Cotoneum

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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

Main article: Garischismic clan § Cotoneum
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 ET 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) 		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

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