Parakleismic
| Parakleismic |
3136/3125, 4375/4374 (7-limit)
Parakleismic is a microtemperament generated by a ~6/5 minor third, much like catakleismic but a good tuning has the generator flat, instead of sharp, than just. The sixth generator step is half a syntonic comma flat of the 3rd harmonic. Consequently, the 12th generator step after octave reduction represents 10/9 instead of 9/8, and the 13th generator step after octave reduction represents 4/3. This results in the parakleisma being tempered out.
Note that 5/4 is found at the 14th generator step (as the octave complement of 8/5) and is already split in halves. Letting each part represent 28/25 gives rise to the canonical 7-limit extension where it tempers out 3136/3125 and 4375/4374.
Extensions for harmonic 11 include undecimal parakleismic, mapping it to +36 steps, paralytic, to -82 steps, parkleismic, to -63 steps, and paradigmic, to +17 steps.
See Ragismic microtemperaments #Parakleismic for technical data. See Parakleismic extensions for a discussion on 11- and 13-limit extensions.
Interval chain
In the following table, odd harmonics and subharmonics 1–9 are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.00 | 1/1 |
| 1 | 315.19 | 6/5 |
| 2 | 630.38 | 36/25 |
| 3 | 945.57 | 140/81 |
| 4 | 60.76 | 28/27 |
| 5 | 375.96 | 56/45 |
| 6 | 691.15 | 112/75 |
| 7 | 1006.34 | 25/14 |
| 8 | 121.53 | 15/14 |
| 9 | 436.72 | 9/7 |
| 10 | 751.91 | 54/35 |
| 11 | 1067.10 | 50/27 |
| 12 | 182.29 | 10/9 |
| 13 | 497.49 | 4/3 |
| 14 | 812.68 | 8/5 |
| 15 | 1127.87 | 48/25 |
| 16 | 243.06 | 144/125 |
| 17 | 558.25 | 112/81 |
| 18 | 873.44 | 224/135 |
| 19 | 1188.63 | 125/63, 448/225, 486/245 |
| 20 | 303.82 | 25/21 |
| 21 | 619.01 | 10/7 |
| 22 | 934.21 | 12/7 |
| 23 | 49.40 | 36/35 |
| 24 | 364.59 | 100/81 |
| 25 | 679.78 | 40/27 |
| 26 | 994.97 | 16/9 |
| 27 | 110.16 | 16/15 |
| 28 | 425.35 | 32/25 |
| 29 | 740.54 | 75/49 |
| 30 | 1055.73 | 90/49 |
| 31 | 170.93 | 54/49 |
| 32 | 486.12 | 250/189, 324/245 |
| 33 | 801.31 | 100/63 |
| 34 | 1116.50 | 40/21 |
| 35 | 231.69 | 8/7 |
| 36 | 546.88 | 48/35 |
| 37 | 862.07 | 288/175, 400/243 |
| 38 | 1177.26 | 160/81 |
| 39 | 292.46 | 32/27 |
| 40 | 607.65 | 64/45 |
* In 7-limit CWE tuning, octave reduced
Tunings
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged interval)* |
Generator (¢) |
Comments |
|---|---|---|---|
| 16\61 | 314.754 | 61d val, lower bound of 9-odd-limit diamond monotone | |
| 15/14 | 314.930 | ||
| 21\80 | 315.000 | ||
| 9/7 | 315.009 | ||
| 7/5 | 315.118 | ||
| 7/6 | 315.142 | ||
| 26\99 | 315.152 | ||
| 21/20 | 315.163 | ||
| 49/48 | 315.163 | ||
| 36/35 | 315.164 | ||
| 8/7 | 315.176 | 7-odd-limit minimax (error = 1.217 ¢) | |
| 80/63 | 315.183 | 9-odd-limit minimax (error = 1.345 ¢) | |
| 10/9 | 315.200 | ||
| 57\217 | 315.207 | ||
| 4/3 | 315.234 | ||
| 16/15 | 315.249 | 5-odd-limit minimax (error = 0.196 ¢) | |
| 31\118 | 315.254 | ||
| 5/4 | 315.263 | ||
| 25/24 | 315.289 | ||
| 6/5 | 315.641 | ||
| 28/27 | 315.740 | ||
| 5\19 | 315.789 | Upper bound of 9-odd-limit diamond monotone |
* Besides the octave