Cotoneum: Difference between revisions

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Cut table to 41 generators. The mappings of every interval do not need to be listed.
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{{Todo|inline=1|improve synopsis|improve readability|comment=Review if tables of these sizes are necessary.}}
{{Infobox regtemp
{{Infobox regtemp
| Title = Cotoneum
| Title = Cotoneum
Line 14: Line 13:
| Odd limit 2 = 21 | Mistuning 2 = 2.48 | Complexity 2 = 176
| Odd limit 2 = 21 | Mistuning 2 = 2.48 | Complexity 2 = 176
}}
}}
'''Cotoneum''' is a [[rank]]-2 [[regular temperament|temperament]] for the 7- through 19-limit. It is a member of the [[hemimage temperaments]], [[quince clan]], and [[garischismic clan]]. The generator of cotoneum is a perfect fifth sharp by about 0.4-0.5 cents, and it maps [[8/7]] to the double-augmented unison (+14 fifths), [[tempering out]] the [[garischisma]]. However, unlike in [[garibaldi]], the schisma is not tempered out, meaning 5/4 is not found at the diminished fourth. Instead, 5/4 is found at the sextuple-diminished octave (–49 fifths). It is a weak extension of the [[2.5.7 subgroup|2.5.7-subgroup]] temperament [[mercy]], with its secor-sized generator mapped to the augmented unison.
'''Cotoneum''' is a [[rank]]-2 [[regular temperament|temperament]] for the 7- through 19-limit. It is a member of the [[hemimage temperaments]], [[quince clan]], and [[garischismic clan]]. The generator of cotoneum is a perfect fifth sharp by about 0.4–0.5 cents, and it maps [[8/7]] to the double-augmented unison (+14 fifths), [[tempering out]] the [[garischisma]]. However, unlike in [[garibaldi]], the schisma is not tempered out, meaning 5/4 is not found at the diminished fourth. Instead, 5/4 is found at the sextuple-diminished octave (–49 fifths). It is a weak extension of the [[2.5.7 subgroup|2.5.7-subgroup]] temperament [[mercy]], with its secor-sized generator mapped to the augmented unison.


It can seen as a detemperament of [[41edo|41 equal temperament]], with the [[countercomp comma|41-comma]] shrunk down to about 5 cents, representing important intervals such as the [[schisma]], [[5120/5103]], [[325/324]], [[352/351]], [[385/384]], [[513/512]], etc.
It can seen as a detemperament of [[41edo|41 equal temperament]], with the [[countercomp comma|41-comma]] shrunk down to about 5 cents, representing many important intervals such as the [[schisma]], [[5120/5103]], [[243/242]], [[273/272]], [[325/324]], [[352/351]], [[385/384]], [[513/512]], [[896/891]], etc.


[[217edo]] is an excellent tuning for cotoneum, with a fifth generator of 127\217, and [[mos scale]]s of 12, 17, 29, 41, 53, 94, 135, or 176 notes are available.
[[217edo]] is an excellent tuning for cotoneum, with a fifth generator of 127\217, and [[mos scale]]s of 12, 17, 29, 41, 53, 94, 135, or 176 notes are available.
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== Interval chain ==
== Interval chain ==
Odd harmonics and subharmonics 1–21 are in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! Number of <br>fifth
! Fifths
! Cents <br>value*
! Cents <br>value*
! Approximate Ratios
! Approximate Ratios
Line 32: Line 33:
| 0
| 0
| 0.000
| 0.000
| 1/1
| '''1/1'''
|-
|-
| 1
| 1
| 702.308
| 702.308
| 3/2
| '''3/2'''
|-
|-
| 2
| 2
| 204.615
| 204.615
| 9/8
| '''9/8'''
|-
|-
| 3
| 3
Line 84: Line 85:
| 13
| 13
| 730.001
| 730.001
| 32/21
| '''32/21'''
|-
|-
| 14
| 14
| 232.308
| 232.308
| 8/7
| '''8/7'''
|-
|-
| 15
| 15
Line 124: Line 125:
| 23
| 23
| 553.078
| 553.078
| 11/8
| '''11/8'''
|-
|-
| 24
| 24
Line 148: Line 149:
| 29
| 29
| 1166.925
| 1166.925
| 51/26, 96/49, <br>108/55, 112/57
| 51/26, 96/49,<br>108/55, 112/57
|-
|-
| 30
| 30
Line 196: Line 197:
| 41
| 41
| 1194.618
| 1194.618
|  
| 351/176, 891/448
|-
| 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, <br>78/77, 81/80, <br>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
|}
|}
<nowiki>*</nowiki> in 19-limit POTE tuning
<nowiki>*</nowiki> in 19-limit POTE tuning

Revision as of 00:39, 19 May 2026

Cotoneum
Subgroups 2.3.5.7, 2.3.5.7.11.13, 2.3.5.7.11.13.17.19
Comma basis 10976/10935, 823543/819200 (7-limit);
364/363, 441/440, 3584/3575, 10976/10935 (13-limit);
343/342, 364/363, 441/440, 595/594, 1216/1215,
1729/1728 (19-limit)
Reduced mapping ⟨1; 1 -49 -14 23 61 89 -44]
ET join 41 & 217
Generators (CWE) ~3/2 = 702.308 ¢
MOS scales 12L 17s, 12L 29s, 41L 12s, 41L 53s
Ploidacot monocot
Pergen (P8, P5)
Minimax error 15-odd-limit: 2.48 ¢;
21-odd-limit: 2.48 ¢
Target scale size 15-odd-limit: 135 notes;
21-odd-limit: 176 notes

Cotoneum is a rank-2 temperament for the 7- through 19-limit. It is a member of the hemimage temperaments, quince clan, and garischismic clan. The generator of cotoneum is a perfect fifth sharp by about 0.4–0.5 cents, and it maps 8/7 to the double-augmented unison (+14 fifths), tempering out the garischisma. However, unlike in garibaldi, the schisma is not tempered out, meaning 5/4 is not found at the diminished fourth. Instead, 5/4 is found at the sextuple-diminished octave (–49 fifths). It is a weak extension of the 2.5.7-subgroup temperament mercy, with its secor-sized generator mapped to the augmented unison.

It can seen as a detemperament of 41 equal temperament, with the 41-comma shrunk down to about 5 cents, representing many important intervals such as the schisma, 5120/5103, 243/242, 273/272, 325/324, 352/351, 385/384, 513/512, 896/891, etc.

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".

For technical data, see Garischismic clan #Cotoneum.

Interval chain

Odd harmonics and subharmonics 1–21 are in bold.

Fifths 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 1194.618 351/176, 891/448

* 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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