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'''Parakleismic''' is the microtemperament tempering out the [[parakleisma]] in the 5-limit, and [[3136/3125]] and [[4375/4374]] in the 7-limit.  
{{Infobox regtemp
| Title = Parakleismic
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[Parakleisma|1224440064/1220703125]] (5-limit); <br>[[3136/3125]], [[4375/4374]] (7-limit)
| Edo join 1 = 19 | Edo join 2 = 99
| Mapping = 1; 13 14 35
| Generators = 5/3
| Generators tuning = 884.81
| Optimization method = CWE
| MOS scales = [[4L 15s]], [[19L 4s]], [[19L 23s]]
| Odd limit 1 = 9 | Mistuning 1 = 1.34 | Complexity 1 = 42
}}
'''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 [[3/1|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 [[tempering out|tempered out]].  


Parakleismic is much alike [[catakleismic]] but a good tuning has the generator flat, instead of sharp, than the just version. The sixth generator step is half a [[syntonic comma]] flat of the harmonic 3. The 12th step is mapped to [[10/9]] instead of [[9/8]], and the 13th step is mapped to [[4/3]] instead of [[27/20]].  
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 includes ''undecimal parakleismic'', mapping it to +36 steps, ''paralytic'', to -82 steps, ''parkleismic'', to -63 steps, and ''paradigmic'', to +17 steps.  
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 [[Ragismic microtemperaments #Parakleismic]] for technical data. See [[Parakleismic extensions]] for a discussion on [[11-limit|11-]] and [[13-limit]] extensions.  


== 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
|-
| 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 CWE tuning, octave reduced
 
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator<br>(¢)
! 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
| 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
|}
|}
<nowiki>*</nowiki> in 7-limit POTE tuning
<nowiki>*</nowiki> Besides the octave


[[Category:Temperament]]
[[Category:Parakleismic| ]] <!-- main article -->
[[Category:Parakleismic| ]] <!-- main article -->
[[Category:Hemimean]]
[[Category:Rank-2 temperaments]]
[[Category:Hemimean clan]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Hemimage]]
[[Category:Hemimage temperaments]]
[[Category:Canousmic temperaments]]
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