Parakleismic: Difference between revisions

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@@ Line 2: / Line 2: @@
 | Title = Parakleismic
 | 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
 | Mapping = 1; 13 14 35
@@ Line 11: / Line 11: @@
 | Odd limit 1 = 9 | Mistuning 1 = 1.34 | Complexity 1 = 42
 }}
-'''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 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 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]].
+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 ==
@@ Line 193: / Line 193: @@
 <nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced
-== Tuning spectrum ==
+== Tunings ==
+=== Tuning spectrum ===
 {| 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
 |
 | 314.754
-| Lower bound of 9-odd-limit diamond monotone
+| 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¢)
+| 7-odd-limit minimax (error = 1.217 ¢)
 |-
 |
 | 80/63
 | 315.183
-| 9-odd-limit minimax (error = 1.345¢)
+| 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¢)
+| 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
@@ Line 305: / Line 311: @@
 | Upper bound of 9-odd-limit diamond monotone
 |}
+<nowiki>*</nowiki> Besides the octave
 [[Category:Parakleismic| ]] <!-- main article -->