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'''Misty''' is the [[regular temperament]] [[tempering out]] the [[misty comma]]. It equates the [[Pythagorean comma]] with the [[diesis]], and splits this interval into three equal parts, one representing the [[schisma]]~[[diaschisma]], and two representing the [[syntonic comma]]. Consequently, the octave is also split into three. This temperament, supported by [[12edo|12et]], is notably in the [[schismic-Pythagorean equivalence continuum]], with ''n'' = 3.  
{{Infobox regtemp
| Title = Misty
| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.17.19
| Comma basis = [[67108864/66430125]] (5-limit); <br>[[3136/3125]], [[5120/5103]] (7-limit); <br>[[256/255]], [[324/323]], [[400/399]], [[476/475]]<br>(2.3.5.7.17.19)
| Edo join 1 = 12 | Edo join 2 = 99
| Mapping = 3; 1 -4 -10 3 1
| Generators = 3/2
| Generators tuning = 703.1
| Optimization method = CWE
| MOS scales = [[3L 9s]], [[12L 3s]], [[12L 15s]], [[12L 27s]]
| Odd limit 1 = 9 | Mistuning 1 = 1.96 | Complexity 1 = 39
}}
'''Misty''' is a [[regular temperament|temperament]] with a 1/3-octave [[period]], generated by a [[3/2|perfect fifth]], and four [[4/3|perfect fourths]] [[octave reduction|octave reduced]] (i.e. a minor sixth, [[~]][[128/81]]) minus a period give the [[~]][[5/4]], [[tempering out]] the [[misty comma]].  


In the 7-limit, the canonical extension tempers out [[3136/3125]] and [[5120/5103]]. Possible tunings include [[87edo]], [[99edo]] and [[111edo]].  
Misty entails a mildly sharp perfect fifth. For example, a perfect fifth 3 cents sharp of [[12edo]]'s (about 1 cent sharp of just) generates a minor sixth 12 cents flat of 8\12, and subtracting 4\12 from it will produce a 5/4 in excellent tune. [[Edo]]s that provide such a fifth include [[87edo]], [[99edo]] and [[111edo]]. Such a fifth stacked six times octave reduced (i.e. an augmented fourth) is close in size to [[10/7]], which gives rise to the [[7-limit]] [[extension]] where it tempers out [[5120/5103]]. This places [[7/4]] six fifths further on the generator chain and implies the [[5/4]] is split into two equal parts each for [[28/25]], tempering out [[3136/3125]], and that [[63/50]] is mapped to the 1/3-octave period, tempering out [[250047/250000]].  


See [[Misty family]] for more technical data.  
It is easy to extend misty to the no-11 no-13 [[19-limit]], where it merges [[16/15]] with [[17/16]], [[18/17]] with [[19/18]], and [[20/19]] with [[21/20]], tempering out [[256/255]] ({{S|16}}), [[324/323]] ({{S|18}}), and [[400/399]] ({{S|20}}). This lowers the overall accuracy, but supplies more harmonic resources.  


== Interval chain ==
See [[Misty family #Misty]] and [[Misty family #Septimal misty|#Septimal misty]] for technical data. See [[Misty extensions]] for a discussion on [[11-limit|11-]] and [[13-limit]] extensions.
{| class="wikitable center-1 right-2"
 
! rowspan="2" |#
== Intervals ==
! colspan="2" |Period 0
=== Interval chain ===
! colspan="2" |Period 1
In the following table, odd harmonics 1–21 are in '''bold'''.
! colspan="2" |Period 2
 
{| class="wikitable center-1 right-2 right-4 right-6"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 2
|-
|-
! Cents*
! Cents*
! Approximate Ratios
! Approx. ratios
! Cents*
! Cents*
! Approximate Ratios
! Approx. ratios
! Cents*
! Cents*
! Approximate Ratios
! Approx. ratios
|-
|-
| 0
| 0
Line 23: Line 41:
| '''1/1'''
| '''1/1'''
| 400.0
| 400.0
| 63/50
| 24/19, 34/27
| 800.0
| 800.0
| 100/63
| 19/12, 27/17
|-
|-
| 1
| 1
| 96.9
| 703.1
| 135/128
| '''3/2'''
| 496.9
| 1103.1
| '''4/3'''
| 17/9, 36/19
| 896.9
| 303.1
| 42/25
| '''19/16''', 25/21
|-
|-
| 2
| 2
| 193.7
| 206.1
| 28/25
| '''9/8'''
| 593.7
| 606.1
| 45/32
| 17/12
| 993.7
| 1006.1
| 16/9
| 25/14, 34/19
|-
|-
| 3
| 3
| 290.6
| 909.2
| 32/27
| 27/16
| 690.6
| 109.2
| 112/75
| '''16/15''', '''17/16'''
| 1090.6
| 509.2
| 15/8
| 51/38, 75/56
|-
|-
| 4
| 4
| 387.4
| 412.2
| '''5/4'''
| 19/15
| 787.4
| 812.2
| 63/40
| '''8/5'''
| 1187.4
| 12.2
| 125/63, 448/225
| 126/125, 225/224
|-
| 5
| 1115.3
| 19/10, 40/21
| 315.3
| 6/5
| 715.3
| 68/45
|-
| 6
| 618.3
| 10/7
| 1018.3
| 9/5
| 218.3
| 17/15
|-
| 7
| 121.4
| 15/14
| 521.4
| 27/20
| 921.4
| 17/10
|-
| 8
| 824.4
| 45/28
| 24.4
| 64/63, 81/80
| 424.4
| 32/25
|-
| 9
| 327.5
| 76/63, 135/112
| 727.5
| '''32/21'''
| 1127.5
| 48/25
|-
| 10
| 1030.5
| 38/21
| 230.5
| '''8/7'''
| 630.5
| 36/25
|-
| 11
| 533.6
| 19/14
| 933.6
| 12/7
| 133.6
| 27/25
|-
| 12
| 36.7
| 50/49, 57/56
| 436.7
| 9/7
| 836.7
| 34/21
|}
|}
<nowiki>*</nowiki> in 7-limit CTE tuning
<nowiki/>* In 7-limit CWE tuning, octave reduced
 
