Misty family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit

FloraC (talk | contribs)

4 user groups
27,047 edits
VisualWikitext
Move the large chunk of subgroup extensions to the bottom
Inherits 5-limit
Tag: Undo
 
(13 intermediate revisions by 2 users not shown)
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
Line 12: Line 12:


{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}
: mapping generators: ~512/405, ~3
: mapping generators: ~512/405, ~3


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: ~512/405 = 1\3, ~3/2 = 703.2481
* [[WE]]: ~512/405 = 399.8595{{c}}, ~3/2 = 702.8645{{c}}
* [[POTE]]: ~512/405 = 1\3, ~3/2 = 703.111
: [[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 }}


{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc }}


[[Badness]]: 0.106540
[[Badness]] (Sintel): 2.50


=== Overview to extensions ===
=== Overview to extensions ===
Line 31: Line 32:
{{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 ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 37: Line 43:
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


: mapping generators: ~63/50, ~3
[[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
 
{{Mapping|legend=2| 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}}
 
{{Optimal ET sequence|legend=0| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
 
Badness (Sintel): 0.688
 
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19
 
Comma list: 256/255, 324/323, 400/399, 476/475
 
{{Mapping|legend=2| 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 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448
Badness (Sintel): 0.637


{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}
=== Murky ===
Murky (87 & 99ef) tempers out [[441/440]] and [[896/891]], which joins misty with [[parapyth]]/[[pele]].


[[Badness]]: 0.036802
This extension used to be known as ''undecimal misty'', but was decanonicalized in 2026 in favor of the no-11 no-13 temperament.  


=== Undecimal misty ===
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: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}
{{Mapping|legend=0| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~44/35 = 1\3, ~3/2 = 703.3127
* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 703.210
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.2385{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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 65: Line 108:
Comma list: 196/195, 352/351, 364/363, 625/624
Comma list: 196/195, 352/351, 364/363, 625/624


Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}
{{Mapping|legend=0| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
* WE: ~44/35 = 399.8888{{c}}, ~3/2 = 703.1080{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 703.303
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3258{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}
{{Optimal ET sequence|legend=0| 12f, …, 87, 186ef }}


Badness: 0.027820
Badness (Sintel): 1.15


===== 17-limit =====
===== 17-limit =====
Line 80: 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: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}
{{Mapping|legend=0| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~34/27 = 1\3, ~3/2 = 703.3686
* WE: ~34/27 = 399.8046{{c}}, ~3/2 = 702.8894{{c}}
* POTE: ~34/27 = 1\3, ~3/2 = 703.233
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.2597{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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 95: 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: {{mapping| 3 0 26 56 77 92 -2 8 | 0 1 -4 -10 -14 -17 3 1 }}
{{Mapping|legend=0| 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 }}


Optimal tunings:
Badness (Sintel): 1.21
* CTE: ~24/19 = 1\3, ~3/2 = 703.3626
* POTE: ~34/27 = 1\3, ~3/2 = 703.253


Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}
=== Hazy ===
Subgroup: 2.3.5.7.11


Badness: 0.019878
Comma list: 176/175, 1375/1372, 5120/5103


==== Mystic ====
{{Mapping|legend=0| 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
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|legend=0| 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


Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}
Comma list: 385/384, 2200/2187, 3136/3125


Optimal tunings:
{{Mapping|legend=0| 3 0 26 56 -61 | 0 1 -4 -10 15 }}
* CTE: ~44/35 = 1\3, ~3/2 = 703.3285
* POTE: ~44/35 = 1\3, ~3/2 = 703.311


Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}
Optimal tunings:  
* WE: ~63/50 = 400.0072{{c}}, ~3/2 = 702.3068{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 703.2941{{c}}


Badness: 0.036872
{{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|legend=0| 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|legend=0| 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|legend=0| 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 ===
=== Semimist ===
Line 125: Line 243:
Comma list: 3136/3125, 5120/5103, 9801/9800
Comma list: 3136/3125, 5120/5103, 9801/9800


