Tertiaseptal

Todo: improve synopsis, improve readability Add infobox. Cut huge tables to reasonable sizes.

Tertiaseptal is a temperament for the 7, 11, 13, and 17 limit. EDOs that support tertiaseptal include 31edo, 140edo, 171edo and 311edo.

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

Gencom: [2 68/65; 243/242 375/374 441/440 625/624 3584/3575]

Gencom mapping: [⟨1 3 2 3 7 1 1], ⟨0 -22 5 -3 -55 42 48]]

eigenmonzo (unchanged-interval)	generator (¢)	comments
8/7	77.0580
13/10	77.1409
14/13	77.1489
16/13	77.1554
15/13	77.1569
16/15	77.1607
13/12	77.1652
18/13	77.1700
20/17	77.1777
17/14	77.1790
17/15	77.1798
15/14	77.1827
4/3	77.1839
18/17	77.1853
24/17	77.1857
7/5	77.1860	7 and 9-odd-limit minimax
17/16	77.1866
10/9	77.1919
9/7	77.1931
6/5	77.1985	5-odd-limit minimax
7/6	77.2037
13/11	77.2083	13, 15 and 17-odd-limit minimax
22/17	77.2198
11/8	77.2488
11/10	77.2499	11-odd-limit minimax
14/11	77.2598
5/4	77.2627
15/11	77.2882
12/11	77.2920
17/13	77.4046
11/9	77.5084

Tertia

Gencom: [2 22/21; 352/351 385/384 561/560 625/624 715/714]

Gencom mapping: [⟨1 3 2 3 5 1 1], ⟨0 -22 5 -3 -24 42 48]]

eigenmonzo (unchanged interval)	generator (¢)	comments
12/11	75.3185
15/11	76.7073
14/11	77.0242
11/8	77.0284
8/7	77.0580
11/10	77.0688
13/11	77.1092
22/17	77.1339
13/10	77.1409
14/13	77.1489
16/13	77.1554
15/13	77.1569
16/15	77.1607
13/12	77.1652
18/13	77.1700	13, 15 and 17-odd-limit minimax
20/17	77.1777
17/14	77.1790
17/15	77.1798
15/14	77.1827
4/3	77.1839	11-odd-limit minimax
18/17	77.1853
24/17	77.1857
7/5	77.1860	7 and 9-odd-limit minimax
17/16	77.1866
10/9	77.1919
9/7	77.1931
6/5	77.1985	5-odd-limit minimax
7/6	77.2037
5/4	77.2627
11/9	77.3704
17/13	77.4046

Hemitert

Gencom: [2 45/44; 2401/2400 3025/3024 65625/65536]

Gencom mapping: [⟨1 3 2 3 6], ⟨0 -44 10 -6 -79]]

eigenmonzo (unchanged interval)	generator (¢)	comments
8/7	38.5290
16/15	38.5803
12/11	38.5896
11/8	38.5909
15/14	38.5913
4/3	38.5919
7/5	38.5930	7, 9 and 11-odd-limit minimax
11/10	38.5955
10/9	38.59596
14/11	38.59600
9/7	38.5965
15/11	38.5989
6/5	38.5992	5-odd-limit minimax
11/9	38.6009
7/6	38.6019
5/4	38.6314