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Misty family: Difference between revisions

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Extensions: hazy (99 & 111), smoky (87 & 99), and ashy (99ef & 111)
 
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The '''misty family''' tempers out the [[misty comma]], 67108864/66430125 = {{monzo| 26 -12 -3 }}.
{{Technical data page}}
 
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 }}


Subgroup: 2.3.5
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


[[Comma list]]: 67108864/66430125
[[Comma list]]: 67108864/66430125


[[Mapping]]: [{{val| 3 0 26 }}, {{val| 0 1 -4 }}]
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}
: mapping generators: ~512/405, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~512/405 = 399.8595{{c}}, ~3/2 = 702.8645{{c}}
: [[error map]]: {{val| -0.421 +0.488 +0.261 }}
* [[CWE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.1489{{c}}
: error map: {{val| 0.000 +1.194 +1.091 }}


[[POTE generator]]: ~3/2 = 703.111
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc }}


{{Val list|legend=1| 12, 51, 63, 75, 87, 99, 285, 384 }}
[[Badness]] (Sintel): 2.50


[[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 ==
{{Main| Misty }}
{{Main| Misty }}


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


[[Comma list]]: 3136/3125, 5120/5103
[[Comma list]]: 3136/3125, 5120/5103


[[Mapping]]: [{{val| 3 0 26 56 }}, {{val| 0 1 -4 -10 }}]
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}
 
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 399.8579{{c}}, ~3/2 = 702.7714{{c}}
: [[error map]]: {{val| -0.426 +0.390 +0.611 -0.236 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.0551{{c}}
: error map: {{val| 0.000 +1.100 +1.466 +0.623 }}
 
{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}
 
[[Badness]] (Sintel): 0.931
 
==== 2.3.5.7.17-subgroup ====
Subgroup: 2.3.5.7.17
 
Comma list: 256/255, 1701/1700, 2025/2023
 
Subgroup-val mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}
: mapping generators: ~34/27, ~3
 
Optimal tunings:
* WE: ~34/27 = 399.8023{{c}}, ~3/2 = 702.6087{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 702.9837{{c}}


{{Multival|legend=1| 3 -12 -30 -26 -56 -36 }}
{{Optimal ET sequence|legend=0| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}


[[POTE generator]]: ~3/2 = 703.021
Badness (Sintel): 0.688


{{Val list|legend=1| 12, 87, 99 }}
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19


[[Badness]]: 0.036802
Comma list: 256/255, 324/323, 400/399, 476/475
 
Subgroup-val mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}
 
Optimal tunings:
* WE: ~24/19 = 399.7556{{c}}, ~3/2 = 702.4862{{c}}
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 702.9418{{c}}
 
{{Optimal ET sequence|legend=0| 12, 87, 99, 210gh, 309cdghh }}
 
Badness (Sintel): 0.637
 
=== Murky ===
Murky (87 & 99ef) tempers out [[441/440]] and [[896/891]], which joins misty with [[parapyth]]/[[pele]].  
 
This extension used to be known as ''undecimal misty'', but was decanonicalized in 2026 in favor of the no-11 no-13 temperament.


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 3136/3125, 4375/4356
Comma list: 441/440, 896/891, 3136/3125


Mapping: [{{val| 3 0 26 56 77 }}, {{val| 0 1 -4 -10 -14 }}]
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


POTE generator: ~3/2 = 703.210
Optimal tunings:  
* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.2385{{c}}


Vals: {{Val list| 12, 87, 99e, 186e }}
{{Optimal ET sequence|legend=0| 12, …, 87, 99e, 186e }}


Badness: 0.035932
Badness (Sintel): 1.19


==== 13-limit ====
==== 13-limit ====
Line 55: Line 108:
Comma list: 196/195, 352/351, 364/363, 625/624
Comma list: 196/195, 352/351, 364/363, 625/624


Mapping: [{{val| 3 0 26 56 77 92 }}, {{val| 0 1 -4 -10 -14 -17 }}]
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}


POTE generator: ~3/2 = 703.303
Optimal tunings:  
* WE: ~44/35 = 399.8888{{c}}, ~3/2 = 703.1080{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3258{{c}}