Mapping: {{mapping| 6 0 52 112 173 | 0 1 -4 -10 -16 }}
{{Mapping|legend=0| 6 0 52 112 173 | 0 1 -4 -10 -16 }}
 
: mapping generators: ~55/49, ~3
: mapping generators: ~55/49, ~3


Optimal tunings:
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 703.0864
* WE: ~55/49 = 199.9260{{c}}, ~3/2 = 702.7162{{c}}
* POTE: ~55/49 = 1\6, ~3/2 = 702.9764
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0062{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 174e, 186, 198, 408cde, 606bcde }}
{{Optimal ET sequence|legend=0| 12, 174e, 186, 198, 408cde, 606bcde }}


Badness: 0.068509
Badness (Sintel): 2.26


==== 13-limit ====
==== 13-limit ====
Line 142: Line 259:
Comma list: 352/351, 847/845, 1716/1715, 3136/3125
Comma list: 352/351, 847/845, 1716/1715, 3136/3125


Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}
{{Mapping|legend=0| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 703.1045
* WE: ~55/49 = 199.9293{{c}}, ~3/2 = 702.7602{{c}}
* POTE: ~55/49 = 1\6, ~3/2 = 703.0087
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0343{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 174ef, 186, 198 }}
{{Optimal ET sequence|legend=0| 12f, 174ef, 186, 198 }}


Badness: 0.044073
Badness (Sintel): 1.82


=== Hemimist ===
=== Hemimist ===
Line 157: Line 274:
Comma list: 3025/3024, 3136/3125, 5120/5103
Comma list: 3025/3024, 3136/3125, 5120/5103


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


Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721
Optimal tunings:
* WE: ~63/50 = 399.8922{{c}}, ~400/231 = 951.2704{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~400/231 = 951.5363{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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 172: Line 290:
Comma list: 352/351, 676/675, 847/845, 3025/3024
Comma list: 352/351, 676/675, 847/845, 3025/3024


Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}
{{Mapping|legend=0| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.5810
Optimal tunings:
* WE: ~63/50 = 399.8818{{c}}, ~26/15 = 951.2442{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 951.5358{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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
Line 189: Line 309:
{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}
{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}


: mapping generators: ~80/63, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~80/63 = 399.7918{{c}}, ~3/2 = 703.7511{{c}}
Optimal tunings:
: [[error map]]: {{val| -0.624 +1.172 -4.232 +4.712 }}
* CTE: ~80/63 = 1\3, ~3/2 = 704.3694
* [[CWE]]: ~80/63 = 400.0000{{c}}, ~3/2 = 704.1865{{c}}
* POTE: ~80/63 = 1\3, ~3/2 = 704.118
: error map: {{val| 0.000 +2.232 -3.060 +6.055 }}


{{Optimal ET sequence|legend=1| 12, 39c, 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 204: Line 324:
Comma list: 100/99, 225/224, 2420/2401
Comma list: 100/99, 225/224, 2420/2401


Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
{{Mapping|legend=0| 3 0 26 37 58 | 0 1 -4 -6 -10 }}


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


Optimal tunings:
{{Optimal ET sequence|legend=0| 12, 39ce, 51, 63, 138de }}
* CTE: ~44/35 = 1\3, ~3/2 = 704.6253
* POTE: ~44/35 = 1\3, ~3/2 = 704.411


Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}
Badness (Sintel): 1.70
 
Badness: 0.051348
 
== Subgroup extensions ==
=== Misty (2.3.5.7.17) ===
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
 
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 tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.0963
 
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
 
Badness: 0.0186
 
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19
 
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 tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.0778
 
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 210gh, 309cdghh }}
 
Badness: 0.0132


[[Category:Misty family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Misty family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]

Latest revision as of 10:38, 6 October 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The misty family tempers out the misty comma (monzo: [26 -12 -3⟩, ratio: 67108864/66430125).

Misty

Main article: Misty

Misty features a third-octave period, which represents the ptolemaic diminished fourth (512/405). Four 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 schismas, making it a member of the schismic–Pythagorean equivalence continuum with n = 3.

