156edo

156edo is the equal division of the octave into 156 parts of 7.6923 cents each. It tempers out 531441/524288 (pythagorean comma) and 1220703125/1207959552 (ditonic comma) in the 5-limit, as well as 1224440064/1220703125 (parakleisma); 225/224, 250047/250000, and 589824/588245 in the 7-limit. Using the patent val, it tempers out 441/440, 1375/1372, 4375/4356, and 65536/65219 in the 11-limit; 351/350, 364/363, 625/624, 1625/1617, and 13122/13013 in the 13-limit. Using the 156e val, it tempers out 385/384, 540/539, 1331/1323, and 78408/78125 in the 11-limit; 351/350, 625/624, 847/845, and 1001/1000 in the 13-limit.

156edo

Welcome to 156edo

156edo = 12edo × 13edo

Distinctively interesting choice for an edo, due to its divisibility by both 12 and 13. This means it can be used to play 12edo music and 13edo music at the same time, using the 156/13 and 156/12 vals respectively.

7.692

156edo has a step size of 7.6923076923077 cents. Microtonal enthusiasts might notice it seems close to the 225/224 ratio, but it isn't in the patent val, because 156edo tempers out 225/224. The edo step size closest to 225/224, taking into account the primes that compose it, is in 171edo. You have to use the 2*156 val, sacrificing 3 for a new 9, to make this approximation come true. It is up to the microtonalist to decide whether this tradeoff is worth it.

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 156edo (ordered by absolute error).

Interval, complement	Error (abs., in cents)
13/12, 24/13	0.111
6/5, 5/3	0.257
13/10, 20/13	0.368
8/7, 7/4	0.405
15/13, 26/15	1.587
5/4, 8/5	1.698
18/13, 13/9	1.844
4/3, 3/2	1.955
16/13, 13/8	2.066
7/5, 10/7	2.103
14/11, 11/7	2.123
10/9, 9/5	2.212
7/6, 12/7	2.36
14/13, 13/7	2.471
11/8, 16/11	2.528
16/15, 15/8	3.653
9/8, 16/9	3.91
15/14, 28/15	4.058
11/10, 20/11	4.227
9/7, 14/9	4.315
12/11, 11/6	4.483
13/11, 22/13	4.594
15/11, 22/15	6.182
11/9, 18/11	6.438