=== As a detemperament of 12et ===
[[File: Misty 12et Detempering.png|thumb|Misty as a 75-tone 12et detempering]]


== Tuning spectra ==
Misty is naturally a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 75-tone detempered scale, with a generator range of -12 to +12, which covers most of the intervals in the 2.3.5.7.17.19-subgroup 21-odd-limit. Each category is divided into six or seven qualities separated by 4 generator steps, which represent the generic half-comma step. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, misty gives us more than a dozen of qualities for each diatonic category.
* 7-limit POTE tuning: ~3/2 = 703.0212
 
* 7-limit CTE tuning: ~3/2 = 703.1448
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 703.2481{{c}}
| CWE: ~3/2 = 703.1489{{c}}
| POTE: ~3/2 = 703.1114{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 703.1448{{c}}
| CWE: ~3/2 = 703.0551{{c}}
| POTE: ~3/2 = 703.0212{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.17.19-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 703.0778{{c}}
| CWE: ~3/2 = 702.9418{{c}}
| POTE: ~3/2 = 702.9156{{c}}
|}


=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! ET<br>Generator
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged Interval)]]
! Edo<br>Generator
! Generator<br>(¢)
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator (¢)
! Comments
! Comments
|-
|-
Line 77: Line 210:
|-
|-
|  
|  
| 4/3
| 3/2
| 701.955
| 701.955
|
|-
|
| 81/80
| 702.688
|  
|  
|-
|-
Line 91: Line 229:
|  
|  
|-
|-
| 123\210
|  
|  
| 28/27
| 702.857
| 702.849
| 210gh val
|  
|-
|-
|  
|  
Line 107: Line 245:
|-
|-
|  
|  
| 10/9
| 9/5
| 702.933
| 702.933
| 9-odd-limit minimax (error = 1.955¢)
| 9-odd-limit minimax (error = 1.955{{c}})
|-
|-
|  
|  
Line 122: Line 260:
|-
|-
|  
|  
| 36/35
| 35/18
| 703.048
| 703.048
|  
|  
Line 134: Line 272:
| 21/20
| 21/20
| 703.107
| 703.107
|
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 703.117
| 703.117
| 7-odd-limit minimax (error = 1.217¢)
| 7-odd-limit minimax (error = 1.217{{c}})
|-
|-
|  
|  
| 6/5
| 5/3
| 703.128
| 703.128
| 5-odd-limit minimax (error = 1.173¢)
| 5-odd-limit minimax (error = 1.173{{c}})
|-
| 109\186
|
| 703.226
| 186gh val
|-
|
| 21/16
| 703.247
|
|-
|-
|  
|  
| 25/24
| 25/24
| 703.259
| 703.259
|
|-
|
| 63/32
| 703.408
|  
|  
|-
|-
Line 162: Line 315:
|-
|-
|  
|  
| 16/15
| 15/8
| 703.910
| 703.910
|  
|  
Line 169: Line 322:
|  
|  
| 704.000
| 704.000
|  
| 75d val
|-
|-
| 37\63
| 37\63
|  
|  
| 704.762
| 704.762
| Upper bound of 9-odd-limit diamond monotone
| 63d val, upper bound of 9-odd-limit diamond monotone
|}
|}


{{IoT}}
[[Category:Misty| ]] <!-- main article -->
[[Category:Misty| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Misty family]]
[[Category:Misty family]]
[[Category:Hemimean clan]]
[[Category:Hemimean clan]]
[[Category:Hemifamity temperaments]]
[[Category:Hemifamity temperaments]]

Latest revision as of 12:16, 29 March 2026

Misty
Subgroups 2.3.5, 2.3.5.7, 2.3.5.7.17.19
Comma basis 67108864/66430125 (5-limit);
3136/3125, 5120/5103 (7-limit);
256/255, 324/323, 400/399, 476/475
(2.3.5.7.17.19)
Reduced mapping ⟨3; 1 -4 -10 3 1]
ET join 12 & 99
Generators (CWE) ~3/2 = 703.1 ¢
MOS scales 3L 9s, 12L 3s, 12L 15s, 12L 27s
Ploidacot triploid monocot
Minimax error 9-odd-limit: 1.96 ¢
Target scale size 9-odd-limit: 39 notes

Misty is a temperament with a 1/3-octave period, generated by a perfect fifth, and four perfect fourths octave reduced (i.e. a minor sixth, ~128/81) minus a period give the ~5/4, tempering out the misty comma.

