Cotoneum: Difference between revisions
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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]], | | 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. | | 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]] | ||
Revision as of 08:03, 20 May 2026
| Cotoneum |
364/363, 441/440, 3584/3575,
10976/10935 (13-limit);
343/342, 364/363, 441/440, 595/594,
1216/1215, 1729/1728 (19-limit)
21-odd-limit: 2.48 ¢
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 0.3–0.4 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 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 5-6 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.
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 | 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
- 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