Misty family: Difference between revisions

The '''misty family''' [[tempering out|tempers out]] the [[misty comma]] ({{monzo|legend=1| 26 -12 -3 }}, [[ratio]]: 67108864/66430125).  

== 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 }}.  

[[Subgroup]]: 2.3.5

[[Comma list]]: 67108864/66430125

: mapping generators: ~512/405, ~3

* [[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}}

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

[[Badness]] (Sintel): 2.50

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.

{{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]])}.  

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}

[[Optimal tuning]]s:

: [[error map]]: {{val| -0.426 +0.390 +0.611 -0.236 }}

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 703.0551{{c}}

{{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 }}

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

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 tunings:  

* 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.  

Subgroup: 2.3.5.7.11

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

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

* WE: ~44/35 = 399.8727{{c}}, ~3/2 = 702.9867{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.2385{{c}}

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

Badness (Sintel): 1.19

Subgroup: 2.3.5.7.11.13

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Temperament families]]

[[Category:Misty family| ]] <!-- main article -->

[[Category:Rank 2]]