Misty family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
CompactStar (talk | contribs)
5,020 edits
No edit summary
 
(9 intermediate revisions by 5 users not shown)
Line 1: Line 1:
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]].  
{{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 second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
Line 14: Line 15:
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}


[[Optimal tuning]] ([[POTE]]): ~512/405 = 1\3, ~3/2 = 703.111
: mapping generators: ~512/405, ~3


{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384 }}
[[Optimal tuning]]s:
* [[CTE]]: ~512/405 = 1\3, ~3/2 = 703.2481
* [[POTE]]: ~512/405 = 1\3, ~3/2 = 703.111
 
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}


[[Badness]]: 0.106540
[[Badness]]: 0.106540
Line 29: Line 34:
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


{{Multival|legend=1| 3 -12 -30 -26 -56 -36 }}
: mapping generators: ~63/50, ~3


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~3/2 = 703.021
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448


{{Optimal ET sequence|legend=1| 12, 87, 99 }}
{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}


[[Badness]]: 0.036802
[[Badness]]: 0.036802


=== 11-limit ===
=== 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
 
Comma list: 256/255, 1701/1700, 2025/2023
 
Sval mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}
 
: sval 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
 
Sval mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}
 
: sval mapping generators: ~24/19, ~3
 
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
 
=== Undecimal misty ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 44: Line 81:
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 703.210
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3127
* POTE: ~44/35 = 1\3, ~3/2 = 703.210


{{Optimal ET sequence|legend=1| 12, 87, 99e, 186e }}
Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}


Badness: 0.035932
Badness: 0.035932
Line 57: Line 96:
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 703.303
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
* POTE: ~44/35 = 1\3, ~3/2 = 703.303


{{Optimal ET sequence|legend=1| 12f, 87, 99ef, 186ef, 273ef }}
Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}


Badness: 0.027820
Badness: 0.027820
Line 70: Line 111:
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}


Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 703.233
Optimal tunings:
* CTE: ~34/27 = 1\3, ~3/2 = 703.3686
* POTE: ~34/27 = 1\3, ~3/2 = 703.233


{{Optimal ET sequence|legend=1| 12f, 87, 99ef, 186efg }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efg }}


Badness: 0.023908
Badness: 0.023908
Line 83: Line 126:
Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 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 tuning (POTE): ~34/27 = 1\3, ~3/2 = 703.253
Optimal tunings:
* CTE: ~24/19 = 1\3, ~3/2 = 703.3626
* POTE: ~34/27 = 1\3, ~3/2 = 703.253


{{Optimal ET sequence|legend=1| 12f, 87, 99ef, 186efgh }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}


Badness: 0.019878
Badness: 0.019878
Line 96: Line 141:
Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}
Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 703.311
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3285
* POTE: ~44/35 = 1\3, ~3/2 = 703.311


{{Optimal ET sequence|legend=1| 12, 87, 186e, 273e }}
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}


Badness: 0.036872
Badness: 0.036872
Line 111: Line 158:
: mapping generators: ~55/49, ~3
: mapping generators: ~55/49, ~3


Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 702.9764
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.0864
* POTE: ~55/49 = 1\6, ~3/2 = 702.9764


{{Optimal ET sequence|legend=1| 12, 174e, 186, 198, 408cde, 606bcde }}
Optimal ET sequence: {{Optimal ET sequence| 12, 174e, 186, 198, 408cde, 606bcde }}


Badness: 0.068509
Badness: 0.068509
Line 124: Line 173:
Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}
Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}


Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 703.0087
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.1045
* POTE: ~55/49 = 1\6, ~3/2 = 703.0087


{{Optimal ET sequence|legend=1| 12f, 174ef, 186, 198 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 174ef, 186, 198 }}


Badness: 0.044073
Badness: 0.044073
Line 139: Line 190:
: mapping generators: ~63/50, ~400/231
: mapping generators: ~63/50, ~400/231


Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.527
Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721


{{Optimal ET sequence|legend=1| 24d, 87, 111, 198 }}
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}


Badness: 0.047439
Badness: 0.047439
Line 152: Line 203:
Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}
Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.525
Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.5810


{{Optimal ET sequence|legend=1| 24d, 87, 111, 198 }}
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}


Badness: 0.022923
Badness: 0.022923
Line 165: Line 216:
[[Comma list]]: 225/224, 156250/151263
[[Comma list]]: 225/224, 156250/151263


{{Mapping|legend=1| 3 5 6 7 | 0 -1 4 6 }}
{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}


{{Multival|legend=1| 3 -12 -18 -26 -37 -8 }}
: mapping generators: ~80/63, ~3


[[Optimal tuning]] ([[POTE]]): ~80/63 = 1\3, ~3/2 = 704.118
Optimal tunings:
* CTE: ~80/63 = 1\3, ~3/2 = 704.3694
* POTE: ~80/63 = 1\3, ~3/2 = 704.118


