Misty family - Revision history

FloraC: Extensions: hazy (99 & 111), smoky (87 & 99), and ashy (99ef & 111)

2026-02-12T06:22:21Z

Extensions: hazy (99 & 111), smoky (87 & 99), and ashy (99ef & 111)

@@ Line 84: / Line 84: @@
 Badness (Sintel): 0.637
-=== Undecimal misty ===
+=== Murky ===
 Subgroup: 2.3.5.7.11
@@ Line 144: / Line 148: @@
 Badness (Sintel): 1.21
-==== Mystic ====
+=== Hazy ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 441/440, 896/891, 1001/1000
+Comma list: 352/351, 540/539, 729/728, 3136/3125
-Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}
+Mapping: {{mapping| 3 0 26 56 -80 -65 | 0 1 -4 -10 19 16 }}
 Optimal tunings:
-* WE: ~44/35 = 399.9543{{c}}, ~3/2 = 702.2310{{c}}
+* WE: ~63/50 = 399.7470{{c}}, ~3/2 = 702.3782{{c}}
-* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3140{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.8135{{c}}
-{{Optimal ET sequence|legend=0| 12, 87, 186e, 273e }}
+{{Optimal ET sequence|legend=0| 12e, 99ef, 111, 210ef, 321cdeff, 531ccddeeffff }}
-Badness (Sintel): 1.52
+Badness (Sintel): 1.72
 === Semimist ===

FloraC: Expand

2026-01-04T16:24:14Z

Expand

← Older revision		Revision as of 16:24, 4 January 2026
Line 5:		Line 5:
	{{Main\| Misty }}		{{Main\| Misty }}

	Misty features a third-octave period, which represents the [[512/405\|ptolemaic diminished fourth (512/405)]]. Four [[4/3\|perfect fourths]] (i.e. a minor sixth, ~[[128/81]]) minus a period give the interval class of 5. Misty has a [[ploidacot]] of triploid monocot, and equates the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap\| ''n'' {{=}} 3 }}.		Misty features a third-octave period, which represents the [[512/405\|ptolemaic diminished fourth (512/405)]]. Four [[4/3\|perfect fourths]] (i.e. a minor sixth, [[~]][[128/81]]) minus a period give the interval class of 5. Misty has a [[ploidacot]] of triploid monocot, and equates the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap\| ''n'' {{=}} 3 }}.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
Line 31:		Line 31:
	== Septimal misty ==		== Septimal misty ==
	{{Main\| Misty }}		{{Main\| Misty }}

			Septimal misty tempers out not only [[3136/3125]] and [[5120/5103]], but [[250047/250000]], mapping [[~]][[63/50]] to the 1/3-octave period. It can be described as {{nowrap\| 12 & 99 }}, and finds [[7/4]] by [[5/4]] plus six extra fourths octave reduced. [[99edo]] makes for an excellent tuning, and [[87edo]] and [[111edo]] are notable alternatives.

			It extends best to the no-11 no-13 [[19-limit]], where the ~[[16/15]] stands in for ~[[17/16]], and the ~[[25/21]], a fifth minus a period, stands in for ~[[19/16]]. The [[S-expression]]-based comma list for the 2.3.5.7.17-subgroup extension is {([[5120/5103\|S8/S9]]), [[2025/2023\|S15/S17]], [[256/255\|S16]], [[1701/1700\|S18/S20]], ([[2500/2499\|S50]])}, and for the 2.3.5.7.17.19-subgroup extension, {([[5120/5103\|S8/S9]]), [[256/255\|S16]], [[324/323\|S18]], [[400/399\|S20]], [[1445/1444\|S17/S19]], ([[2500/2499\|S50]])}.

	=== 7-limit ===		=== 7-limit ===
Line 50:		Line 54:

	==== 2.3.5.7.17-subgroup ====		==== 2.3.5.7.17-subgroup ====
	The [[S-expression]]-based comma list for this extension is {([[5120/5103\|S8/S9]]), [[2025/2023\|S15/S17]], [[256/255\|S16]], [[1701/1700\|S18/S20]], ([[2500/2499\|S50]])}, and for the 2.3.5.7.17.19-subgroup extension, {([[5120/5103\|S8/S9]]), [[256/255\|S16]], [[324/323\|S18]], [[400/399\|S20]], [[1445/1444\|S17/S19]], ([[2500/2499\|S50]])}.

