Misty - Revision history

FloraC: Text replacement - "Category:Hemifamity temperaments" to "Category:Aberschismic temperaments"

2026-06-06T12:39:48Z

Text replacement - "Category:Hemifamity temperaments" to "Category:Aberschismic temperaments"

← Older revision		Revision as of 12:39, 6 June 2026
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	[[Category:Misty family]]		[[Category:Misty family]]
	[[Category:Hemimean clan]]		[[Category:Hemimean clan]]
	[[Category:~~Hemifamity~~ temperaments]]		[[Category:Aberschismic temperaments]]

FloraC: + as a detemp

2026-03-29T12:16:20Z

+ as a detemp

← Older revision		Revision as of 12:16, 29 March 2026
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	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.		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'''.		In the following table, odd harmonics 1–21 are in '''bold'''.

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	\|}		\|}
	<nowiki/>* In 7-limit CWE tuning, octave reduced		<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 ==		== Tunings ==

FloraC: Expand with a tuning example

2026-03-11T09:27:19Z

Expand with a tuning example

← Older revision		Revision as of 09:27, 11 March 2026
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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''' 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.		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.

FloraC: /* Tunings */ + more edo tunings

2026-03-11T08:40:53Z

Tunings: + more edo tunings

← Older revision		Revision as of 08:40, 11 March 2026
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	\| 702.778		\| 702.778
	\|		\|
			\|-
			\| 123\210
			\|
			\| 702.857
			\| 210gh val
	\|-		\|-
	\|		\|
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	\| 703.128		\| 703.128
	\| 5-odd-limit minimax (error = 1.173{{c}})		\| 5-odd-limit minimax (error = 1.173{{c}})
			\|-
			\| 109\186
			\|
			\| 703.226
			\| 186gh val
	\|-		\|-
	\|		\|

FloraC: /* Tunings */ fix tuning

2026-03-10T17:50:37Z

Tunings: fix tuning

← Older revision		Revision as of 17:50, 10 March 2026
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	! Tenney		! Tenney
	\| CTE: ~3/2 = 703.0778{{c}}		\| CTE: ~3/2 = 703.0778{{c}}
	\| CWE: ~3/2 = ~~703~~.9418{{c}}		\| CWE: ~3/2 = 702.9418{{c}}
	\| POTE: ~3/2 = 702.9156{{c}}		\| POTE: ~3/2 = 702.9156{{c}}
	\|}		\|}

FloraC: + mos scales

2026-02-16T09:43:32Z

+ mos scales

← Older revision		Revision as of 09:43, 16 February 2026
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	\| Generators tuning = 703.1		\| Generators tuning = 703.1
	\| Optimization method = CWE		\| Optimization method = CWE
	\| MOS scales = [[3L 9s]], [[12L 3s]]		\| MOS scales = [[3L 9s]], [[12L 3s]], [[12L 15s]], [[12L 27s]]
	\| Odd limit 1 = 9 \| Mistuning 1 = 1.96 \| Complexity 1 = 39		\| Odd limit 1 = 9 \| Mistuning 1 = 1.96 \| Complexity 1 = 39
	}}		}}

FloraC: Fix formatting. + links

2026-02-12T08:00:13Z

Fix formatting. + links

← Older revision		Revision as of 08:00, 12 February 2026
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	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 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]].

	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. 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.		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 ==		== Interval chain ==

