Misty: Difference between revisions

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In the 7-limit, the canonical extension tempers out [[3136/3125]] and [[5120/5103]]. Possible tunings include [[87edo]], [[99edo]] and [[111edo]].  
In the 7-limit, the canonical extension tempers out [[3136/3125]] and [[5120/5103]]. Possible tunings include [[87edo]], [[99edo]] and [[111edo]].  


See [[Misty family]] for more technical data.  
See [[Misty family]] for technical data.  


== Interval chain ==
== Interval chain ==
{| class="wikitable center-1 right-2"
In the following table, odd harmonics 1–21 are in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4 right-6"
|-
|-
! rowspan="2" | #
! rowspan="2" | #
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|-
|-
! Cents*
! Cents*
! Approximate Ratios
! Approx. ratios
! Cents*
! Cents*
! Approximate Ratios
! Approx. ratios
! Cents*
! Cents*
! Approximate Ratios
! Approx. ratios
|-
|-
| 0
| 0
Line 24: Line 26:
| '''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 />* In 7-limit CTE tuning
<nowiki/>* In 7-limit CWE tuning, octave reduced


== Tunings ==
== Tunings ==
Line 67: Line 133:


=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-

Revision as of 07:10, 12 February 2026

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 parts of 512/405 each. This temperament, supported by 12et, is notably in the schismic–Pythagorean equivalence continuum, with n = 3.

In the 7-limit, the canonical extension tempers out 3136/3125 and 5120/5103. Possible tunings include 87edo, 99edo and 111edo.

See Misty family for technical data.

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

Tunings

  • 7-limit POTE tuning: ~3/2 = 703.0212
  • 7-limit CTE tuning: ~3/2 = 703.1448

Tuning spectrum

Edo
Generator 		Eigenmonzo
(rnchanged-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
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¢)
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
37\63 704.762 Upper bound of 9-odd-limit diamond monotone
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