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'''Cotoneum temperament''' is temperament for the 7, 11, 13, 17, and 19 [[Harmonic limit|prime limits]]. It is a member of [[hemimage temperaments]], [[quince clan]], and [[garischismic clan]]. [[217edo|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.
{{Infobox regtemp
| Title = 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);<br>[[364/363]], [[441/440]], [[3584/3575]], <br>[[10976/10935]] (13-limit);<br>[[343/342]], [[364/363]], [[441/440]], [[595/594]], <br>[[1216/1215]], [[1729/1728]] (19-limit)
| Edo join 1 = 41 | Edo join 2 = 217
| Mapping = 1; 1 -49 -14 23 61 89 -44
| Generators = 3/2
| Generators tuning = 702.31
| Optimization method = CWE
| MOS scales = [[12L 17s]], [[12L 29s]], [[41L 12s]], [[41L 53s]]
| Pergen = (P8, P5)
| Odd limit 1 = 15 | Mistuning 1 = 2.48 | Complexity 1 = 135
| 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. The generator of cotoneum is a [[3/2|perfect fifth]] sharp by about 0.3–0.4 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 as a diminished fourth. Instead, 5/4 is found as a 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 is a member of the [[hemimage temperaments]], [[quince clan]], and [[garischismic clan]].  


See [[Hemimage temperaments #Cotoneum]] for more technical data.
It can seen as a detemperament of [[41edo|41 equal temperament]], with the [[countercomp comma|41-comma]] shrunk down to about 5–6 cents for a generic aberschisma, which represents the [[schisma]] and [[aberschisma]].
 
This generic aberschisma takes on more important roles from the 11-limit onwards, where it represents [[176/175]], [[243/242]], [[385/384]], [[540/539]] and [[896/891]]. In the 13-limit it represents [[352/351]], in the 17-limit [[273/272]], and in the 19-limit the undevicesimal schisma of [[513/512]].
 
[[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.
 
The temperament was named by [[Xenllium]] in 2021. ''Cotoneum'' is Latin for "quince".
 
For technical data, see [[Garischismic clan #Cotoneum]].


== 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 <br>generators
! #
! Cents <br>value*
! Cents*
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
| 0.000
| 0.00
| 1/1
| '''1/1'''
|-
|-
| 1
| 1
| 702.308
| 702.31
| 3/2
| '''3/2'''
|-
|-
| 2
| 2
| 204.615
| 204.62
| 9/8
| '''9/8'''
|-
|-
| 3
| 3
| 906.923
| 906.92
| 27/16
| 27/16
|-
|-
| 4
| 4
| 409.231
| 409.23
| 19/15
| 19/15
|-
|-
| 5
| 5
| 1111.539
| 1111.54
| 19/10
| 19/10
|-
|-
| 6
| 6
| 613.846
| 613.85
|  
| 57/40
|-
|-
| 7
| 7
| 116.154
| 116.15
|  
| 77/72
|-
|-
| 8
| 8
| 818.462
| 818.46
|  
| 77/48
|-
|-
| 9
| 9
| 320.770
| 320.77
|  
| 77/64
|-
|-
| 10
| 10
| 1023.077
| 1023.08
| 65/36
| 65/36
|-
|-
| 11
| 11
| 525.385
| 525.38
| 65/48
| 65/48
|-
|-
| 12
| 12
| 27.693
| 27.69
| 56/55, 64/63, <br>65/64, 66/65
| 56/55, 64/63, 65/64, 66/65
|-
|-
| 13
| 13
| 730.001
| 730.00
| 32/21
| '''32/21'''
|-
|-
| 14
| 14
| 232.308
| 232.31
| 8/7
| '''8/7'''
|-
|-
| 15
| 15
| 934.616
| 934.62
| 12/7
| 12/7
|-
|-
| 16
| 16
| 436.924
| 436.92
| 9/7
| 9/7
|-
|-
| 17
| 17
| 1139.232
| 1139.23
| 27/14
| 27/14
|-
|-
| 18
| 18
| 641.539
| 641.54
|  
| 81/56
|-
|-
| 19
| 19
| 143.847
| 143.85
|  
| 88/81
|-
|-
| 20
| 20
| 846.155
| 846.15
| 44/27
| 44/27
|-
|-
| 21
| 21
| 348.463
| 348.46
| 11/9
| 11/9
|-
|-
| 22
| 22
| 1050.770
| 1050.77
| 11/6
| 11/6
|-
|-
| 23
| 23
| 553.078
| 553.08
| 11/8
| '''11/8'''
|-
|-
| 24
| 24
| 55.386
| 55.38
| 33/32, 65/63
| 33/32
|-
|-
| 25
| 25
| 757.694
| 757.69
| 65/42
| 65/42
|-
|-
| 26
| 26
| 260.001
| 260.00
| 64/55, 65/56
| 64/55, 65/56
|-
|-
| 27
| 27
| 962.309
| 962.31
|  
| 68/39, 96/55
|-
|-
| 28
| 28
| 464.617
| 464.62
| 17/13
| 17/13
|-
|-
| 29
| 29
| 1166.925
| 1166.92
| 51/26, 96/49, <br>108/55, 112/57
| 51/26, 96/49, 108/55, 112/57
|-
|-
| 30
| 30
| 669.232
| 669.23
| 28/19
| 28/19
|-
|-
| 31
| 31
| 171.540
| 171.54
| 21/19
| 21/19
|-
|-
| 32
| 32
| 873.848
| 873.85
|  
| 63/38
|-
|-
| 33
| 33
| 376.156
| 376.15
|  
| 56/45
|-
|-
| 34
| 34
| 1078.463
| 1078.46
| 28/15
| 28/15
|-
|-
| 35
| 35
| 580.771
| 580.77
| 7/5
| 7/5
|-
|-
| 36
| 36
| 83.079
| 83.08
| 21/20, 22/21
| 21/20, 22/21
|-
|-
| 37
| 37
| 785.387
| 785.38
| 11/7
| 11/7
|-
|-
| 38
| 38
| 287.694
| 287.69
| 13/11
| 13/11
|-
|-
| 39
| 39
| 990.002
| 990.00
| 39/22
| 39/22
|-
|-
| 40
| 40
| 492.310
| 492.31
|  
| 117/88
|-
|-
| 41
| 41
| 1194.618
| 1194.62
|  
| 351/176, 484/243, 539/270
|}
<nowiki/>* In 19-limit CWE tuning, octave reduced
 
