114edo

← 113edo

114edo

115edo →

Prime factorization	2 × 3 × 19
Step size	10.5263¢
Fifth	67\114 (705.263¢)
Semitones (A1:m2)	13:7 (136.8¢ : 73.68¢)
Consistency limit	7
Distinct consistency limit	7

114 equal divisions of the octave (abbreviated 114edo or 114ed2), also called 114-tone equal temperament (114tet) or 114 equal temperament (114et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 114 equal parts of about 10.5 ¢ each. Each step represents a frequency ratio of 2^1/114, or the 114th root of 2.

In the 5-limit it tempers out 2048/2025, in the 7-limit 245/243, in the 11-limit 121/120, 176/175 and 117440512/117406179, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for shrutar temperament; it is in fact the optimal patent val for shrutar in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

Harmonics

Approximation of odd harmonics in 114edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+3.31	+3.16	-0.40	-3.91	-3.95	+1.58	-4.06	+0.31	-2.78	+2.90	+3.30
	relative (%)	+31	+30	-4	-37	-38	+15	-39	+3	-26	+28	+31
Steps (reduced)		181 (67)	265 (37)	320 (92)	361 (19)	394 (52)	422 (80)	445 (103)	466 (10)	484 (28)	501 (45)	516 (60)

Intervals

Steps	Cents	Ups and downs notation	Approximate ratios
0	0	D	1/1
1	10.5263	↑D, ↓6E♭
2	21.0526	↑↑D, ↓5E♭
3	31.5789	↑3D, ↓4E♭	49/48, 56/55
4	42.1053	↑4D, ↓3E♭	40/39
5	52.6316	↑5D, ↓↓E♭	33/32, 36/35, 65/63
6	63.1579	↑6D, ↓E♭
7	73.6842	↑7D, E♭	25/24
8	84.2105	↑8D, ↓12E	21/20
9	94.7368	↑9D, ↓11E	55/52
10	105.263	↑10D, ↓10E	35/33, 52/49
11	115.789	↑11D, ↓9E
12	126.316	↑12D, ↓8E	14/13
13	136.842	D♯, ↓7E	13/12, 27/25
14	147.368	↑D♯, ↓6E
15	157.895	↑↑D♯, ↓5E	35/32
16	168.421	↑3D♯, ↓4E
17	178.947	↑4D♯, ↓3E	10/9, 72/65
18	189.474	↑5D♯, ↓↓E	39/35, 49/44
19	200	↑6D♯, ↓E	55/49
20	210.526	E
21	221.053	↑E, ↓6F
22	231.579	↑↑E, ↓5F	8/7, 55/48
23	242.105	↑3E, ↓4F
24	252.632	↑4E, ↓3F
25	263.158	↑5E, ↓↓F	7/6, 64/55
26	273.684	↑6E, ↓F
27	284.211	F	33/28
28	294.737	↑F, ↓6G♭
29	305.263	↑↑F, ↓5G♭	25/21
30	315.789	↑3F, ↓4G♭	6/5
31	326.316	↑4F, ↓3G♭
32	336.842	↑5F, ↓↓G♭	40/33
33	347.368	↑6F, ↓G♭	49/40
34	357.895	↑7F, G♭	16/13
35	368.421	↑8F, ↓12G	26/21
36	378.947	↑9F, ↓11G
37	389.474	↑10F, ↓10G	5/4
38	400	↑11F, ↓9G	63/50
39	410.526	↑12F, ↓8G	33/26, 80/63
40	421.053	F♯, ↓7G	14/11
41	431.579	↑F♯, ↓6G	50/39
42	442.105	↑↑F♯, ↓5G
43	452.632	↑3F♯, ↓4G	13/10
44	463.158	↑4F♯, ↓3G	55/42, 64/49
45	473.684	↑5F♯, ↓↓G	21/16
46	484.211	↑6F♯, ↓G
47	494.737	G	4/3
