Parakleismic: Difference between revisions

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| Title = Parakleismic
| Title = Parakleismic
| Subgroups = 2.3.5, 2.3.5.7
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[1224440064/1220703125]] (5-limit); <br>[[3136/3125]], [[4375/4374]] (7-limit)
| Comma basis = [[Parakleisma|1224440064/1220703125]] (5-limit); <br>[[3136/3125]], [[4375/4374]] (7-limit)
| Edo join 1 = 19 | Edo join 2 = 99
| Edo join 1 = 19 | Edo join 2 = 99
| Mapping = 1; 13 14 35
| Mapping = 1; 13 14 35
Line 193: Line 193:
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced


== Tuning spectrum ==
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! EDO<br>generator
! Edo<br>generator
! [[eigenmonzo|eigenmonzo<br>(unchanged interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! generator<br>(¢)
! Generator<br>(¢)
! comments
! Comments
|-
|-
| 16\61
| 16\61
|  
|  
| 314.754
| 314.754
| Lower bound of 9-odd-limit diamond monotone
| 61d val, lower bound of 9-odd-limit diamond monotone
|-
|-
|  
|  
| 15/14
| 15/14
| 314.930
| 314.930
|
|  
|-
|-
| 21\80
| 21\80
|  
|  
| 315.000
| 315.000
|
|  
|-
|-
|  
|  
| 9/7
| 9/7
| 315.009
| 315.009
|
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 315.118
| 315.118
|
|  
|-
|-
|  
|  
| 7/6
| 7/6
| 315.142
| 315.142
|
|  
|-
|-
| 26\99
| 26\99
|  
|  
| 315.152
| 315.152
|
|  
|-
|-
|  
|  
| 21/20
| 21/20
| 315.163
| 315.163
|
|  
|-
|-
|  
|  
| 49/48
| 49/48
| 315.163
| 315.163
|
|  
|-
|-
|  
|  
| 36/35
| 36/35
| 315.164
| 315.164
|
|  
|-
|-
|  
|  
| 8/7
| 8/7
| 315.176
| 315.176
| 7-odd-limit minimax (error = 1.217¢)
| 7-odd-limit minimax (error = 1.217 ¢)
|-
|-
|  
|  
| 80/63
| 80/63
| 315.183
| 315.183
| 9-odd-limit minimax (error = 1.345¢)
| 9-odd-limit minimax (error = 1.345 ¢)
|-
|-
|  
|  
| 10/9
| 10/9
| 315.200
| 315.200
|
|
|-
| 57\217
|
| 315.207
|  
|-
|-
|  
|  
| 4/3
| 4/3
| 315.234
| 315.234
|
|  
|-
|-
|  
|  
| 16/15
| 16/15
| 315.249
| 315.249
| 5-odd-limit minimax (error = 0.196¢)
| 5-odd-limit minimax (error = 0.196 ¢)
|-
|-
| 31\118
| 31\118
|  
|  
| 315.254
| 315.254
|
|  
|-
|-
|  
|  
| 5/4
| 5/4
| 315.263
| 315.263
|
|  
|-
|-
|  
|  
| 25/24
| 25/24
| 315.289
| 315.289
|
|  
|-
|-
|  
|  
| 6/5
| 6/5
| 315.641
| 315.641
|
|  
|-
|-
|  
|  
| 28/27
| 28/27
| 315.740
| 315.740
|
|  
|-
|-
| 5\19
| 5\19
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| Upper bound of 9-odd-limit diamond monotone
| Upper bound of 9-odd-limit diamond monotone
|}
|}
<nowiki>*</nowiki> Besides the octave


[[Category:Parakleismic| ]] <!-- main article -->
[[Category:Parakleismic| ]] <!-- main article -->

Latest revision as of 22:12, 12 May 2026

Parakleismic
Subgroups 2.3.5, 2.3.5.7
Comma basis 1224440064/1220703125 (5-limit);
3136/3125, 4375/4374 (7-limit)
Reduced mapping ⟨1; 13 14 35]
ET join 19 & 99
Generators (CWE) ~5/3 = 884.81 ¢
MOS scales 4L 15s, 19L 4s, 19L 23s
Ploidacot theta-13-cot
Minimax error 9-odd-limit: 1.34 ¢
Target scale size 9-odd-limit: 42 notes

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

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