Misty: Difference between revisions

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'''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]].

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]].

Misty entails a mildly sharp perfect fifth. For example, a perfect fifth 3 cents sharp of [[12edo]]'s (about 1 cent sharp of just) generates a minor sixth 12 cents flat of 8\12, and subtracting 4\12 from it will produce a 5/4 in excellent tune. [[Edo]]s that provide such a fifth include [[87edo]], [[99edo]] and [[111edo]]. Such a fifth stacked six times 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 places [[7/4]] six fifths further on the generator chain and 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]].

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]]~~.

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.

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

== Interval chain ==

== Intervals ==

=== Interval chain ===

In the following table, odd harmonics 1–21 are in '''bold'''.

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<nowiki/>* In 7-limit CWE tuning, octave reduced

=== As a detemperament of 12et ===

[[File: Misty 12et Detempering.png|thumb|Misty as a 75-tone 12et detempering]]

Misty is naturally a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 75-tone detempered scale, with a generator range of -12 to +12, which covers most of the intervals in the 2.3.5.7.17.19-subgroup 21-odd-limit. Each category is divided into six or seven qualities separated by 4 generator steps, which represent the generic half-comma step. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, misty gives us more than a dozen of qualities for each diatonic category.

== Tunings ==

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 '''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]].
-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]].
+Misty entails a mildly sharp perfect fifth. For example, a perfect fifth 3 cents sharp of [[12edo]]'s (about 1 cent sharp of just) generates a minor sixth 12 cents flat of 8\12, and subtracting 4\12 from it will produce a 5/4 in excellent tune. [[Edo]]s that provide such a fifth include [[87edo]], [[99edo]] and [[111edo]]. Such a fifth stacked six times 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 places [[7/4]] six fifths further on the generator chain and 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]].
-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]].
+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.
 See [[Misty family #Misty]] and [[Misty family #Septimal misty|#Septimal misty]] for technical data. See [[Misty extensions]] for a discussion on [[11-limit|11-]] and [[13-limit]] extensions.
-== Interval chain ==
+== Intervals ==
+=== Interval chain ===
 In the following table, odd harmonics 1–21 are in '''bold'''.
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 |}
 <nowiki/>* In 7-limit CWE tuning, octave reduced
+=== As a detemperament of 12et ===
+[[File: Misty 12et Detempering.png|thumb|Misty as a 75-tone 12et detempering]]
+Misty is naturally a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 75-tone detempered scale, with a generator range of -12 to +12, which covers most of the intervals in the 2.3.5.7.17.19-subgroup 21-odd-limit. Each category is divided into six or seven qualities separated by 4 generator steps, which represent the generic half-comma step. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, misty gives us more than a dozen of qualities for each diatonic category.
 == Tunings ==