Misty: Difference between revisions

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| Odd limit 1 = 9 | Mistuning 1 = 1.96 | Complexity 1 = 39

}}

'''Misty''' is ~~the~~ [[regular temperament]] [[~~tempering out~~]] ~~the~~ [[~~misty comma~~]]~~. It equates the~~ [[~~Pythagorean comma~~]] ~~with the~~ [[~~diesis~~]], ~~and splits this interval into three equal parts, one representing the~~ [[~~schisma~~]]~[[~~diaschisma~~]]~~, and two representing~~ the [[~~syntonic comma~~]]~~. Consequently, the octave is also split into three parts of~~ [[~~512~~/~~405~~]] ~~each. This temperament~~, ~~supported by~~ [[~~12edo|12et~~]]~~, is notably in~~ the [[~~schismic–Pythagorean equivalence continuum~~]]~~, with {{nowrap|''n'' {{=}} 3}}~~.

'''Misty''' is a [[regular temperament|temperament]] with a 1/3-octave [[period]], generated by a [[3/2|perfect fifth]], and four [[4/3|perfect fourths]] [[octave reduction|octave reduced]] (i.e. a minor sixth, [[~]][[128/81]]) minus a period give the [[~]][[5/4]], [[tempering out]] the [[misty comma]].

In the 7-limit~~, the canonical~~ extension tempers out [[~~3136~~/~~3125~~]] ~~and~~ [[~~5120~~/~~5103~~]]~~. Possible tunings include~~ [[~~87edo~~]], [[~~99edo~~]] and [[~~111edo~~]].

Misty entails a mildly sharp perfect fifth such that six of them octave reduced (i.e. an augmented fourth) is close in size to [[10/7]], which gives rise to the [[7-limit]] [[extension]] where it tempers out [[5120/5103]]. This implies the [[5/4]] is split into two equal parts each for [[28/25]], tempering out [[3136/3125]], and that [[63/50]] is mapped to the 1/3-octave period, tempering out [[250047/250000]].

See [[Misty family #Misty]] and [[Misty family #Septimal misty|#Septimal misty]] for technical data.

It is easy to extend misty to the no-11 no-13 [[19-limit]], where it merges [[16/15]] with [[17/16]], [[18/17]] with [[19/18]], and [[20/19]] with [[21/20]], tempering out [[256/255]] ({{S|16}}), [[324/323]] {{S|18}}, and [[400/399]] ({{S|20}}). This lowers the overall accuracy, but supplies more harmonic resources. Tunings for misty include [[87edo]], [[99edo]] and [[111edo]].

See [[Misty family #Misty]] and [[Misty family #Septimal misty|#Septimal misty]] for technical data. See [[Misty extensions]] for a discussion on 11- and 13-limit extensions.

== Interval chain ==

@@ Line 11: / Line 11: @@
 | Odd limit 1 = 9 | Mistuning 1 = 1.96 | Complexity 1 = 39
 }}
-'''Misty''' is the [[regular temperament]] [[tempering out]] the [[misty comma]]. It equates the [[Pythagorean comma]] with the [[diesis]], and splits this interval into three equal parts, one representing the [[schisma]]~[[diaschisma]], and two representing the [[syntonic comma]]. Consequently, the octave is also split into three parts of [[512/405]] each. This temperament, supported by [[12edo|12et]], is notably in the [[schismic–Pythagorean equivalence continuum]], with {{nowrap|''n'' {{=}} 3}}.
+'''Misty''' is a [[regular temperament|temperament]] with a 1/3-octave [[period]], generated by a [[3/2|perfect fifth]], and four [[4/3|perfect fourths]] [[octave reduction|octave reduced]] (i.e. a minor sixth, [[~]][[128/81]]) minus a period give the [[~]][[5/4]], [[tempering out]] the [[misty comma]].
-In the 7-limit, the canonical extension tempers out [[3136/3125]] and [[5120/5103]]. Possible tunings include [[87edo]], [[99edo]] and [[111edo]].
+Misty entails a mildly sharp perfect fifth such that six of them octave reduced (i.e. an augmented fourth) is close in size to [[10/7]], which gives rise to the [[7-limit]] [[extension]] where it tempers out [[5120/5103]]. This implies the [[5/4]] is split into two equal parts each for [[28/25]], tempering out [[3136/3125]], and that [[63/50]] is mapped to the 1/3-octave period, tempering out [[250047/250000]].
-See [[Misty family #Misty]] and [[Misty family #Septimal misty|#Septimal misty]] for technical data.
+It is easy to extend misty to the no-11 no-13 [[19-limit]], where it merges [[16/15]] with [[17/16]], [[18/17]] with [[19/18]], and [[20/19]] with [[21/20]], tempering out [[256/255]] ({{S|16}}), [[324/323]] {{S|18}}, and [[400/399]] ({{S|20}}). This lowers the overall accuracy, but supplies more harmonic resources. Tunings for misty include [[87edo]], [[99edo]] and [[111edo]].
+See [[Misty family #Misty]] and [[Misty family #Septimal misty|#Septimal misty]] for technical data. See [[Misty extensions]] for a discussion on 11- and 13-limit extensions.
 == Interval chain ==