Parakleismic: Difference between revisions

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== Interval chain ==
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–9 are in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! #
! #
! Cents*
! Cents*
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
| 0.0
| 0.00
| '''1/1'''
| '''1/1'''
|-
|-
| 1
| 1
| 315.2
| 315.19
| 6/5
| 6/5
|-
|-
| 2
| 2
| 630.4
| 630.38
| 36/25
| 36/25
|-
|-
| 3
| 3
| 945.5
| 945.57
| 140/81
| 140/81
|-
|-
| 4
| 4
| 60.7
| 60.76
| 28/27
| 28/27
|-
|-
| 5
| 5
| 375.9
| 375.96
| 56/45
| 56/45
|-
|-
| 6
| 6
| 691.1
| 691.15
| 112/75
| 112/75
|-
|-
| 7
| 7
| 1006.3
| 1006.34
| 25/14
| 25/14
|-
|-
| 8
| 8
| 121.4
| 121.53
| 15/14
| 15/14
|-
|-
| 9
| 9
| 436.6
| 436.72
| 9/7
| 9/7
|-
|-
| 10
| 10
| 751.8
| 751.91
| 54/35
| 54/35
|-
|-
| 11
| 11
| 1067.0
| 1067.10
| 50/27
| 50/27
|-
|-
| 12
| 12
| 182.2
| 182.29
| 10/9
| 10/9
|-
|-
| 13
| 13
| 497.4
| 497.49
| '''4/3'''
| '''4/3'''
|-
|-
| 14
| 14
| 812.5
| 812.68
| '''8/5'''
| '''8/5'''
|-
|-
| 15
| 15
| 1127.7
| 1127.87
| 48/25
| 48/25
|-
|-
| 16
| 16
| 242.9
| 243.06
| 144/125
| 144/125
|-
|-
| 17
| 17
| 558.1
| 558.25
| 112/81
| 112/81
|-
|-
| 18
| 18
| 873.7
| 873.44
| 224/135
| 224/135
|-
|-
| 19
| 19
| 1188.4
| 1188.63
| 125/63, 448/225, 486/245
| 125/63, 448/225, 486/245
|-
|-
| 20
| 20
| 303.6
| 303.82
| 25/21
| 25/21
|-
|-
| 21
| 21
| 618.8
| 619.01
| 10/7
| 10/7
|-
|-
| 22
| 22
| 934.0
| 934.21
| 12/7
| 12/7
|-
|-
| 23
| 23
| 49.2
| 49.40
| 36/35
| 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
|}
|}
<nowiki>*</nowiki> in 7-limit POTE tuning
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced


== Tuning spectrum ==
== Tuning spectrum ==

Revision as of 10:50, 14 March 2026

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

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

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
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