FloraC: /* Tunings */ + norm-based tunings

2026-02-12T07:59:21Z

Tunings: + norm-based tunings

← Older revision		Revision as of 07:59, 12 February 2026
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	== Tunings ==		== Tunings ==
	* 7-limit POTE ~~tuning~~: ~3/2 = 703.0212		=== Norm-based tunings ===
	* 7-~~limit~~ CTE ~~tuning~~: ~3/2 = 703.~~1448~~		{\| class="wikitable mw-collapsible mw-collapsed"
			\|+ style="font-size: 105%; white-space: nowrap;" \| 5-limit norm-based tunings
			\|-
			! rowspan="2" \|
			! colspan="3" \| Euclidean
			\|-
			! Constrained
			! Constrained & skewed
			! Destretched
			\|-
			! Tenney
			\| CTE: ~3/2 = 703.2481{{c}}
			\| CWE: ~3/2 = 703.1489{{c}}
			\| POTE: ~3/2 = 703.1114{{c}}
			\|}
			{\| class="wikitable mw-collapsible mw-collapsed"
			\|+ style="font-size: 105%; white-space: nowrap;" \| 7-limit norm-based tunings
			\|-
			! rowspan="2" \|
			! colspan="3" \| Euclidean
			\|-
			! Constrained
			! Constrained & skewed
			! Destretched
			\|-
			! Tenney
			\| CTE: ~3/2 = 703.1448{{c}}
			\| CWE: ~3/2 = 703.0551{{c}}
			\| POTE: ~3/2 = 703.0212{{c}}
			\|}
			{\| class="wikitable mw-collapsible mw-collapsed"
			\|+ style="font-size: 105%; white-space: nowrap;" \| 2.3.5.7.17.19-subgroup norm-based tunings
			\|-
			! rowspan="2" \|
			! colspan="3" \| Euclidean
			\|-
			! Constrained
			! Constrained & skewed
			! Destretched
			\|-
			! Tenney
			\| CTE: ~3/2 = 703.0778{{c}}
			\| CWE: ~3/2 = 703.9418{{c}}
			\| POTE: ~3/2 = 702.9156{{c}}
			\|}

	=== Tuning spectrum ===		=== Tuning spectrum ===
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	\|-		\|-
	! Edo<br>Generator		! Edo<br>Generator
	! [[Eigenmonzo\|Eigenmonzo<br>(~~rnchanged~~-interval)]]		! [[Eigenmonzo\|Eigenmonzo<br>(Unchanged-interval)]]
	! Generator~~<br>~~(¢)		! Generator (¢)
	! Comments		! Comments
	\|-		\|-
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	\| 9/5		\| 9/5
	\| 702.933		\| 702.933
	\| 9-odd-limit minimax (error = 1.~~955¢~~)		\| 9-odd-limit minimax (error = 1.955{{c}})
	\|-		\|-
	\|		\|
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	\| 7/4		\| 7/4
	\| 703.117		\| 703.117
	\| 7-odd-limit minimax (error = 1.~~217¢~~)		\| 7-odd-limit minimax (error = 1.217{{c}})
	\|-		\|-
	\|		\|
	\| 5/3		\| 5/3
	\| 703.128		\| 703.128
	\| 5-odd-limit minimax (error = 1.~~173¢~~)		\| 5-odd-limit minimax (error = 1.173{{c}})
	\|-		\|-
	\|		\|
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	\|		\|
	\| 704.000		\| 704.000
	\|		\| 75d val
	\|-		\|-
	\| 37\63		\| 37\63
	\|		\|
	\| 704.762		\| 704.762
	\| ~~Upper~~ bound of 9-odd-limit diamond monotone		\| 63d val, upper bound of 9-odd-limit diamond monotone
	\|}		\|}

FloraC: Rework intro

2026-02-12T07:52:59Z

Rework intro

← Older revision		Revision as of 07:52, 12 February 2026
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	\| Odd limit 1 = 9 \| Mistuning 1 = 1.96 \| Complexity 1 = 39		\| 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 ==		== Interval chain ==

FloraC: + infobox

2026-02-12T07:21:58Z

+ infobox

← Older revision		Revision as of 07:21, 12 February 2026
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			{{Infobox regtemp
			\| Title = Misty
			\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.17.19
			\| Comma basis = [[67108864/66430125]] (5-limit); <br>[[3136/3125]], [[5120/5103]] (7-limit); <br>[[256/255]], [[324/323]], [[400/399]], [[476/475]]<br>(2.3.5.7.17.19)
			\| Edo join 1 = 12 \| Edo join 2 = 99
			\| Mapping = 3; 1 -4 -10 3 1
			\| Generators = 3/2
			\| Generators tuning = 703.1
			\| Optimization method = CWE
			\| MOS scales = [[3L 9s]], [[12L 3s]]
			\| 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 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}}.

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

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

	== Interval chain ==		== Interval chain ==
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	\| '''8/5'''		\| '''8/5'''
	\| 12.2		\| 12.2
	\| 126/125, 225/224~~, …~~		\| 126/125, 225/224
	\|-		\|-
	\| 5		\| 5