View source for Cotoneum

{{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]]. 

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 ==
Odd harmonics and subharmonics 1–21 are in '''bold'''.

{| class="wikitable center-1 right-2"
! #
! Cents*
! 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, <br>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,<br>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
| 175/88, 484/243,<br>351/176, 768/385
|}
<nowiki>*</nowiki> in 19-limit POTE tuning

== 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 ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo tuning<br>(n\EDO)
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-Interval)]]
! Generator<br>(¢)
! Comments
|-
| '''[[53edo|31\53]]'''
| 
| '''701.8868'''
| '''Lower bound of 9-odd-limit [[diamond monotone]]'''<br>53cffgggh val
|-
| 
| [[4/3]]
| 701.9550
| 
|-
| '''[[94edo|55\94]]'''
| 
| '''702.1277'''
| '''Lower bound of 11-odd-limit diamond monotone'''<br>94cfggh val
|-
| 
| [[9/7]]
| 702.1928
| 
|-
| 
| [[7/6]]
| 702.2086
| 
|-
| '''[[135edo|79\135]]'''
| 
| '''702.2222'''
| '''Lower bound of 13- and 15-odd-limit diamnod monotone''' <br>135cfgh val
|-
| 
| [[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
| 
|-
| '''[[176edo|103\176]]'''
| 
| '''702.2727'''
| '''Lower bound of 17- through 21-odd-limit diamond monotone'''
|-
| 
| [[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- through 21-odd-limit minimax
|-
| 
| [[17/13]]
| 702.3010
| 
|-
| 
| [[17/16]]
| 702.3029
| 
|-
| 
| [[16/13]]
| 702.3037
| 
|-
| [[217edo|127\217]]
| 
| 702.3041
| 
|-
| 
| [[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
| 
|-
| [[258edo|151\258]]
| 
| 702.3256
| 
|-
| 
| [[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
| 
|-
| '''[[41edo|24\41]]'''
| 
| '''702.4390'''
| '''Upper bound of 11- through 21-odd-limit diamond monotone'''
|}

== Scales ==
* [[Cotoneum5]] - proper [[2L 3s]]
* [[Cotoneum7]] - improper [[5L 2s]]
* [[Cotoneum12]] - proper [[5L 7s]]
* [[Cotoneum17]] - improper [[12L 5s]]
* [[Cotoneum29]] - improper [[12L 17s]]
* [[Cotoneum41]] - proper [[12L 29s]]
* [[Cotoneum53]] - improper [[41L 12s]]

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