Cotoneum
Cotoneum is a temperament for the 7, 11, 13, 17, and 19 prime limits. It is a member of hemimage temperaments, quince clan, and garischismic clan. The fifth of cotoneum in size is a garibaldi fifth, but eight of them are not used to reach 5th subharmonic. This has the effect of making the Pythagorean sextuple-augmented unison a meantone fifth in size, four of them being ~1.4 cent sharp of 5th harmonic in the 19-limit POTE tuning for instance. 217EDO is an excellent tuning for cotoneum, with a fifth generator of 127\217, and MOS of 12, 17, 29, 41, 53, 94, 135, or 176 notes are available.
Temperament data
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
Mapping: [⟨1 2 -18 -3 13 29 41 -14], ⟨0 -1 49 14 -23 -61 -89 44]]
- 7-limit: ~4/3 = 497.68302
- 11-limit: ~4/3 = 497.69730
- 13-limit: ~4/3 = 497.69399
- 17-limit: ~4/3 = 497.69334
- 19-limit: ~4/3 = 497.69225
- 7-limit: ~2 = 1200.03878, ~4/3 = 497.69910
- 11-limit: ~2 = 1199.86324, ~4/3 = 497.64058
- 13-limit: ~2 = 1199.89014, ~4/3 = 497.64843
- 17-limit: ~2 = 1199.89426, ~4/3 = 497.64949
- 19-limit: ~2 = 1199.87700, ~4/3 = 497.64124
Diamond monotone ranges:
- 7-odd-limit: ~4/3 = [497.14286, 498.46154] (29\70 to 27\65)
- 9-odd-limit: ~4/3 = [497.14286, 498.11321] (29\70 to 22\53)
- 11-odd-limit: ~4/3 = [497.56098, 497.87234] (17\41 to 39\94)
- 13 and 15-odd-limit: ~4/3 = [497.56098, 497.77778] (17\41 to 56\135)
- 17, 19, and 21-odd-limit: ~4/3 = [497.56098, 497.72727] (17\41 to 73\176)
Diamond tradeoff ranges:
- 7, 9, 11, and 13-odd-limit: ~4/3 = [497.64251, 498.04500]
- 15 and 17-odd-limit: ~4/3 = [497.63067, 498.04500]
- 19 and 21-odd-limit: ~4/3 = [497.62290, 498.04500]
Diamond monotone and tradeoff ranges:
- 7 and 9-odd-limit: ~4/3 = [497.64251, 498.04500]
- 11-odd-limit: ~4/3 = [497.64251, 497.87234]
- 13-odd-limit: ~4/3 = [497.64251, 497.77778]
- 15-odd-limit: ~4/3 = [497.63067, 497.77778]
- 17-odd-limit: ~4/3 = [497.63067, 497.72727]
- 19 and 21-odd-limit: ~4/3 = [497.62290, 497.72727]
- 7-limit: 0.105632
- 11-limit: 0.050966
- 13-limit: 0.036951
- 17-limit: 0.029495
- 19-limit: 0.021811
Interval chain
Number of fifth |
Cents value* |
Approximate Ratios |
---|---|---|
0 | 0.000 | 1/1 |
1 | 702.308 | 3/2 |
2 | 204.615 | 9/8 |
3 | 906.923 | 27/16 |
4 | 409.231 | 19/15 |
5 | 1111.539 | 19/10 |
6 | 613.846 | 57/40 |
7 | 116.154 | 77/72 |
8 | 818.462 | 77/48 |
9 | 320.770 | 77/64 |
10 | 1023.077 | 65/36 |
11 | 525.385 | 65/48 |
12 | 27.693 | 56/55, 64/63, 65/64, 66/65 |
13 | 730.001 | 32/21 |
14 | 232.308 | 8/7 |
15 | 934.616 | 12/7 |
16 | 436.924 | 9/7 |
17 | 1139.232 | 27/14 |
18 | 641.539 | 81/56 |
19 | 143.847 | |
20 | 846.155 | 44/27 |
21 | 348.463 | 11/9 |
22 | 1050.770 | 11/6 |
23 | 553.078 | 11/8 |
24 | 55.386 | 33/32, 65/63 |
25 | 757.694 | 65/42 |
26 | 260.001 | 64/55, 65/56 |
27 | 962.309 | 68/39, 96/55 |
28 | 464.617 | 17/13 |
29 | 1166.925 | 51/26, 96/49, 108/55, 112/57 |
30 | 669.232 | 28/19 |
31 | 171.540 | 21/19 |
32 | 873.848 | 63/38 |
33 | 376.156 | 56/45 |
34 | 1078.463 | 28/15 |
35 | 580.771 | 7/5 |
36 | 83.079 | 21/20, 22/21 |
37 | 785.387 | 11/7 |
38 | 287.694 | 13/11 |
39 | 990.002 | 39/22 |
40 | 492.310 | |
41 | 1194.618 | |
42 | 696.925 | |
43 | 199.233 | 64/57 |
44 | 901.541 | 32/19 |
45 | 403.849 | 24/19 |
46 | 1106.156 | 36/19 |
47 | 608.464 | 27/19, 64/45 |
48 | 110.772 | 16/15 |
49 | 813.080 | 8/5 |
50 | 315.387 | 6/5 |
51 | 1017.695 | 9/5 |
