Parakleismic

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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 alike 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

ET
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