	Subgroup: 2.3.5.7.17		Subgroup: 2.3.5.7.17

Line 220:		Line 222:

	== Fog ==		== Fog ==
	~~The~~ fog temperament ~~({{nowrap\| 12 & 63 }})~~ tempers out the marvel comma, [[225/224]].		Named by [[Xenllium]] in 2021, the fog temperament tempers out the marvel comma, [[225/224]], and may be described as {{nowrap\| 12 & 63 }}.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: - CTE & POTE tunings

2026-01-04T15:55:02Z

- CTE & POTE tunings

← Older revision		Revision as of 15:55, 4 January 2026
Line 19:		Line 19:
	* [[CWE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.1489{{c}}		* [[CWE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.1489{{c}}
	: error map: {{val\| 0.000 +1.194 +1.091 }}		: error map: {{val\| 0.000 +1.194 +1.091 }}
	~~<!-- * [[CTE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.2481{{c}}~~
	* [[POTE]]: ~512/405 = 400.000{{c}}, ~3/2 = 703.111{{c}} -->

	{{Optimal ET sequence\|legend=1\| 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc }}		{{Optimal ET sequence\|legend=1\| 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc }}
Line 46:		Line 44:
	* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.0551{{c}}		* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.0551{{c}}
	: error map: {{val\| 0.000 +1.100 +1.466 +0.623 }}		: error map: {{val\| 0.000 +1.100 +1.466 +0.623 }}
	~~<!-- * [[CTE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.1448{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 12, …, 87, 99 }}		{{Optimal ET sequence\|legend=1\| 12, …, 87, 99 }}
Line 65:		Line 62:
	* WE: ~34/27 = 399.8023{{c}}, ~3/2 = 702.6087{{c}}		* WE: ~34/27 = 399.8023{{c}}, ~3/2 = 702.6087{{c}}
	* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 702.9837{{c}}		* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 702.9837{{c}}
	~~<!-- * CTE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.0963{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}		{{Optimal ET sequence\|legend=0\| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
Line 81:		Line 77:
	* WE: ~24/19 = 399.7556{{c}}, ~3/2 = 702.4862{{c}}		* WE: ~24/19 = 399.7556{{c}}, ~3/2 = 702.4862{{c}}
	* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 702.9418{{c}}		* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 702.9418{{c}}
	~~<!-- * CTE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.0778{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 12, 87, 99, 210gh, 309cdghh }}		{{Optimal ET sequence\|legend=0\| 12, 87, 99, 210gh, 309cdghh }}
Line 97:		Line 92:
	* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}		* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}
	* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.2385{{c}}		* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.2385{{c}}
	~~<!-- * CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3127{{c}}~~
	* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.210{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12, …, 87, 99e, 186e }}		{{Optimal ET sequence\|legend=0\| 12, …, 87, 99e, 186e }}
Line 114:		Line 107:
	* WE: ~44/35 = 399.8888{{c}}, ~3/2 = 703.1080{{c}}		* WE: ~44/35 = 399.8888{{c}}, ~3/2 = 703.1080{{c}}
	* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3258{{c}}		* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3258{{c}}
	~~<!-- * CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3859{{c}}~~
	* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.303{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12f, …, 87, 186ef }}		{{Optimal ET sequence\|legend=0\| 12f, …, 87, 186ef }}
Line 131:		Line 122:
	* WE: ~34/27 = 399.8046{{c}}, ~3/2 = 702.8894{{c}}		* WE: ~34/27 = 399.8046{{c}}, ~3/2 = 702.8894{{c}}
	* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.2597{{c}}		* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.2597{{c}}
	~~<!-- * CTE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.3686{{c}}~~
	* POTE: ~34/27 = 400.000{{c}}, ~3/2 = 703.233{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12f, 87, 99ef, 186efg }}		{{Optimal ET sequence\|legend=0\| 12f, 87, 99ef, 186efg }}
