Parakleismic: Difference between revisions

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| 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 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 ==