35/26, 39/29	^G, ^^A♭♭
34	529.9	19/14, 34/25	^^G, ^3A♭♭
35	545.5	26/19, 37/27	^3G, v3A♭
36	561	18/13, 29/21	v3G♯, vvA♭
37	576.6	39/28	vvG♯, vA♭
38	592.2	31/22, 38/27	vG♯, A♭
39	607.8	27/19, 37/26	G♯, ^A♭
40	623.4	33/23	^G♯, ^^A♭
41	639	13/9	^^G♯, ^3A♭
42	654.5	19/13, 35/24	^3G♯, v3A
43	670.1	25/17, 28/19	v3G𝄪, vvA
44	685.7		vvG𝄪, vA
45	701.3	3/2	A
46	716.9		^A, ^^B♭♭
47	732.5	26/17, 29/19, 32/21	^^A, ^3B♭♭
48	748.1	20/13, 37/24	^3A, v3B♭
49	763.6	14/9	v3A♯, vvB♭
50	779.2	11/7	vvA♯, vB♭
51	794.8	19/12	vA♯, B♭
52	810.4	8/5	A♯, ^B♭
53	826	29/18, 37/23	^A♯, ^^B♭
54	841.6	13/8	^^A♯, ^3B♭
55	857.1	23/14	^3A♯, v3B
56	872.7		v3A𝄪, vvB
57	888.3		vvA𝄪, vB
58	903.9	27/16, 32/19	B
59	919.5	17/10	^B, ^^C♭
60	935.1	12/7	^^B, ^3C♭
61	950.6	26/15	^3B, v3C
62	966.2	7/4	v3B♯, vvC
63	981.8	30/17, 37/21	vvB♯, vC
64	997.4	16/9	C
65	1013		^C, ^^D♭♭
66	1028.6	29/16, 38/21	^^C, ^3D♭♭
67	1044.2		^3C, v3D♭
68	1059.7	24/13, 35/19	v3C♯, vvD♭
69	1075.3		vvC♯, vD♭
70	1090.9	15/8	vC♯, D♭
71	1106.5	36/19	C♯, ^D♭
72	1122.1	21/11	^C♯, ^^D♭
73	1137.7	27/14	^^C♯, ^3D♭
74	1153.2	35/18, 37/19, 39/20	^3C♯, v3D
75	1168.8		v3C𝄪, vvD
76	1184.4		vvC𝄪, vD
77	1200	2/1	D

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 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio (Reduced)	Temperaments
1	4\77	62.34	28/27	Unicorn / alicorn / camahueto / qilin
1	5\77	77.92	21/20	Valentine
1	9\77	140.26	13/12	Tsaharuk
1	15\77	233.77	8/7	Guiron
1	16\77	249.35	15/13	Hemischis (77e)
1	20\77	311.69	6/5	Oolong
1	23\77	358.44	16/13	Restles
1	31\77	483.12	45/34	Hemiseven
1	32\77	498.70	4/3	Grackle
1	34\77	529.87	512/375	Tuskaloosa Muscogee
7	32\77 (1\77)	498.70 (15.58)	4/3 (81/80)	Absurdity
11	32\77 (3\77)	498.70 (46.75)	4/3 (36/35)	Hendecatonic

Music

Jake Freivald

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

Joel Grant Taylor

• Star 1-GrimaldiA+Bmod

Chris Vaisvil

• 77et Star