Table of 156edo intervals

Step	Five limit	Seven limit	Eleven limit	Thirteen limit
1	15625/15552	1029/1024	242/243	144/143
2	81/80	81/80	81/80	78/77
3	3125/3072	245/243	121/120	65/64
4	2048/2025	50/49	45/44	45/44
5	16875/16384	49/48	49/48	49/48
6	128/125	36/35	36/35	36/35
7	250/243	250/243	33/32	33/32
8	648/625	648/625	80/77	27/26
9	25/24	25/24	25/24	25/24
10	6561/6250	256/245	81/77	81/77
11	135/128	21/20	21/20	21/20
12	16384/15625	729/686	128/121	96/91
13	256/243	200/189	35/33	35/33
14	62500/59049	343/324	343/324	143/135
15	16/15	15/14	15/14	15/14
16	3125/2916	343/320	77/72	77/72
17	27/25	27/25	27/25	14/13
18	625/576	175/162	121/112	13/12
19	2187/2000	160/147	12/11	12/11
20	1125/1024	35/32	35/32	35/32
21	2048/1875	54/49	54/49	54/49
22	800/729	441/400	11/10	11/10
23	3456/3125	972/875	135/121	72/65
24	10/9	10/9	10/9	10/9
25	17496/15625	384/343	384/343	143/128
26	9/8	9/8	9/8	9/8
27	15625/13824	1944/1715	121/108	91/81
28	729/640	500/441	25/22	25/22
29	9375/8192	245/216	245/216	91/80
30	256/225	8/7	8/7	8/7
31	2500/2187	147/128	55/48	55/48
32	144/125	81/70	81/70	15/13
33	125/108	125/108	121/105	52/45
34	729/625	400/343	64/55	64/55
35	75/64	7/6	7/6	7/6
36	32768/28125	288/245	288/245	168/143
37	1215/1024	147/125	33/28	13/11
38	18432/15625	1458/1225	144/121	108/91
39	32/27	25/21	25/21	25/21
40	15625/13122	343/288	343/288	143/120
41	6/5	6/5	6/5	6/5
42	3125/2592	875/729	77/64	65/54
43	243/200	243/200	40/33	40/33
44	625/512	98/81	98/81	39/32
45	4096/3375	60/49	27/22	27/22
46	8000/6561	49/40	11/9	11/9
47	768/625	216/175	150/121	16/13
48	100/81	100/81	99/80	26/21
49	3888/3125	1280/1029	96/77	81/65
50	5/4	5/4	5/4	5/4
51	19683/15625	432/343	432/343	169/135
52	81/64	63/50	44/35	44/35
53	15625/12288	1225/972	121/96	91/72
54	512/405	80/63	14/11	14/11
55	25000/19683	245/192	245/192	143/112
56	32/25	9/7	9/7	9/7
57	625/486	625/486	77/60	77/60
58	162/125	162/125	100/77	84/65
59	125/96	35/27	35/27	13/10
60	6561/5000	64/49	64/49	64/49
61	675/512	21/16	21/16	21/16
62	4096/3125	324/245	160/121	120/91
63	320/243	250/189	33/25	33/25
64	20736/15625	1715/1296	162/121	143/108
65	4/3	4/3	4/3	4/3
66	15625/11664	343/256	343/256	169/126
67	27/20	27/20	27/20	27/20
68	3125/2304	875/648	121/90	65/48
69	2187/1600	200/147	15/11	15/11
70	5625/4096	49/36	49/36	49/36
71	512/375	48/35	48/35	48/35
72	1000/729	343/250	11/8	11/8
73	864/625	243/175	168/121	18/13
74	25/18	25/18	25/18	25/18
75	4374/3125	480/343	108/77	108/77
76	45/32	7/5	7/5	7/5
77	65536/46875	486/343	486/343	128/91
78	729/512	567/400	99/70	55/39
79	46875/32768	343/243	343/243	91/64
80	64/45	10/7	10/7	10/7
81	3125/2187	343/240	77/54	77/54
82	36/25	36/25	36/25	36/25
83	625/432	350/243	121/84	13/9
84	729/500	500/343	16/11	16/11
85	375/256	35/24	35/24	35/24
86	8192/5625	72/49	72/49	72/49
87	3200/2187	147/100	22/15	22/15
88	4608/3125	729/490	180/121	96/65
89	40/27	40/27	40/27	40/27
90	23328/15625	512/343	512/343	143/96
91	3/2	3/2	3/2	3/2
92	15625/10368	2401/1600	121/81	121/81
93	243/160	189/125	50/33	50/33
94	3125/2048	245/162	121/80	91/60
95	1024/675	32/21	32/21	32/21
96	10000/6561	49/32	49/32	49/32
97	192/125	54/35	54/35	20/13
98	125/81	125/81	77/50	65/42
99	972/625	972/625	120/77	81/52
100	25/16	14/9	14/9	14/9
101	19683/12500	384/245	243/154	169/108
102	405/256	63/40	11/7	11/7
103	24576/15625	1944/1225	192/121	144/91
104	128/81	100/63	35/22	35/22
105	31250/19683	343/216	343/216	143/90
106	8/5	8/5	8/5	8/5
107	3125/1944	1029/640	77/48	77/48
108	81/50	81/50	81/50	21/13
109	625/384	175/108	121/75	13/8
110	6561/4000	80/49	18/11	18/11
111	3375/2048	49/30	44/27	44/27
112	1024/625	81/49	81/49	64/39
113	400/243	400/243	33/20	33/20
114	5184/3125	1458/875	128/77	108/65
115	5/3	5/3	5/3	5/3
116	26244/15625	576/343	539/324	240/143
117	27/16	27/16	27/16	27/16
118	15625/9216	1225/729	121/72	91/54
119	2048/1215	250/147	56/33	22/13
120	28125/16384	245/144	245/144	143/84
121	128/75	12/7	12/7	12/7
122	1250/729	343/200	55/32	55/32
123	216/125	216/125	210/121	45/26
124	125/72	125/72	121/70	26/15
125	2187/1250	256/147	96/55	96/55
126	225/128	7/4	7/4	7/4
127	16384/9375	432/245	432/245	160/91
128	1280/729	441/250	44/25	39/22
129	27648/15625	1715/972	216/121	143/81
130	16/9	16/9	16/9	16/9
131	15625/8748	343/192	343/192	143/80
132	9/5	9/5	9/5	9/5
133	3125/1728	875/486	231/128	65/36
134	729/400	625/343	20/11	20/11
135	1875/1024	49/27	49/27	49/27
136	2048/1125	64/35	64/35	64/35
137	4000/2187	147/80	11/6	11/6
138	1152/625	324/175	224/121	24/13
139	50/27	50/27	50/27	13/7
140	5832/3125	640/343	144/77	144/77
141	15/8	15/8	15/8	15/8
142	59049/31250	648/343	539/288	169/90
143	243/128	189/100	66/35	49/26
144	15625/8192	1225/648	121/64	91/48
145	256/135	40/21	21/11	21/11
146	12500/6561	245/128	154/81	143/75
147	48/25	27/14	27/14	25/13
148	625/324	625/324	77/40	52/27
149	243/125	243/125	64/33	64/33
150	125/64	35/18	35/18	35/18
151	19683/10000	96/49	96/49	96/49
152	2025/1024	49/25	49/25	49/25
153	6144/3125	486/245	240/121	128/65
154	160/81	125/63	99/50	77/39
155	31104/15625	1715/864	243/121	143/72
156	2/1	2/1	2/1	2/1