Cotoneum: Difference between revisions

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| Odd limit 2 = 21 | Mistuning 2 = 2.48 | Complexity 2 = 176
| Odd limit 2 = 21 | Mistuning 2 = 2.48 | Complexity 2 = 176
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'''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 0.3–0.4 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 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

Revision as of 09:10, 20 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 = 702.31 ¢
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. The generator of cotoneum is a 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 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 41 equal temperament, with the 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 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.

# 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,
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 1194.618 175/88, 484/243,
351/176, 768/385

* in 19-limit POTE tuning

Tuning spectrum

Edo tuning
(n\EDO) 		Eigenmonzo
(Unchanged-Interval) 		Generator
(¢) 		Comments
31\53 701.8868 Lower bound of 9-odd-limit diamond monotone
53cffgggh val
4/3 701.9550
55\94 702.1277 Lower bound of 11-odd-limit diamond monotone
94cfggh val
9/7 702.1928
7/6 702.2086
79\135 702.2222 Lower bound of 13- and 15-odd-limit diamnod monotone
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
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
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
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
24\41 702.4390 Upper bound of 11- through 21-odd-limit diamond monotone

Scales

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