== Notation ==
Cotoneum can be notated just like [[cassaschismic]], with accidentals for the generic comma and the generic aberschisma. As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step. The only difference is that the aberschisma step which is independent in cassaschismic is equated with the 41-comma here. In other words, we have C–^↑↑E ~ C–↓↓E, implying ~11/9 (double-comma-up minor third) + an aberschisma-up = ~27/22 (double-comma-down major third).
 
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
| 42
! rowspan="2" |  
| 696.925
! colspan="3" | Euclidean
|  
|-
|-
| 43
! Constrained
| 199.233
! Constrained & skewed
|
! Destretched
|-
|-
| 44
! Tenney
| 901.541
| CTE: ~3/2 = 702.3149{{C}}
| 32/19
| CWE: ~3/2 = 702.3164{{C}}
| POTE: ~3/2 = 702.3170{{C}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
|-
| 45
! rowspan="2" |  
| 403.849
! colspan="3" | Euclidean
| 24/19
|-
|-
| 46
! Constrained
| 1106.156
! Constrained & skewed
| 36/19
! Destretched
|-
|-
| 47
! Tenney
| 608.464
| CTE: ~3/2 = 702.3063{{C}}
| 27/19, 64/45
| CWE: ~3/2 = 702.3061{{C}}
| POTE: ~3/2 = 702.3060{{C}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 19-limit norm-based tunings
|-
|-
| 48
! rowspan="2" |  
| 110.772
! colspan="3" | Euclidean
| 16/15
|-
|-
| 49
! Constrained
| 813.080
! Constrained & skewed
| 8/5
! Destretched
|-
|-
| 50
! Tenney
| 315.387
| CTE: ~3/2 = 702.3069{{C}}
| 6/5
| CWE: ~3/2 = 702.3077{{C}}
| POTE: ~3/2 = 702.3077{{C}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
|-
| 51
! Edo generator
| 1017.695
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
| 9/5
! Generator (¢)
! Comments
|-
|-
| 52
| '''[[53edo|31\53]]'''
| 520.003
| 27/20
|-
| 53
| 22.310
| 81/80
|-
| 54
| 724.618
| 38/25
|-
| 55
| 226.926
|  
|  
| '''701.8868'''
| '''Lower bound of 9-odd-limit [[diamond monotone]]'''<br>53cffgggh val
|-
|-
| 56
| 929.234
|  
|  
|-
| [[4/3]]
| 57
| 701.9550
| 431.541
|  
|  
|-
|-
| 58
| '''[[94edo|55\94]]'''
| 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
|  
|  
| '''702.1277'''
| '''Lower bound of 11-odd-limit diamond monotone'''<br>94cfggh val
|-
|-
| 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
|  
|  
|-
| [[9/7]]
| 78
| 702.1928
| 780.004
|  
|  
|-
|-
| 79
| 282.312
|  
|  
|-
| [[7/6]]
| 80
| 702.2086
| 984.620
|  
|  
|-
|-
| 81
| '''[[135edo|79\135]]'''
| 486.927
|  
|  
| '''702.2222'''
| '''Lower bound of 13- and 15-odd-limit diamnod monotone''' <br>135cfgh val
|-
|-
| 82
| 1189.235
|  
|  
|-
| [[8/7]]
