Septidiasemi

Septidiasemi, in this article, is the rank-two regular temperament for the 2.3.5.7.13.17 subgroup defined by tempering out 2401/2400 and 2152828125/2147483648 in the 7-limit; 2205/2197 and 4096/4095 in the 13-limit; 833/832, 1275/1274, and 2025/2023 in the 17-limit.

It can be seen as implying a rank-two tuning which is generated by a large semitone of about 119.3 cents which represents 15/14 (septimal diatonic semitone) and from this it derives its name. Equal temperaments that support septidiasemi include 10EDO (generator 1\10), 151EDO (generator 15\151), 161EDO (generator 16\161), and 171EDO (generator 17\171).

See Breedsmic temperaments #Septidiasemi for more technical data.

Intervals

Generator steps	Cents*	Approximate ratios†
0	0.000	1/1
1	119.297	15/14
2	238.593	39/34
3	357.890	16/13
4	477.186	112/85, 120/91
5	596.483	24/17
6	715.779	68/45
7	835.076	34/21
8	954.372	26/15
9	1073.669	13/7
10	1192.965	448/225, 255/128, 544/273, 576/289
11	112.262	16/15
12	231.559	8/7
13	350.855	60/49, 49/40
14	470.152	21/16
15	589.448	45/32
16	708.745	128/85
17	828.041	21/13
18	947.338
19	1066.634	(close to 50/27)
20	1185.931	119/60, 135/68
21	105.227	17/16
22	224.524
23	343.820	39/32
24	463.117	17/13
25	582.414	7/5
26	701.710	3/2
27	821.007
28	940.303
29	1059.600	24/13
30	1178.896	(close to 160/81)
31	98.193	18/17
32	217.489	17/15
33	336.786	17/14
34	456.082	13/10
35	575.379
36	694.676	112/75
37	813.972	8/5
38	933.269	12/7
39	1052.565
40	1171.862	63/32, 128/65
41	91.158
42	210.455
43	329.751
44	449.048
45	568.344
46	687.641
47	806.937
48	926.234
49	1045.531	64/35
50	1164.827
51	84.124	21/20
52	203.420	9/8
53	322.717
54	442.013
55	561.310	18/13
56	680.606
57	799.903
58	919.199	17/10
59	1038.496
60	1157.793	39/20
61	77.089
62	196.386	28/25
63	315.682	6/5
64	434.979	9/7
65	554.275	(close to 11/8)
66	673.572
67	792.868
68	912.165
69	1031.461
70	1150.758
71	70.054
72	189.351
73	308.648
74	427.944	32/25
75	547.241
76	666.537
77	785.834
78	905.130	27/16
79	1024.427
80	1143.723
81	63.020	(close to 28/27)
82	182.316	(close to 10/9)
83	301.613
84	420.910
85	540.206
86	659.503
87	778.799
88	898.096	42/25
89	1017.392	9/5
90	1136.689	27/14

^* in 2.3.5.7.13.17 POTE tuning

^† 2.3.5.7.13.17 ratio interpretations

See also

• 15/14 - septimal diatonic semitone