48	505.263	↑G, ↓6A♭
49	515.789	↑↑G, ↓5A♭	35/26, 66/49
50	526.316	↑3G, ↓4A♭	65/48
51	536.842	↑4G, ↓3A♭
52	547.368	↑5G, ↓↓A♭	11/8, 48/35
53	557.895	↑6G, ↓A♭
54	568.421	↑7G, A♭	25/18
55	578.947	↑8G, ↓12A	7/5
56	589.474	↑9G, ↓11A
57	600	↑10G, ↓10A
58	610.526	↑11G, ↓9A
59	621.053	↑12G, ↓8A	10/7
60	631.579	G♯, ↓7A	36/25
61	642.105	↑G♯, ↓6A
62	652.632	↑↑G♯, ↓5A	16/11, 35/24
63	663.158	↑3G♯, ↓4A
64	673.684	↑4G♯, ↓3A
65	684.211	↑5G♯, ↓↓A	49/33, 52/35
66	694.737	↑6G♯, ↓A
67	705.263	A	3/2
68	715.789	↑A, ↓6B♭
69	726.316	↑↑A, ↓5B♭	32/21
70	736.842	↑3A, ↓4B♭	49/32
71	747.368	↑4A, ↓3B♭	20/13
72	757.895	↑5A, ↓↓B♭	65/42
73	768.421	↑6A, ↓B♭	39/25
74	778.947	↑7A, B♭	11/7
75	789.474	↑8A, ↓12B	52/33, 63/40
76	800	↑9A, ↓11B
77	810.526	↑10A, ↓10B	8/5
78	821.053	↑11A, ↓9B
79	831.579	↑12A, ↓8B	21/13
80	842.105	A♯, ↓7B	13/8
81	852.632	↑A♯, ↓6B	80/49
82	863.158	↑↑A♯, ↓5B	33/20
83	873.684	↑3A♯, ↓4B
84	884.211	↑4A♯, ↓3B	5/3
85	894.737	↑5A♯, ↓↓B	42/25
86	905.263	↑6A♯, ↓B
87	915.789	B	56/33
88	926.316	↑B, ↓6C
89	936.842	↑↑B, ↓5C	12/7, 55/32
90	947.368	↑3B, ↓4C
91	957.895	↑4B, ↓3C
92	968.421	↑5B, ↓↓C	7/4
93	978.947	↑6B, ↓C
94	989.474	C
95	1000	↑C, ↓6D♭
96	1010.53	↑↑C, ↓5D♭	70/39
97	1021.05	↑3C, ↓4D♭	9/5, 65/36
98	1031.58	↑4C, ↓3D♭
99	1042.11	↑5C, ↓↓D♭	64/35
100	1052.63	↑6C, ↓D♭
101	1063.16	↑7C, D♭	24/13, 50/27
102	1073.68	↑8C, ↓12D	13/7
103	1084.21	↑9C, ↓11D
104	1094.74	↑10C, ↓10D	49/26, 66/35
105	1105.26	↑11C, ↓9D
106	1115.79	↑12C, ↓8D	40/21
107	1126.32	C♯, ↓7D	48/25
108	1136.84	↑C♯, ↓6D
109	1147.37	↑↑C♯, ↓5D	35/18, 64/33
110	1157.89	↑3C♯, ↓4D	39/20
111	1168.42	↑4C♯, ↓3D	55/28
112	1178.95	↑5C♯, ↓↓D
113	1189.47	↑6C♯, ↓D
114	1200	D	2/1

Period of 19-limit Shrutar

Degree	Cents	Difference from 68edo
2	21.05263	3.40557¢
3	31.57895	-3.71517¢
5	52.63158	-0.3096¢
7	73.68421	3.096¢
8	84.21053	-4.02477¢
10	105.26316	-0.619195¢
12	126.31579	2.78638¢
13	136.842105	-4.334365¢
15	157.89474	-0.9288¢
17	178.94737	2.47678¢
18	189.47369	-4.644¢
20	210.52632	-1.23839¢
22	231.57895	2.16718¢
23	242.10526	-4.953560372
25	263.157895	-1.548¢
27	284.21053	1.857585¢
29	305.26316	5.26316¢
30	315.78947	-1.857585¢
32	336.842105	1.548¢
34	357.89474	4.95356¢
35	368.42105	-2.16718¢
37	389.47368	1.23839¢
39	410.52632	4.64396¢
40	421.05263	-2.47678¢
42	442.10526	0.92879¢
44	463.157895	4.334365¢
45	473.68421	-2.78638¢
47	494.73684	0.619195¢
49	515.78947	4.02477¢
50	526.31579	-3.095975¢
52	547.36842	0.3096¢
54	568.42105	3.71517¢
55	578.94737	-3.40557¢