Misty entails a mildly sharp perfect fifth. For example, a perfect fifth 3 cents sharp of 12edo's (about 1 cent sharp of just) generates a minor sixth 12 cents flat of 8\12, and subtracting 4\12 from it will produce a 5/4 in excellent tune. Edos that provide such a fifth include 87edo, 99edo and 111edo. Such a fifth stacked six times octave reduced (i.e. an augmented fourth) is close in size to 10/7, which gives rise to the 7-limit extension where it tempers out 5120/5103. This places 7/4 six fifths further on the generator chain and implies the 5/4 is split into two equal parts each for 28/25, tempering out 3136/3125, and that 63/50 is mapped to the 1/3-octave period, tempering out 250047/250000.

It is easy to extend misty to the no-11 no-13 19-limit, where it merges 16/15 with 17/16, 18/17 with 19/18, and 20/19 with 21/20, tempering out 256/255 (S16), 324/323 (S18), and 400/399 (S20). This lowers the overall accuracy, but supplies more harmonic resources.

See Misty family #Misty and #Septimal misty for technical data. See Misty extensions for a discussion on 11- and 13-limit extensions.

Intervals

Interval chain

In the following table, odd harmonics 1–21 are in bold.

# Period 0 Period 1 Period 2
Cents* Approx. ratios Cents* Approx. ratios Cents* Approx. ratios
0 0.0 1/1 400.0 24/19, 34/27 800.0 19/12, 27/17
1 703.1 3/2 1103.1 17/9, 36/19 303.1 19/16, 25/21
2 206.1 9/8 606.1 17/12 1006.1 25/14, 34/19
3 909.2 27/16 109.2 16/15, 17/16 509.2 51/38, 75/56
4 412.2 19/15 812.2 8/5 12.2 126/125, 225/224
5 1115.3 19/10, 40/21 315.3 6/5 715.3 68/45
6 618.3 10/7 1018.3 9/5 218.3 17/15
7 121.4 15/14 521.4 27/20 921.4 17/10
8 824.4 45/28 24.4 64/63, 81/80 424.4 32/25
9 327.5 76/63, 135/112 727.5 32/21 1127.5 48/25
10 1030.5 38/21 230.5 8/7 630.5 36/25
11 533.6 19/14 933.6 12/7 133.6 27/25
12 36.7 50/49, 57/56 436.7 9/7 836.7 34/21

* In 7-limit CWE tuning, octave reduced

As a detemperament of 12et

Misty as a 75-tone 12et detempering

Misty is naturally a detemperament of the 12 equal temperament. The diagram on the right shows a 75-tone detempered scale, with a generator range of -12 to +12, which covers most of the intervals in the 2.3.5.7.17.19-subgroup 21-odd-limit. Each category is divided into six or seven qualities separated by 4 generator steps, which represent the generic half-comma step. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, misty gives us more than a dozen of qualities for each diatonic category.

Tunings

Norm-based tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 703.2481 ¢ CWE: ~3/2 = 703.1489 ¢ POTE: ~3/2 = 703.1114 ¢
7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 703.1448 ¢ CWE: ~3/2 = 703.0551 ¢ POTE: ~3/2 = 703.0212 ¢
2.3.5.7.17.19-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 703.0778 ¢ CWE: ~3/2 = 702.9418 ¢ POTE: ~3/2 = 702.9156 ¢

Tuning spectrum

Edo
Generator 		Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
7\12 700.000 Lower bound of 9-odd-limit diamond monotone
3/2 701.955
81/80 702.688
65\111 702.703
15/14 702.778
123\210 702.857 210gh val
7/5 702.915
9/7 702.924
9/5 702.933 9-odd-limit minimax (error = 1.955 ¢)
7/6 703.012
58\99 703.030
35/18 703.048
49/48 703.062
21/20 703.107
7/4 703.117 7-odd-limit minimax (error = 1.217 ¢)
5/3 703.128 5-odd-limit minimax (error = 1.173 ¢)
109\186 703.226 186gh val
21/16 703.247
25/24 703.259
63/32 703.408
5/4 703.422
51\87 703.448
15/8 703.910
44\75 704.000 75d val
37\63 704.762 63d val, upper bound of 9-odd-limit diamond monotone
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