| 83
| 702.2267
| 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
|  
|  
|-
|-
| 91
| 310.005
|  
|  
|-
| [[14/11]]
| 92
| 702.2295
| 1012.313
|  
|  
|-
|-
| 93
| 514.620
|  
|  
|-
| [[11/8]]
| 94
| 702.2312
| 16.928
| 121/120
|-
| 95
| 719.236
|  
|  
|-
|-
| 96
| 221.544
|  
|  
|-
| [[22/21]]
| 97
| 702.2371
| 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
|  
|  
|-
| [[20/19]]
| 105
| 702.2399
| 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
|  
|  
|-
| [[12/11]]
| 113
| 702.2438
| 160.775
|  
|  
|-
|-
| 114
| 863.083
|  
|  
|-
| [[21/16]]
| 115
| 702.2476
| 365.391
|  
|  
|-
|-
| 116
| 1067.698
|  
|  
|-
| [[11/9]]
| 117
| 702.2575
| 570.006
|  
|  
|-
|-
| 118
| '''[[176edo|103\176]]'''
| 72.314
|  
|  
| '''702.2727'''
| '''Lower bound of 17- through 21-odd-limit diamond monotone'''
|-
|-
| 119
| 774.622
|  
|  
|-
| [[14/13]]
| 120
| 702.2894
| 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
|  
|  
| [[11/10]]
| 702.2917
| 11- and 13-odd-limit minimax
|-
|-
| 127
| 393.084
|  
|  
|-
| [[17/14]]
| 128
| 702.2925
| 1095.391
|  
|  
|-
|-
| 129
| 597.699
|  
|  
|-
| [[26/21]]
| 130
| 702.2939
| 100.007
|  
|  
|-
|-
| 131
| 802.315
|  
|  
|-
| [[22/19]]
| 132
| 702.2956
| 304.622
|  
|  
|-
|-
| 133
| 1006.930
| 34/19
|-
| 134
| 509.238
|  
|  
|-
| [[21/17]]
| 135
| 702.2958
| 11.546
| 126/125
|-
| 136
| 713.853
|  
|  
|-
|-
| 137
| 216.161
| 17/15
|-
| 138
| 918.469
| 17/10
|}
<nowiki>*</nowiki> in 19-limit POTE tuning
== Tuning spectrum ==
Gencom: [2 4/3; 343/342 364/363 441/440 595/594 1216/1215 1729/1728]
Gencom map: [{{val| 1 2 -18 -3 13 29 41 -14 }}, {{val| 0 -1 49 14 -23 -61 -89 44 }}]
{| class="wikitable center-1 right-2"
|-
! Eigenmonzo
! Fifth <br>generator
! Comments
|-
| 4/3
| 701.9550
|  
|  
| [[15/11]]
| 702.2965
| 15- through 21-odd-limit minimax
|-
|-
| 9/7
| 702.1928
|  
|  
|-
| [[17/13]]
| 7/6
| 702.3010
| 702.2086
|  
|  
|-
|-
| 8/7
| 702.2267
|  
|  
|-
| [[17/16]]
| 14/11
| 702.3029
| 702.2295
|  
|  
|-
|-
| 11/8
| 702.2312
|  
|  
|-
| [[16/13]]
| 22/21
| 702.3037
| 702.2371
|  
|  
|-
|-
| 20/19
| [[217edo|127\217]]
| 702.2399
|  
|  
|-
| 702.3041
| 12/11
| 702.2438
|  
|  
|-
|-
| 21/16
| 702.2476
|  
|  
| [[10/9]]
| 702.3058
| 9-odd-limit minimax
|-
|-
| 11/9
| 702.2575
|  
|  
|-
| [[24/17]]
| 14/13
| 702.3068
| 702.2894
|  
|  
|-
|-
| 11/10
| 702.2917
| 11 and 13-odd-limit minimax
|-
| 17/14
| 702.2925
|  
|  
|-
| [[20/17]]
| 26/21
| 702.3090
| 702.2939
|  
|  
|-
|-
| 22/19