Subgroup: 2.3.5

Comma list: 67108864/66430125

Mapping: [⟨3 0 26], ⟨0 1 -4]]

mapping generators: ~512/405, ~3

Optimal tunings:

  • WE: ~512/405 = 399.8595 ¢, ~3/2 = 702.8645 ¢
error map: ⟨-0.421 +0.488 +0.261]
  • CWE: ~512/405 = 400.0000 ¢, ~3/2 = 703.1489 ¢
error map: ⟨0.000 +1.194 +1.091]

Optimal ET sequence: 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc

Badness (Sintel): 2.50

Overview to extensions

The second comma of the 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

Main article: 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 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 {(S8/S9), S15/S17, S16, S18/S20, (S50)}, and for the 2.3.5.7.17.19-subgroup extension, {(S8/S9), S16, S18, S20, S17/S19, (S50)}.

7-limit

Subgroup: 2.3.5.7

Comma list: 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]

Optimal tunings:

  • WE: ~63/50 = 399.8579 ¢, ~3/2 = 702.7714 ¢
error map: ⟨-0.426 +0.390 +0.611 -0.236]
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 703.0551 ¢
error map: ⟨0.000 +1.100 +1.466 +0.623]

Optimal ET sequence: 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: [⟨3 0 26 56 -2], ⟨0 1 -4 -10 3]]

mapping generators: ~34/27, ~3

Optimal tunings:

  • WE: ~34/27 = 399.8023 ¢, ~3/2 = 702.6087 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~3/2 = 702.9837 ¢

Optimal ET sequence: 12, 87, 99, 309cdg, 408cdgg, 507cdggg

Badness (Sintel): 0.688

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 256/255, 324/323, 400/399, 476/475

Subgroup-val mapping: [⟨3 0 26 56 -2 8], ⟨0 1 -4 -10 3 1]]

Optimal tunings:

  • WE: ~24/19 = 399.7556 ¢, ~3/2 = 702.4862 ¢
  • CWE: ~24/19 = 400.0000 ¢, ~3/2 = 702.9418 ¢

Optimal ET sequence: 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.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 26 56 77], ⟨0 1 -4 -10 -14]]

Optimal tunings:

  • WE: ~44/35 = 399.8727 ¢, ~3/2 = 702.9867 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.2385 ¢

Optimal ET sequence: 12, …, 87, 99e, 186e

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 625/624

Mapping: [⟨3 0 26 56 77 92], ⟨0 1 -4 -10 -14 -17]]

Optimal tunings:

  • WE: ~44/35 = 399.8888 ¢, ~3/2 = 703.1080 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.3258 ¢

Optimal ET sequence: 12f, …, 87, 186ef

Badness (Sintel): 1.15

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 352/351, 364/363, 375/374, 595/594

Mapping: [⟨3 0 26 56 77 92 -2], ⟨0 1 -4 -10 -14 -17 3]]

Optimal tunings:

  • WE: ~34/27 = 399.8046 ¢, ~3/2 = 702.8894 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~3/2 = 703.2597 ¢

Optimal ET sequence: 12f, 87, 99ef, 186efg

Badness (Sintel): 1.22

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594

Mapping: [⟨3 0 26 56 77 92 -2 8], ⟨0 1 -4 -10 -14 -17 3 1]]

Optimal tunings:

  • WE: ~24/19 = 399.7530 ¢, ~3/2 = 702.7601 ¢
  • CWE: ~24/19 = 400.0000 ¢, ~3/2 = 703.2216 ¢

Optimal ET sequence: 12f, 87, 99ef, 186efgh

Badness (Sintel): 1.21

Hazy

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 5120/5103

Mapping: [⟨3 0 26 56 96], ⟨0 1 -4 -10 -18]]

Optimal tunings:

  • WE: ~63/50 = 399.8097 ¢, ~3/2 = 702.4039 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 702.7731 ¢

