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| Title = Cotoneum
| Title = Cotoneum
| Subgroups = 2.3.5.7, 2.3.5.7.11.13, 2.3.5.7.11.13.17.19
| 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]], [[10976/10935]] (13-limit);<br>[[343/342]], [[364/363]], [[441/440]], [[595/594]], [[1216/1215]],<br> [[1729/1728]] (19-limit)
| 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
| Edo join 1 = 41 | Edo join 2 = 217
| Mapping = 1; 1 -49 -14 23 61 89 -44
| Mapping = 1; 1 -49 -14 23 61 89 -44
| Generators = 3/2
| Generators = 3/2
| Generators tuning = 702.308
| Generators tuning = 702.31
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[12L 17s]], [[12L 29s]], [[41L 12s]], [[41L 53s]]
| MOS scales = [[12L 17s]], [[12L 29s]], [[41L 12s]], [[41L 53s]]
Line 13: 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. 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]].  


It can seen as a detemperament of [[41edo|41 equal temperament]], with the [[countercomp comma|41-comma]] shrunk down to about 5-6 cents, representing many important intervals such as the [[schisma]], [[5120/5103]], [[176/175]], [[243/242]], [[273/272]], [[325/324]], [[352/351]], [[385/384]], [[513/512]], [[896/891]], etc.
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.
[[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 [[User:Xenllium|Xenllium]] in 2021. ''Cotoneum'' is Latin for "quince".  
The temperament was named by [[Xenllium]] in 2021. ''Cotoneum'' is Latin for "quince".  


For technical data, see [[Garischismic clan #Cotoneum]].
For technical data, see [[Garischismic clan #Cotoneum]].
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{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! Fifths
! #
! 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
| 57/40
|-
|-
| 7
| 7
| 116.154
| 116.15
| 77/72
| 77/72
|-
|-
| 8
| 8
| 818.462
| 818.46
| 77/48
| 77/48
|-
|-
| 9
| 9
| 320.770
| 320.77
| 77/64
| 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
| 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
| 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
| 63/38
|-
|-
| 33
| 33
| 376.156
| 376.15
| 56/45
| 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
| 175/88, 484/243,<br>351/176, 768/385
| 351/176, 484/243, 539/270
|}
|}
<nowiki>*</nowiki> in 19-limit POTE tuning
<nowiki/>* In 19-limit CWE tuning, octave reduced


== Tuning spectrum ==
== Notation ==
Gencom: [2 4/3; 343/342 364/363 441/440 595/594 1216/1215 1729/1728]
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).


Gencom mapping: [{{val| 1 2 -18 -3 13 29 41 -14 }}, {{val| 0 -1 49 14 -23 -61 -89 44 }}]
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.3149{{C}}
| 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
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.3063{{C}}
| 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
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.3069{{C}}
| CWE: ~3/2 = 702.3077{{C}}
| POTE: ~3/2 = 702.3077{{C}}
|}


{| class="wikitable center-1 center-2"
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-Interval)]]
! Edo generator
! Generator<br>(¢)
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator (¢)
! Comments
! Comments
|-
|-
| 4/3
| '''[[53edo|31\53]]'''
|
| '''701.8868'''
| '''Lower bound of 9-odd-limit [[diamond monotone]]'''<br>53cffgggh val
|-
|
| [[4/3]]
| 701.9550
| 701.9550
|  
|  
|-
|-
| 9/7
| '''[[94edo|55\94]]'''
|
| '''702.1277'''
| '''Lower bound of 11-odd-limit diamond monotone'''<br>94cfggh val
|-
|
| [[9/7]]
| 702.1928
| 702.1928
|  
|  
|-
|-
| 7/6
|  
| [[7/6]]
| 702.2086
| 702.2086
|  
|  
|-
|-
| 8/7
| '''[[135edo|79\135]]'''
|
| '''702.2222'''
| '''Lower bound of 13- and 15-odd-limit diamnod monotone''' <br>135cfgh val
|-
|
| [[8/7]]
| 702.2267
| 702.2267
|  
|  
|-
|-
| 14/11
|  
| [[14/11]]
| 702.2295
| 702.2295
|  
|  
|-
|-
| 11/8
|  
| [[11/8]]
| 702.2312
| 702.2312
|  
|  
|-
|-
| 22/21
|  
| [[22/21]]
| 702.2371
| 702.2371
|  
|  
|-
|-
| 20/19
|  
| [[20/19]]
| 702.2399
| 702.2399
|  
|  
|-
|-
| 12/11
|  
| [[12/11]]
| 702.2438
| 702.2438
|  
|  
|-
|-
| 21/16
|  
| [[21/16]]
| 702.2476
| 702.2476
|  
|  
|-
|-
| 11/9
|  
| [[11/9]]
| 702.2575
| 702.2575
|  
|  
|-
|-
| 14/13
| '''[[176edo|103\176]]'''
|
| '''702.2727'''
| '''Lower bound of 17- through 21-odd-limit diamond monotone'''
|-
|
| [[14/13]]
| 702.2894
| 702.2894
|  
|  
|-
|-
| 11/10
|  
| [[11/10]]
| 702.2917
| 702.2917
| 11 and 13-odd-limit minimax
| 11- and 13-odd-limit minimax
|-
|-
| 17/14
|  
| [[17/14]]
| 702.2925
| 702.2925
|  
|  
|-
|-
| 26/21
|  
| [[26/21]]
| 702.2939
| 702.2939
|  
|  
|-
|-
| 22/19
|  
| [[22/19]]
| 702.2956
| 702.2956
|  
|  
|-
|-
| 21/17
|  
| [[21/17]]
| 702.2958
| 702.2958
|  
|  
|-
|-
| 15/11
|  
| [[15/11]]
| 702.2965
| 702.2965
| 15, 17, 19, and 21-odd-limit minimax  
| 15- through 21-odd-limit minimax
|-
|-
| 17/13
|  
| [[17/13]]
| 702.3010
| 702.3010
|  
|  
|-
|-
| 17/16
|  
| [[17/16]]
| 702.3029
| 702.3029
|  
|  
|-
|-
| 16/13
|  
| [[16/13]]
| 702.3037
| 702.3037
|  
|  
|-
|-
| 10/9
| [[217edo|127\217]]
|
| 702.3041
|
|-
|
| [[10/9]]
| 702.3058
| 702.3058
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
| 24/17
|  
| [[24/17]]
| 702.3068
| 702.3068
|  
|  
|-
|-
| 20/17
|  
| [[20/17]]
| 702.3090
| 702.3090
|  
|  
|-
|-
| 13/12
|  
| [[13/12]]
| 702.3095
| 702.3095
|  
|  
|-
|-
| 18/17
|  
| [[18/17]]
| 702.3109
| 702.3109
|  
|  
|-
|-
| 13/10
|  
| [[13/10]]
| 702.3110
| 702.3110
|  
|  
|-
|-
| 19/15
|  
| [[19/15]]
| 702.3111
| 702.3111
|  
|  
|-
|-
| 17/15
|  
| [[17/15]]
| 702.3116
| 702.3116
|  
|  
|-
|-
| 19/17
|  
| [[19/17]]
| 702.3116
| 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'''
|}
|}


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