Tertiaseptal

Tertiaseptal is a temperament for the 7, 11, 13, and 17 limit. EDOs that support tertiaseptal include 31edo, 140edo, and 171edo.

See Breedsmic temperaments #Tertiaseptal for more information.

Interval chain

Tertiaseptal and tertia

generator	cents value a (octave-reduced)	17-limit ratio (octave-reduced)
		tertiaseptal (31&171)	tertia (31&140)
1	77.2	117/112, 256/245, 68/65	117/112, 256/245, 68/65, 22/21
2	154.4	130/119, 35/32	12/11, 130/119, 35/32
3	231.6	8/7
4	308.8	117/98, 140/117
5	386.0	5/4
6	463.1	17/13
7	540.3	175/128	15/11, 175/128
8	617.5	10/7
9	694.7	112/75
10	771.9	25/16
11	849.1	44/27, 80/49, 49/30, 85/52, 18/11	80/49, 49/30, 85/52
12	926.3	128/75
13	1003.5	25/14
14	1080.7	28/15
15	1157.9	39/20	39/20, 88/45
16	35.1	55/54, 52/51, 51/50, 50/49, 49/48, 45/44	56/55, 52/51, 51/50, 50/49, 49/48
17	112.3	16/15
18	189.4	39/35
19	266.6	7/6
20	343.8	39/32	39/32, 11/9
21	421.0	51/40	14/11, 51/40
22	498.2	4/3
23	575.4	39/28
24	652.6	35/24	16/11, 35/24
25	729.8	32/21
26	807.0	51/32	35/22, 51/32
27	884.2	5/3
28	961.4	68/39
29	1038.6	51/28	20/11, 51/28
30	1115.7	40/21, 21/11	40/21
31	1192.9
32	70.1	26/25, 25/24
33	147.3	49/45, 12/11	49/45
34	224.5	91/80	25/22, 91/80
35	301.7	25/21
36	378.9	56/45, 96/77	56/45
37	456.1	13/10
38	533.3	34/25, 15/11	34/25
39	610.5	64/45
40	687.7	52/35
41	764.8	14/9
42	842.0	13/8
43	919.2	17/10
44	996.4	16/9
45	1073.6	13/7
46	1150.8	68/35, 35/18	64/33, 68/35, 35/18
47	28.0	65/64, 64/63, 56/55	78/77, 65/64, 64/63, 55/54
48	105.2	17/16
49	182.4	10/9
50	259.6	65/56, 64/55	65/56
51	336.8	17/14
52	414.0	80/63, 14/11	80/63
53	491.1	65/49
54	568.3	25/18
55	645.5	16/11
56	722.7	85/56	50/33, 85/56
57	799.9	100/63, 35/22	100/63
58	877.1	128/77
59	954.3	26/15
60	1031.5	136/75, 20/11	136/75
61	1108.7	91/48, 256/135
62	1185.9	208/105	196/99, 208/105
63	63.1	28/27
64	140.3	13/12
65	217.4	17/15, 25/22	17/15
66	294.6	32/27	13/11, 32/27
67	371.8	26/21
68	449.0	35/27
69	526.2	65/48
70	603.4	17/12
71	680.6	40/27
72	757.8	65/42	17/11, 65/42
73	835.0	34/21
74	912.2	56/33
75	989.4	39/22	136/77, 85/48
76	1066.6	50/27
77	1143.7	64/33	85/44
78	20.9	91/90, 85/84, 78/77	100/99, 91/90, 85/84
79	98.1	35/33
80	175.3	195/176
81	252.5	52/45
82	329.7	40/33
83	406.9	91/72
84	484.1	119/90
85	561.3	112/81
86	638.5	13/9
87	715.7	68/45, 50/33	68/45
88	792.8	128/81	52/33, 128/81
89	870.0	119/72
90	947.2	140/81
91	1024.4	65/36
92	1101.6	17/9
93	1178.8	160/81, 196/99, 240/121	65/33, 160/81
94	56.0	91/88	34/33
95	133.2	68/63
96	210.4	112/99
97	287.6	13/11
98	364.8	68/55
99	442.0	128/99
100	519.1	104/77
101	596.3
102	673.5
103	750.7	17/11
104	827.9	160/99
105	905.1
106	982.3	136/77
107	1059.5
108	1136.7	52/27, 85/44	52/27
109	13.9	100/99
110	91.1	128/121, 256/243	104/99, 256/243
111	168.3
112	245.4
113	322.6
114	399.8	34/27
115	477.0
116	554.2
117	631.4
118	708.6
119	785.8	52/33
120	863.0
121	940.2
122	1017.4
123	1094.5
124	1171.7	65/33
125	48.9	34/33