Line 148:		Line 137:
	* WE: ~24/19 = 399.7530{{c}}, ~3/2 = 702.7601{{c}}		* WE: ~24/19 = 399.7530{{c}}, ~3/2 = 702.7601{{c}}
	* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.2216{{c}}		* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.2216{{c}}
	~~<!-- * CTE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.3626{{c}}~~
	* POTE: ~24/19 = 400.000{{c}}, ~3/2 = 703.253{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12f, 87, 99ef, 186efgh }}		{{Optimal ET sequence\|legend=0\| 12f, 87, 99ef, 186efgh }}
Line 165:		Line 152:
	* WE: ~44/35 = 399.9543{{c}}, ~3/2 = 702.2310{{c}}		* WE: ~44/35 = 399.9543{{c}}, ~3/2 = 702.2310{{c}}
	* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3140{{c}}		* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3140{{c}}
	~~<!-- * CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3285{{c}}~~
	* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.311{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12, 87, 186e, 273e }}		{{Optimal ET sequence\|legend=0\| 12, 87, 186e, 273e }}
Line 183:		Line 168:
	* WE: ~55/49 = 199.9260{{c}}, ~3/2 = 702.7162{{c}}		* WE: ~55/49 = 199.9260{{c}}, ~3/2 = 702.7162{{c}}
	* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0062{{c}}		* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0062{{c}}
	~~<!-- * CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0864{{c}}~~
	* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.9764{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12, 174e, 186, 198, 408cde, 606bcde }}		{{Optimal ET sequence\|legend=0\| 12, 174e, 186, 198, 408cde, 606bcde }}
Line 200:		Line 183:
	* WE: ~55/49 = 199.9293{{c}}, ~3/2 = 702.7602{{c}}		* WE: ~55/49 = 199.9293{{c}}, ~3/2 = 702.7602{{c}}
	* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0343{{c}}		* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0343{{c}}
	~~<!-- * CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.1045{{c}}~~
	* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0087{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12f, 174ef, 186, 198 }}		{{Optimal ET sequence\|legend=0\| 12f, 174ef, 186, 198 }}
Line 218:		Line 199:
	* WE: ~63/50 = 399.8922{{c}}, ~400/231 = 951.2704{{c}}		* WE: ~63/50 = 399.8922{{c}}, ~400/231 = 951.2704{{c}}
	* CWE: ~63/50 = 400.0000{{c}}, ~400/231 = 951.5363{{c}}		* CWE: ~63/50 = 400.0000{{c}}, ~400/231 = 951.5363{{c}}
	~~<!-- * POTE: ~63/50 = 400.0000{{c}}, ~400/231 = 951.5721{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 24d, 87, 111, 198 }}		{{Optimal ET sequence\|legend=0\| 24d, 87, 111, 198 }}
Line 234:		Line 214:
	* WE: ~63/50 = 399.8818{{c}}, ~26/15 = 951.2442{{c}}		* WE: ~63/50 = 399.8818{{c}}, ~26/15 = 951.2442{{c}}
	* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.5358{{c}}		* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.5358{{c}}
	~~<!-- * POTE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.5810{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 24d, 87, 111, 198 }}		{{Optimal ET sequence\|legend=0\| 24d, 87, 111, 198 }}
Line 254:		Line 233:
	* [[CWE]]: ~80/63 = 400.0000{{c}}, ~3/2 = 704.1865{{c}}		* [[CWE]]: ~80/63 = 400.0000{{c}}, ~3/2 = 704.1865{{c}}
	: error map: {{val\| 0.000 +2.232 -3.060 +6.055 }}		: error map: {{val\| 0.000 +2.232 -3.060 +6.055 }}
	~~<!-- * CTE: ~80/63 = 400.0000{{c}}, ~3/2 = 704.3694{{c}}~~
	* POTE: ~80/63 = 400.000{{c}}, ~3/2 = 704.118{{c}} -->

	{{Optimal ET sequence\|legend=1\| 12, 39c, 51, 63, 75 }}		{{Optimal ET sequence\|legend=1\| 12, 39c, 51, 63, 75 }}
Line 271:		Line 248:
	* WE: ~44/35 = 399.8078{{c}}, ~3/2 = 704.0728{{c}}		* WE: ~44/35 = 399.8078{{c}}, ~3/2 = 704.0728{{c}}
	* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.4697{{c}}		* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.4697{{c}}
	~~<!-- * CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.6253{{c}}~~
	* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 704.411{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12, 39ce, 51, 63, 138de }}		{{Optimal ET sequence\|legend=0\| 12, 39ce, 51, 63, 138de }}

