49edo

49-EDO, or 49 equal temperament divides the octave into 49 equal parts of 24.490 cents each.

Theory

49edo is very much on the sharp side of things, with sharp tunings of 3 (it is the first square equal division with a "real" 3 of size coprime to its cardinality), 5, 7, and 11. It is the optimal patent val for superpyth temperament in the 7 and 11 limits, archytas (7-limit) and ares (11-limit) planar temperaments and almost identical to the e-based analog of Lucy tuning. It tempers out 64/63, 245/243 and 3125/3087 in the 7-limit, and 100/99 and 1375/1372 in the 11-limit.

Intervals

#	Cents	Approximate Ratios
0	0.000	1/1
1	24.490	50/49
2	48.980	81/80, 28/27, 36/35, 49/48
3	73.469	25/24, 22/21, 33/32
4	97.959	16/15, 21/20
5	122.449	15/14
6	146.939	12/11
7	171.429	10/9, 11/10
8	195.918
9	220.408	9/8, 8/7
10	244.898
11	269.388	7/6
12	293.878
13	318.367	6/5
14	342.857	11/9
15	367.347	27/22
16	391.837	5/4
17	416.327	14/11
18	440.816	9/7
19	465.306
20	489.796	4/3, 21/16
21	514.286
22	538.776	27/20, 15/11
23	563.265	11/8
24	587.755	7/5
25	612.245	10/7
26	636.735	16/11
27	661.244	40/27, 22/15
28	685.714
29	710.204	3/2, 32/21
30	734.694
31	759.184	14/9
32	783.673	11/7
33	808.163	8/5
34	832.653	44/27
35	857.143	18/11
36	881.633	5/3
37	906.122
38	930.612	12/7
39	955.102
40	979.592	16/9, 7/4
41	1004.082
42	1028.571	9/5, 20/11
43	1053.061	11/6
44	1077.551	28/15
45	1102.041	15/8, 40/21
46	1126.531	48/25, 21/11, 64/33
47	1151.020	160/81, 27/14, 35/18, 96/49
48	1175.510	49/25
49	1200.000	2/1

Just approximation

Selected just intervals

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13
Error	absolute (¢)	0.0	+8.2	+5.5	+10.8	+11.9	-7.9
	relative (%)	0.0	+33.7	+22.6	+44.0	+48.8	-32.2

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 49et.

		3-limit	5-limit	7-limit	11-limit
Octave stretch (¢)		-2.60	-2.53	-2.85	-2.97
Error	absolute (¢)	2.60	2.12	1.92	1.74
	relative (%)	10.62	8.69	7.87	7.11

Rank-2 temperaments

Periods per octave	Generator	Temperaments
1	1\49	Sengagen
1	4\49	Passion
1	6\49	Bohpier
1	11\49	Infraorwell
1	13\49	Catalan
1	16\49	Magus
1	18\49	Clyde
1	19\49	Semisept
1	20\49	Superpyth
7	20\49	Sevond/seville