Optimal ET sequence: 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: [⟨3 0 26 56 96 111], ⟨0 1 -4 -10 -18 -21]]

Optimal tunings:

  • WE: ~63/50 = 399.8089 ¢, ~3/2 = 702.3993 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 702.7661 ¢

Optimal ET sequence: 12f, 87ef, 99, 111

Badness (Sintel): 1.54

Smoky

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 3136/3125

Mapping: [⟨3 0 26 56 -61], ⟨0 1 -4 -10 15]]

Optimal tunings:

  • WE: ~63/50 = 400.0072 ¢, ~3/2 = 702.3068 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 703.2941 ¢

Optimal ET sequence: 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: [⟨3 0 26 56 -61 -46], ⟨0 1 -4 -10 15 12]]

Optimal tunings:

  • WE: ~63/50 = 400.0184 ¢, ~3/2 = 703.3420 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 703.3101 ¢

Optimal ET sequence: 12e, 87, 186, 273

Badness (Sintel): 1.63

Ashy

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56 -80], ⟨0 1 -4 -10 19]]

Optimal tunings:

  • WE: ~63/50 = 399.7776 ¢, ~3/2 = 702.4764 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 702.8652 ¢

Optimal ET sequence: 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: [⟨3 0 26 56 -80 -65], ⟨0 1 -4 -10 19 16]]

Optimal tunings:

  • WE: ~63/50 = 399.7470 ¢, ~3/2 = 702.3782 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 702.8135 ¢

Optimal ET sequence: 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: [⟨6 0 52 112 173], ⟨0 1 -4 -10 -16]]

mapping generators: ~55/49, ~3

Optimal tunings:

  • WE: ~55/49 = 199.9260 ¢, ~3/2 = 702.7162 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 703.0062 ¢

Optimal ET sequence: 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: [⟨6 0 52 112 173 203], ⟨0 1 -4 -10 -16 -19]]

Optimal tunings:

  • WE: ~55/49 = 199.9293 ¢, ~3/2 = 702.7602 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 703.0343 ¢

Optimal ET sequence: 12f, 174ef, 186, 198

Badness (Sintel): 1.82

Hemimist

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]]

mapping generators: ~63/50, ~400/231

Optimal tunings:

  • WE: ~63/50 = 399.8922 ¢, ~400/231 = 951.2704 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~400/231 = 951.5363 ¢

Optimal ET sequence: 24d, 87, 111, 198

Badness (Sintel): 1.57

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 3025/3024

Mapping: [⟨3 0 26 56 8 23], ⟨0 2 -8 -20 1 -5]]

Optimal tunings:

  • WE: ~63/50 = 399.8818 ¢, ~26/15 = 951.2442 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~26/15 = 951.5358 ¢

Optimal ET sequence: 24d, 87, 111, 198

Badness (Sintel): 0.947

Fog

Named by Xenllium in 2021, the fog temperament tempers out the marvel comma, 225/224, and may be described as 12 & 63.

Subgroup: 2.3.5.7

Comma list: 225/224, 156250/151263

Mapping: [⟨3 0 26 37], ⟨0 1 -4 -6]]

Optimal tunings:

  • WE: ~80/63 = 399.7918 ¢, ~3/2 = 703.7511 ¢
error map: ⟨-0.624 +1.172 -4.232 +4.712]
  • CWE: ~80/63 = 400.0000 ¢, ~3/2 = 704.1865 ¢
error map: ⟨0.000 +2.232 -3.060 +6.055]

Optimal ET sequence: 12, 39c, 51, 63, 75

Badness (Sintel): 2.20

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 2420/2401

Mapping: [⟨3 0 26 37 58], ⟨0 1 -4 -6 -10]]

Optimal tunings:

  • WE: ~44/35 = 399.8078 ¢, ~3/2 = 704.0728 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~3/2 = 704.4697 ¢

Optimal ET sequence: 12, 39ce, 51, 63, 138de

Badness (Sintel): 1.70

Retrieved from "https://en.xen.wiki/index.php?title=Misty_family&oldid=238775"