FloraC: Move back subgroup extensions since they're still septimal misty extensions

2026-01-04T15:53:53Z

Move back subgroup extensions since they're still septimal misty extensions

@@ Line 34: / Line 34: @@
 {{Main| Misty }}
 [[Subgroup]]: 2.3.5.7
@@ Line 50: / Line 51: @@
 [[Badness]] (Sintel): 0.931
 === Undecimal misty ===
@@ Line 241: / Line 277: @@
 Badness (Sintel): 1.70
 [[Category:Temperament families]]
 [[Category:Misty family| ]] <!-- main article -->
 [[Category:Rank 2]]

FloraC: Switch to Sintel's badness, WE & CWE tunings

2026-01-04T15:52:15Z

Switch to Sintel's badness, WE & CWE tunings

Show changes

FloraC: Expand

2026-01-04T15:19:37Z

Expand

← Older revision		Revision as of 15:19, 4 January 2026
Line 5:		Line 5:
	{{Main\| Misty }}		{{Main\| Misty }}

	Misty features a third-octave period, which ~~is represented by~~ [[512/405]] ~~and acts as the interval separating~~ ~[[128/81]] ~~from ~~~[[~~5/4~~]]~~. It also~~ equates the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap\| ''n'' {{=}} 3 }}.		Misty features a third-octave period, which represents the [[512/405\|ptolemaic diminished fourth (512/405)]]. Four [[4/3\|perfect fourths]] (i.e. a minor sixth, ~[[128/81]]) minus a period give the interval class of 5. Misty has a [[ploidacot]] of triploid monocot, and equates the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap\| ''n'' {{=}} 3 }}.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5

FloraC: Units & misc. cleanup

2026-01-03T13:40:46Z

Units & misc. cleanup

Show changes

FloraC: Move the large chunk of subgroup extensions to the bottom

2026-01-03T13:33:34Z

Move the large chunk of subgroup extensions to the bottom

@@ Line 44: / Line 44: @@
 [[Badness]]: 0.036802
 === Undecimal misty ===
@@ Line 247: / Line 215: @@
 Badness: 0.051348
 [[Category:Temperament families]]
 [[Category:Misty family| ]] <!-- main article -->
 [[Category:Rank 2]]

FloraC: Rework on intro and head

2026-01-03T13:25:30Z

Rework on intro and head

← Older revision		Revision as of 13:25, 3 January 2026
Line 1:		Line 1:
	{{Technical data page}}		{{Technical data page}}
	The '''misty family''' tempers out the [[misty comma]]~~, 67108864/66430125 =~~ {{monzo\| 26 -12 -3 }}, ~~equating the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the~~ [[~~schismic–Pythagorean equivalence continuum~~]]~~. It features a third-octave period, which is represented by [[512~~/~~405]] and acts as the interval separating ~[[128/81]] from ~[[5/4]].~~		The '''misty family''' [[tempering out\|tempers out]] the [[misty comma]] ({{monzo\|legend=1\| 26 -12 -3 }}, [[ratio]]: 67108864/66430125).

	~~The second comma of the [[Normal lists #Normal interval list\|normal comma list]] defines which 7-limit family member we are looking at.~~
	* Adding 3136/3125 (hemimean) leads to septimal misty.
	* Adding 225/224 (marvel) ~~leads to fog~~.

	== Misty ==		== Misty ==
	{{Main\| Misty }}		{{Main\| Misty }}

			Misty features a third-octave period, which is represented by [[512/405]] and acts as the interval separating ~[[128/81]] from ~[[5/4]]. It also equates the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap\| ''n'' {{=}} 3 }}.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
Line 24:		Line 22:

	[[Badness]]: 0.106540		[[Badness]]: 0.106540

			=== Overview to extensions ===
			The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which 7-limit family member we are looking at.
			* Adding 3136/3125 (hemimean) leads to septimal misty.
			* Adding 225/224 (marvel) leads to fog.

	== Septimal misty ==		== Septimal misty ==

FloraC: Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

2025-11-27T12:16:57Z

Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

← Older revision		Revision as of 12:16, 27 November 2025
Line 246:		Line 246:

	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Misty family\| ]] <!-- main article -->		[[Category:Misty family\| ]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]