Vals: {{Val list| 12f, 87, 99ef, 186ef, 273ef }}
{{Optimal ET sequence|legend=0| 12f, …, 87, 186ef }}


Badness: 0.027820
Badness (Sintel): 1.15


===== 17-limit =====
===== 17-limit =====
Line 68: Line 123:
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594


Mapping: [{{val| 3 0 26 56 77 92 -2 }}, {{val| 0 1 -4 -10 -14 -17 3 }}]
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}


POTE generator: ~3/2 = 703.233
Optimal tunings:  
* WE: ~34/27 = 399.8046{{c}}, ~3/2 = 702.8894{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.2597{{c}}


Vals: {{Val list| 12f, 87, 99ef, 186efg }}
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efg }}


Badness: 0.023908
Badness (Sintel): 1.22


===== 19-limit =====
===== 19-limit =====
Line 81: Line 138:
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594


Mapping: [{{val| 3 0 26 56 77 92 -2 8 }}, {{val| 0 1 -4 -10 -14 -17 3 1 }}]
Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 0 1 -4 -10 -14 -17 3 1 }}
 
Optimal tunings:
* WE: ~24/19 = 399.7530{{c}}, ~3/2 = 702.7601{{c}}
* CWE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.2216{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efgh }}
 
Badness (Sintel): 1.21


POTE generator: ~3/2 = 703.253
=== Hazy ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 12f, 87, 99ef, 186efgh }}
Comma list: 176/175, 1375/1372, 5120/5103


Badness: 0.019878
Mapping: {{mapping| 3 0 26 56 96 | 0 1 -4 -10 -18 }}


==== Mystic ====
Optimal tunings:
* WE: ~63/50 = 399.8097{{c}}, ~3/2 = 702.4039{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.7731{{c}}
 
{{Optimal ET sequence|legend=0| 12, 87e, 99, 111 }}
 
Badness (Sintel): 1.84
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 441/440, 896/891, 1001/1000
Comma list: 176/175, 351/350, 847/845, 1375/1372


Mapping: [{{val| 3 0 26 56 77 -46 }}, {{val| 0 1 -4 -10 -14 12 }}]
Mapping: {{mapping| 3 0 26 56 96 111 | 0 1 -4 -10 -18 -21 }}


POTE generator: ~3/2 = 703.311
Optimal tunings:  
* WE: ~63/50 = 399.8089{{c}}, ~3/2 = 702.3993{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.7661{{c}}


Vals: {{Val list| 12, 87, 186e, 273e }}
{{Optimal ET sequence|legend=0| 12f, 87ef, 99, 111 }}


Badness: 0.036872
Badness (Sintel): 1.54
 
=== Smoky ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 2200/2187, 3136/3125
 
Mapping: {{mapping| 3 0 26 56 -61 | 0 1 -4 -10 15 }}
 
Optimal tunings:
* WE: ~63/50 = 400.0072{{c}}, ~3/2 = 702.3068{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 703.2941{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 87, 186, 273 }}
 
Badness (Sintel): 2.25
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 385/384, 3136/3125
 
Mapping: {{mapping| 3 0 26 56 -61 -46 | 0 1 -4 -10 15 12 }}
 
Optimal tunings:
* WE: ~63/50 = 400.0184{{c}}, ~3/2 = 703.3420{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 703.3101{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 87, 186, 273 }}
 
Badness (Sintel): 1.63
 
=== Ashy ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 3136/3125, 5120/5103
 
Mapping: {{mapping| 3 0 26 56 -80 | 0 1 -4 -10 19 }}
 
Optimal tunings:
* WE: ~63/50 = 399.7776{{c}}, ~3/2 = 702.4764{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.8652{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 99e, 210e }}
 
Badness (Sintel): 1.89
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 729/728, 3136/3125
 
Mapping: {{mapping| 3 0 26 56 -80 -65 | 0 1 -4 -10 19 16 }}
 
Optimal tunings:
* WE: ~63/50 = 399.7470{{c}}, ~3/2 = 702.3782{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.8135{{c}}
 
{{Optimal ET sequence|legend=0| 12e, 99ef, 111, 210ef, 321cdeff, 531ccddeeffff }}
 
