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Misty family: Difference between revisions

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Move back subgroup extensions since they're still septimal misty extensions
Extensions: hazy (99 & 111), smoky (87 & 99), and ashy (99ef & 111)
 
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{{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 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 }}
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== 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 ===
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* [[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 }}
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==== 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


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* 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 }}
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* 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 }}
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Badness (Sintel): 0.637
Badness (Sintel): 0.637


=== Undecimal misty ===
=== 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.
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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* 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 }}
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* 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 }}
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* 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 }}
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* 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 }}
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Badness (Sintel): 1.21
Badness (Sintel): 1.21


==== Mystic ====
=== Hazy ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 1375/1372, 5120/5103
 
Mapping: {{mapping| 3 0 26 56 96 | 0 1 -4 -10 -18 }}
 
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
 
Comma list: 176/175, 351/350, 847/845, 1375/1372
 
Mapping: {{mapping| 3 0 26 56 96 111 | 0 1 -4 -10 -18 -21 }}
 
Optimal tunings:
* WE: ~63/50 = 399.8089{{c}}, ~3/2 = 702.3993{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.7661{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 87ef, 99, 111 }}
 
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
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:  
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}}
<!-- * 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| 12e, 99ef, 111, 210ef, 321cdeff, 531ccddeeffff }}


Badness (Sintel): 1.52
Badness (Sintel): 1.72


=== Semimist ===
=== Semimist ===
Line 183: Line 249:
* 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 264:
* 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 280:
* 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 295:
* 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 241: Line 301:


== 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
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* [[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 329:
* 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 }}
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