Cotoneum
| 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. 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 | 0.00 | 1/1 |
| 1 | 702.31 | 3/2 |
| 2 | 204.62 | 9/8 |
| 3 | 906.92 | 27/16 |
| 4 | 409.23 | 19/15 |
| 5 | 1111.54 | 19/10 |
| 6 | 613.85 | 57/40 |
| 7 | 116.15 | 77/72 |
| 8 | 818.46 | 77/48 |
| 9 | 320.77 | 77/64 |
| 10 | 1023.08 | 65/36 |
| 11 | 525.38 | 65/48 |
| 12 | 27.69 | 56/55, 64/63, 65/64, 66/65 |
| 13 | 730.00 | 32/21 |
| 14 | 232.31 | 8/7 |
| 15 | 934.62 | 12/7 |
| 16 | 436.92 | 9/7 |
| 17 | 1139.23 | 27/14 |
| 18 | 641.54 | 81/56 |
| 19 | 143.85 | 88/81 |
| 20 | 846.15 | 44/27 |
| 21 | 348.46 | 11/9 |
| 22 | 1050.77 | 11/6 |
| 23 | 553.08 | 11/8 |
| 24 | 55.38 | 33/32 |
| 25 | 757.69 | 65/42 |
| 26 | 260.00 | 64/55, 65/56 |
| 27 | 962.31 | 68/39, 96/55 |
| 28 | 464.62 | 17/13 |
| 29 | 1166.92 | 51/26, 96/49, 108/55, 112/57 |
| 30 | 669.23 | 28/19 |
| 31 | 171.54 | 21/19 |
| 32 | 873.85 | 63/38 |
| 33 | 376.15 | 56/45 |
| 34 | 1078.46 | 28/15 |
| 35 | 580.77 | 7/5 |
| 36 | 83.08 | 21/20, 22/21 |
| 37 | 785.38 | 11/7 |
| 38 | 287.69 | 13/11 |
| 39 | 990.00 | 39/22 |
| 40 | 492.31 | 117/88 |
| 41 | 1194.62 | 351/176, 484/243, 539/270 |
* 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
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