Cotoneum

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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

Cotoneum temperament (41&176)

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]]

POTE generator:

  • 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

TOP generators:

  • 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]

Optimal GPV sequences:

Badness:

  • 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

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