Cotoneum
Cotoneum is a temperament for the 7-, 11-, 13-, 17-, and 19-limit. It is a member of hemimage temperaments, quince clan, and garischismic clan. The generator of cotoneum is a garibaldi fifth in size, but eight of them are not used to reach the 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 scales of 12, 17, 29, 41, 53, 94, 135, or 176 notes are available.
The temperament was named by Xenllium in 2021. Cotoneum is Latin for "quince".
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) |
Generator (¢) |
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