Parakleismic

Parakleismic is the microtemperament tempering out the parakleisma in the 5-limit. This article also assumes the canonical mapping for 7, which means tempering out 3136/3125 and 4375/4374 in the 7-limit.

Parakleismic is much like catakleismic but a good tuning has the generator (6/5) flat, instead of sharp, than the just version. The sixth generator step is half a syntonic comma flat of the harmonic 3. Consequently, the 12th generator step is mapped to 10/9 instead of 9/8, and the 13th generator step is mapped to 4/3 instead of 27/20.

Extensions for harmonic 11 includes 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.

Interval chain

#	Cents*	Approximate Ratios
0	0.0	1/1
1	315.2	6/5
2	630.4	36/25
3	945.5	140/81
4	60.7	28/27
5	375.9	56/45
6	691.1	112/75
7	1006.3	25/14
8	121.4	15/14
9	436.6	9/7
10	751.8	54/35
11	1067.0	50/27
12	182.2	10/9
13	497.4	4/3
14	812.5	8/5
15	1127.7	48/25
16	242.9	144/125
17	558.1	112/81
18	873.7	224/135
19	1188.4	125/63, 448/225, 486/245
20	303.6	25/21
21	618.8	10/7
22	934.0	12/7
23	49.2	36/35

* in 7-limit POTE tuning

Tuning spectrum

EDO generator	eigenmonzo (unchanged interval)	generator (¢)	comments
16\61		314.754	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
	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