Badness (Sintel): 1.72
 
=== Semimist ===
Subgroup: 2.3.5.7.11
 
Comma list: 3136/3125, 5120/5103, 9801/9800
 
Mapping: {{mapping| 6 0 52 112 173 | 0 1 -4 -10 -16 }}
: mapping generators: ~55/49, ~3
 
Optimal tunings:
* WE: ~55/49 = 199.9260{{c}}, ~3/2 = 702.7162{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0062{{c}}
 
{{Optimal ET sequence|legend=0| 12, 174e, 186, 198, 408cde, 606bcde }}
 
Badness (Sintel): 2.26
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 1716/1715, 3136/3125
 
Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9293{{c}}, ~3/2 = 702.7602{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0343{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 174ef, 186, 198 }}
 
Badness (Sintel): 1.82


=== Hemimist ===
=== Hemimist ===
Line 107: Line 274:
Comma list: 3025/3024, 3136/3125, 5120/5103
Comma list: 3025/3024, 3136/3125, 5120/5103


Mapping: [{{val| 3 0 26 56 8 }}, {{val| 0 2 -8 -20 1 }}]
Mapping: {{mapping| 3 0 26 56 8 | 0 2 -8 -20 1 }}
: mapping generators: ~63/50, ~400/231


POTE generator: ~11/8 = 551.527
Optimal tunings:  
* WE: ~63/50 = 399.8922{{c}}, ~400/231 = 951.2704{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~400/231 = 951.5363{{c}}


Vals: {{Val list| 24d, 87, 111, 198 }}
{{Optimal ET sequence|legend=0| 24d, 87, 111, 198 }}


Badness: 0.047439
Badness (Sintel): 1.57


==== 13-limit ====
==== 13-limit ====
Line 120: Line 290:
Comma list: 352/351, 676/675, 847/845, 3025/3024
Comma list: 352/351, 676/675, 847/845, 3025/3024


Mapping: [{{val| 3 0 26 56 8 23 }}, {{val| 0 2 -8 -20 1 -5 }}]
Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}


POTE generator: ~11/8 = 551.525
Optimal tunings:  
* WE: ~63/50 = 399.8818{{c}}, ~26/15 = 951.2442{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.5358{{c}}


Vals: {{Val list| 24d, 87, 111, 198 }}
{{Optimal ET sequence|legend=0| 24d, 87, 111, 198 }}


Badness: 0.022923
Badness (Sintel): 0.947


== Fog ==
== Fog ==
The ''fog'' temperament (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


[[Comma list]]: 225/224, 156250/151263
[[Comma list]]: 225/224, 156250/151263


[[Mapping]]: [{{val|3 5 6 7}}, {{val|0 -1 4 6}}]
{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}
 
{{Multival|legend=1| 3 -12 -18 -26 -37 -8 }}


[[POTE generator]]: ~3/2 = 704.118
[[Optimal tuning]]s:  
* [[WE]]: ~80/63 = 399.7918{{c}}, ~3/2 = 703.7511{{c}}
: [[error map]]: {{val| -0.624 +1.172 -4.232 +4.712 }}
* [[CWE]]: ~80/63 = 400.0000{{c}}, ~3/2 = 704.1865{{c}}
: error map: {{val| 0.000 +2.232 -3.060 +6.055 }}


{{Val list|legend=1| 12, 51, 63, 75 }}
{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}


[[Badness]]: 0.086866
[[Badness]] (Sintel): 2.20


=== 11-limit ===
=== 11-limit ===
Line 150: Line 324:
Comma list: 100/99, 225/224, 2420/2401
Comma list: 100/99, 225/224, 2420/2401


Mapping: [{{val|3 5 6 7 8}}, {{val|0 -1 4 6 10}}]
Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}


POTE generator: ~3/2 = 704.411
Optimal tunings:  
* WE: ~44/35 = 399.8078{{c}}, ~3/2 = 704.0728{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.4697{{c}}


Vals: {{Val list| 12, 51, 63, 138de }}
{{Optimal ET sequence|legend=0| 12, 39ce, 51, 63, 138de }}


Badness: 0.051348
Badness (Sintel): 1.70


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Misty family| ]] <!-- main article -->
[[Category:Misty family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Misty_family"