| 702.2956
|  
|  
|-
| [[13/12]]
| 21/17
| 702.3095
| 702.2958
|  
|  
|-
|-
| 15/11
| 702.2965
| 15, 17, 19, and 21-odd-limit minimax
|-
| 17/13
| 702.3010
|  
|  
|-
| [[18/17]]
| 17/16
| 702.3109
| 702.3029
|  
|  
|-
|-
| 16/13
| 702.3037
|  
|  
|-
| [[13/10]]
| 10/9
| 702.3110
| 702.3058
| 9-odd-limit minimax
|-
| 24/17
| 702.3068
|  
|  
|-
|-
| 20/17
| 702.3090
|  
|  
|-
| [[19/15]]
| 13/12
| 702.3111
| 702.3095
|  
|  
|-
|-
| 18/17
| 702.3109
|  
|  
|-
| [[17/15]]
| 13/10
| 702.3116
| 702.3110
|  
|  
|-
|-
| 19/15
| 702.3111
|  
|  
|-
| [[19/17]]
| 17/15
| 702.3116
| 702.3116
|  
|  
|-
|-
| 19/17
| 702.3116
|  
|  
|-
| [[6/5]]
| 6/5
| 702.3128
| 702.3128
| 5 and 7-odd-limit minimax
| 5- and 7-odd-limit minimax
|-
|-
| 19/18
|  
| [[19/18]]
| 702.3130
| 702.3130
|  
|  
|-
|-
| 15/13
|  
| [[15/13]]
| 702.3143
| 702.3143
|  
|  
|-
|-
| 26/19
|  
| [[26/19]]
| 702.3144
| 702.3144
|  
|  
|-
|-
| 18/13
|  
| [[18/13]]
| 702.3156
| 702.3156
|  
|  
|-
|-
| 5/4
|  
| [[5/4]]
| 702.3201
| 702.3201
|  
|  
|-
|-
| 24/19
|  
| [[24/19]]
| 702.3209
| 702.3209
|  
|  
|-
|-
| 16/15
| [[258edo|151\258]]
|
| 702.3256
|
|-
|
| [[16/15]]
| 702.3277
| 702.3277
|  
|  
|-
|-
| 22/17
|  
| [[22/17]]
| 702.3278
| 702.3278
|  
|  
|-
|-
| 19/16
|  
| [[19/16]]
| 702.3292
| 702.3292
|  
|  
|-
|-
| 21/20
|  
| [[21/20]]
| 702.3463
| 702.3463
|  
|  
|-
|-
| 13/11
|  
| [[13/11]]
| 702.3476
| 702.3476
|  
|  
|-
|-
| 7/5
|  
| [[7/5]]
| 702.3575
| 702.3575
|  
|  
|-
|-
| 21/19
|  
| [[21/19]]
| 702.3635
| 702.3635
|  
|  
|-
|-
| 15/14
|  
| [[15/14]]
| 702.3693
| 702.3693
|  
|  
|-
|-
| 19/14
|  
| [[19/14]]
| 702.3771
| 702.3771
|  
|  
|-
| '''[[41edo|24\41]]'''
|
| '''702.4390'''
| '''Upper bound of 11- through 21-odd-limit diamond monotone'''
|}
|}


Line 771: Line 536:
* [[Cotoneum41]] - proper [[12L 29s]]
* [[Cotoneum41]] - proper [[12L 29s]]
* [[Cotoneum53]] - improper [[41L 12s]]
* [[Cotoneum53]] - improper [[41L 12s]]
* [[Cotoneum94]] - improper [[41L 53s]]
* [[Cotoneum135]] - [[41L 94s]] scale. The boundary of propriety is [[176edo|176EDO]].
* [[Cotoneum176]] - [[41L 135s]] scale. The boundary of propriety is [[217edo|217EDO]].


[[Category:Hemimage]]
[[Category:Cotoneum| ]]<!-- main article -->
[[Category:Quince]]
[[Category:Rank-2 temperaments]]
[[Category:Garischismic]]
[[Category:Hemimage temperaments]]
[[Category:Index of temperaments]]
[[Category:Quince clan]]
[[Category:Garischismic clan]]
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