52 | 520.003 | 27/20 |
53 | 22.310 | 76/75, 77/76, 78/77, 81/80, 99/98 |
54 | 724.618 | 38/25 |
55 | 226.926 | |
56 | 929.234 | |
57 | 431.541 | |
58 | 1133.849 | 52/27 |
59 | 636.157 | 13/9 |
60 | 138.465 | 13/12 |
61 | 840.772 | 13/8 |
62 | 343.080 | 39/32 |
63 | 1045.388 | 64/35 |
64 | 547.696 | 48/35 |
65 | 50.003 | 34/33, 36/35 |
66 | 752.311 | 17/11 |
67 | 254.619 | 22/19 |
68 | 956.927 | 33/19 |
69 | 459.234 | 98/75, 99/76 |
70 | 1161.542 | 88/45, 49/25 |
71 | 663.850 | 22/15 |
72 | 166.158 | 11/10 |
73 | 868.465 | 33/20 |
74 | 370.773 | 26/21 |
75 | 1073.081 | 13/7 |
76 | 575.389 | 39/28 |
77 | 77.696 | |
78 | 780.004 | |
79 | 282.312 | |
80 | 984.620 | |
81 | 486.927 | |
82 | 1189.235 | |
83 | 691.543 | 112/75 |
84 | 193.851 | 28/25 |
85 | 896.158 | 42/25 |
86 | 398.466 | 34/27 |
87 | 1100.774 | 17/9 |
88 | 603.082 | 17/12 |
89 | 105.389 | 17/16 |
90 | 807.697 | 51/32 |
91 | 310.005 | |
92 | 1012.313 | |
93 | 514.620 | |
94 | 16.928 | 121/120 |
95 | 719.236 | |
96 | 221.544 | |
97 | 923.851 | |
98 | 426.159 | 32/25 |
99 | 1128.467 | 48/25 |
100 | 630.775 | 36/25 |
101 | 133.082 | 27/25 |
102 | 835.390 | 34/21 |
103 | 337.698 | 17/14 |
104 | 1040.005 | 51/28 |
105 | 542.313 | 26/19 |
106 | 44.621 | 39/38 |
107 | 746.929 | |
108 | 249.236 | 52/45 |
109 | 951.544 | 26/15 |
110 | 453.852 | 13/10 |
111 | 1156.160 | 39/20 |
112 | 658.467 | |
113 | 160.775 | |
114 | 863.083 | |
115 | 365.391 | |
116 | 1067.698 | |
117 | 570.006 | |
118 | 72.314 | |
119 | 774.622 | |
120 | 276.929 | |
121 | 979.237 | 44/25 |
122 | 481.545 | 33/25 |
123 | 1183.853 | 99/50 |
124 | 686.160 | 52/35 |
125 | 188.468 | 39/35 |
126 | 890.776 | |
127 | 393.084 | |
128 | 1095.391 | |
129 | 597.699 | |
130 | 100.007 | |
131 | 802.315 | |
132 | 304.622 | |
133 | 1006.930 | 34/19 |
134 | 509.238 | |
135 | 11.546 | 126/125 |
136 | 713.853 | |
137 | 216.161 | 17/15 |
138 | 918.469 | 17/10 |
* in 19-limit POTE tuning
Tuning spectrum
Gencom: [2 4/3; 343/342 364/363 441/440 595/594 1216/1215 1729/1728]
Gencom mapping: [⟨1 2 -18 -3 13 29 41 -14], ⟨0 -1 49 14 -23 -61 -89 44]]
eigenmonzo (unchanged interval) |
fifth (¢) |
comments |
---|---|---|
4/3 | 701.9550 | |
9/7 | 702.1928 | |
7/6 | 702.2086 | |
8/7 | 702.2267 | |
14/11 | 702.2295 | |
11/8 | 702.2312 | |
22/21 | 702.2371 | |
20/19 | 702.2399 | |
12/11 | 702.2438 | |
21/16 | 702.2476 | |
11/9 | 702.2575 | |
14/13 | 702.2894 | |
11/10 | 702.2917 | 11 and 13-odd-limit minimax |
17/14 | 702.2925 | |
26/21 | 702.2939 | |
22/19 | 702.2956 | |
21/17 | 702.2958 | |
15/11 | 702.2965 | 15, 17, 19, and 21-odd-limit minimax |
17/13 | 702.3010 | |
17/16 | 702.3029 | |
16/13 | 702.3037 | |
10/9 | 702.3058 | 9-odd-limit minimax |
24/17 | 702.3068 | |
20/17 | 702.3090 | |
13/12 | 702.3095 | |
18/17 | 702.3109 | |
13/10 | 702.3110 | |
19/15 | 702.3111 | |
17/15 | 702.3116 | |
19/17 | 702.3116 | |
6/5 | 702.3128 | 5 and 7-odd-limit minimax |
19/18 | 702.3130 | |
15/13 | 702.3143 | |
26/19 | 702.3144 | |
18/13 | 702.3156 | |
5/4 | 702.3201 | |
24/19 | 702.3209 | |
16/15 | 702.3277 | |
22/17 | 702.3278 | |
19/16 | 702.3292 | |
21/20 | 702.3463 | |
13/11 | 702.3476 | |
7/5 | 702.3575 | |
21/19 | 702.3635 | |
15/14 | 702.3693 | |
19/14 | 702.3771 |
Scales
- Cotoneum5 - proper 2L 3s
- Cotoneum7 - improper 5L 2s
- Cotoneum12 - proper 5L 7s
- Cotoneum17 - improper 12L 5s
- Cotoneum29 - improper 12L 17s
- Cotoneum41 - proper 12L 29s
- Cotoneum53 - improper 41L 12s