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


[[Badness]]: 0.086866
[[Badness]]: 0.086866
Line 180: Line 233:
Comma list: 100/99, 225/224, 2420/2401
Comma list: 100/99, 225/224, 2420/2401


Mapping: {{mapping| 3 5 6 7 8 | 0 -1 4 6 10 }}
Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
 
: mapping generators: ~44/35, ~3


Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 704.411
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 704.6253
* POTE: ~44/35 = 1\3, ~3/2 = 704.411


{{Optimal ET sequence|legend=1| 12, 51, 63, 138de }}
Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}


Badness: 0.051348
Badness: 0.051348


[[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]]

Latest revision as of 00:37, 24 June 2025

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, 67108864/66430125 = [26 -12 -3⟩, equating the Pythagorean comma with a stack of three schismas, 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 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.

Misty

Main article: Misty

Subgroup: 2.3.5

Comma list: 67108864/66430125

Mapping: [⟨3 0 26], ⟨0 1 -4]]

mapping generators: ~512/405, ~3

Optimal tunings:

  • CTE: ~512/405 = 1\3, ~3/2 = 703.2481
  • POTE: ~512/405 = 1\3, ~3/2 = 703.111

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

Badness: 0.106540

Septimal misty

Main article: Misty

Subgroup: 2.3.5.7

Comma list: 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]

mapping generators: ~63/50, ~3

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

Optimal ET sequence: 12, …, 87, 99

Badness: 0.036802

2.3.5.7.17 subgroup

The S-expression-based comma list for this 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)}.

Subgroup: 2.3.5.7.17

Comma list: 256/255, 1701/1700, 2025/2023

Sval mapping: [⟨3 0 26 56 -2], ⟨0 1 -4 -10 3]]

sval mapping generators: ~34/27, ~3

Optimal tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.0963

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

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

sval mapping generators: ~24/19, ~3

Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.0778

Optimal ET sequence: 12, 87, 99, 210gh, 309cdghh

Badness: 0.0132

Undecimal misty

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • CTE: ~44/35 = 1\3, ~3/2 = 703.3127
  • POTE: ~44/35 = 1\3, ~3/2 = 703.210

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

Badness: 0.035932

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:

  • CTE: ~44/35 = 1\3, ~3/2 = 703.3859
  • POTE: ~44/35 = 1\3, ~3/2 = 703.303

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

Badness: 0.027820

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:

  • CTE: ~34/27 = 1\3, ~3/2 = 703.3686
  • POTE: ~34/27 = 1\3, ~3/2 = 703.233

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

Badness: 0.023908

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:

  • CTE: ~24/19 = 1\3, ~3/2 = 703.3626
  • POTE: ~34/27 = 1\3, ~3/2 = 703.253

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

Badness: 0.019878

Mystic

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440, 896/891, 1001/1000

Mapping: [⟨3 0 26 56 77 -46], ⟨0 1 -4 -10 -14 12]]

Optimal tunings:

  • CTE: ~44/35 = 1\3, ~3/2 = 703.3285
  • POTE: ~44/35 = 1\3, ~3/2 = 703.311

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

Badness: 0.036872

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:

  • CTE: ~55/49 = 1\6, ~3/2 = 703.0864
  • POTE: ~55/49 = 1\6, ~3/2 = 702.9764

Optimal ET sequence: 12, 174e, 186, 198, 408cde, 606bcde

Badness: 0.068509

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:

  • CTE: ~55/49 = 1\6, ~3/2 = 703.1045
  • POTE: ~55/49 = 1\6, ~3/2 = 703.0087

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

Badness: 0.044073

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 tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721

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

Badness: 0.047439

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 tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.5810

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

Badness: 0.022923

Fog

The fog temperament (12 & 63) tempers out the marvel comma, 225/224.

Subgroup: 2.3.5.7

Comma list: 225/224, 156250/151263

Mapping: [⟨3 0 26 37], ⟨0 1 -4 -6]]

mapping generators: ~80/63, ~3

Optimal tunings:

  • CTE: ~80/63 = 1\3, ~3/2 = 704.3694
  • POTE: ~80/63 = 1\3, ~3/2 = 704.118

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

Badness: 0.086866

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]]

mapping generators: ~44/35, ~3

Optimal tunings:

  • CTE: ~44/35 = 1\3, ~3/2 = 704.6253
  • POTE: ~44/35 = 1\3, ~3/2 = 704.411

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

Badness: 0.051348

Retrieved from "https://en.xen.wiki/index.php?title=Misty_family&oldid=201090"