^a in 7-limit POTE tuning

Hemitert

generator	cents value a (octave-reduced)	11-limit ratio (octave-reduced)
1	38.6	45/44
2	77.2	256/245
3	115.8
4	154.4	35/32
5	193.0
6	231.6	8/7
7	270.2
8	308.8
9	347.4	11/9
10	386.0	5/4
11	424.6
12	463.1	64/49
13	501.7
14	540.3
15	578.9
16	617.5	10/7
17	656.1
18	694.7
19	733.3
20	771.9	25/16
21	810.5
22	849.1
23	887.7
24	926.3
25	964.9
26	1003.5	25/14
27	1042.1
28	1080.7	28/15
29	1119.3	21/11
30	1157.9
31	1196.5
32	35.1	50/49, 49/48
33	73.7
34	112.2	16/15
35	150.8	12/11
36	189.4
37	228.0
38	266.6	7/6
39	305.2
40	343.8
41	382.4
42	421.0
43	459.6
44	498.2	4/3
45	536.8	15/11
46	575.4
47	614.0
48	652.6
49	691.2
50	729.8	32/21
51	768.4
52	807.0
53	845.6
54	884.2	5/3
55	922.8
56	961.4
57	999.9
58	1038.5
59	1077.1
60	1115.7	40/21
61	1154.3
62	1192.9
63	31.5
64	70.1	25/24
65	108.7
66	147.3
67	185.9
68	224.5
69	263.1
70	301.7
71	340.3
72	378.9
73	417.5	14/11
74	456.1
75	494.7
76	533.3
77	571.9
78	610.5
79	649.1	16/11
80	687.6
81	726.2
82	764.8	14/9
83	803.4
84	842.0
85	880.6
86	919.2
87	957.8
88	996.4	16/9
89	1035.0	20/11
90	1073.6
91	1112.2
92	1150.8
93	1189.4
94	28.0	64/63
95	66.6
96	105.2
97	143.8
98	182.4	10/9

^a in 11-limit POTE tuning

Tuning spectra

Tertiaseptal

eigenmonzo (unchanged interval)	septimal whole tone (¢)	major third (¢)	perfect fifth (¢)	comments
8/7	231.1741	385.2902	704.7233
13/10	231.4228	385.7046	702.8998
14/13	231.4468	385.7446	702.7236
16/13	231.4663	385.7771	702.5807
15/13	231.4708	385.7847	702.5475
16/15	231.4820	385.8033	702.4654
13/12	231.4956	385.8260	702.3656
18/13	231.5099	385.8499	702.2606
20/17	231.5331	385.8886	702.0903
17/14	231.5370	385.8950	702.0618
17/15	231.5394	385.8990	702.0445
15/14	231.5480	385.9133	701.9816
4/3	231.5516	385.9193	701.9550
18/17	231.5558	385.9264	701.9239
24/17	231.5572	385.9286	701.9142
7/5	231.5579	385.9299	701.9085	7, 9-limit minimax
17/16	231.5597	385.9329	701.8954
10/9	231.5757	385.9596	701.7779
9/7	231.5792	385.9654	701.7524
6/5	231.5954	385.9924	701.6336	5-limit minimax
7/6	231.6112	386.0187	701.5179
13/11	231.6250	386.0417	701.4164	13, 15, 17-limit minimax
22/17	231.6593	386.0989	701.1648
11/8	231.7463	386.2438	700.5272
11/10	231.7498	386.2496	700.5016	11-limit minimax
14/11	231.7793	386.2988	700.2851
5/4	231.7882	386.3137	700.2197
15/11	231.8645	386.4409	699.6601
12/11	231.8761	386.4602	699.5753
17/13	232.2139	387.0231	697.0983
11/9	232.5251	387.5418	694.8159

Tertia

eigenmonzo (unchanged interval)	septimal whole tone (¢)	major third (¢)	perfect fifth (¢)	comments
12/11	225.9556	376.5926	742.9924
15/11	230.1218	383.5363	712.4404
14/11	231.0726	385.1209	705.4678
11/8	231.0853	385.1421	705.3748
8/7	231.1741	385.2902	704.7233
11/10	231.2065	385.3441	704.4860
13/11	231.3277	385.5462	703.5968
22/17	231.4016	385.6693	703.0552
13/10	231.4228	385.7046	702.8998
14/13	231.4468	385.7446	702.7236
16/13	231.4663	385.7771	702.5807
15/13	231.4708	385.7847	702.5475
16/15	231.4820	385.8033	702.4654
13/12	231.4956	385.8260	702.3656
18/13	231.5099	385.8499	702.2606	13, 15, 17-limit minimax
20/17	231.5331	385.8886	702.0903
17/14	231.5370	385.8950	702.0618
17/15	231.5394	385.8990	702.0445
15/14	231.5480	385.9133	701.9816
4/3	231.5516	385.9193	701.9550	11-limit minimax
18/17	231.5558	385.9264	701.9239
24/17	231.5572	385.9286	701.9142
7/5	231.5579	385.9299	701.9085	7, 9-limit minimax
17/16	231.5597	385.9329	701.8954
10/9	231.5757	385.9596	701.7779
9/7	231.5792	385.9654	701.7524
6/5	231.5954	385.9924	701.6336	5-limit minimax
7/6	231.6112	386.0187	701.5179
5/4	231.7882	386.3137	700.2197
11/9	232.1112	386.8520	697.8513
17/13	232.2139	387.0231	697.0983

Hemitert

eigenmonzo (unchanged interval)	septimal whole tone (¢)	major third (¢)	perfect fifth (¢)	comments
8/7	231.1741	385.2902	704.7233
16/15	231.4820	385.8033	702.4654
12/11	231.5378	385.8963	702.0563
11/8	231.5455	385.9091	701.9999
15/14	231.5480	385.9133	701.9816
4/3	231.5516	385.9193	701.9550
7/5	231.5579	385.9299	701.9085	7, 9, 11-limit minimax
11/10	231.5727	385.9546	701.7998
10/9	231.5757	385.9596	701.7779
14/11	231.5760	385.9600	701.7760
9/7	231.5792	385.9654	701.7524
15/11	231.5934	385.9891	701.6481
6/5	231.5954	385.9924	701.6336	5-limit minimax
11/9	231.6053	386.0088	701.5612
7/6	231.6112	386.0187	701.5179
5/4	231.7882	386.3137	700.2197