Misty: Difference between revisions

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See [[Misty family]].  
'''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 [[12edo|12et]], is notably in the [[schismic–Pythagorean equivalence continuum]], with {{nowrap|''n'' {{=}} 3}}.  


[[Category:Smart redirect]]
In the 7-limit, the canonical extension tempers out [[3136/3125]] and [[5120/5103]]. Possible tunings include [[87edo]], [[99edo]] and [[111edo]].
[[Category:Misty]]
 
See [[Misty family]] for more technical data.
 
== Interval chain ==
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 2
|-
! Cents*
! Approximate Ratios
! Cents*
! Approximate Ratios
! Cents*
! Approximate Ratios
|-
| 0
| 0.0
| '''1/1'''
| 400.0
| 63/50
| 800.0
| 100/63
|-
| 1
| 96.9
| 135/128
| 496.9
| '''4/3'''
| 896.9
| 42/25
|-
| 2
| 193.7
| 28/25
| 593.7
| 45/32
| 993.7
| 16/9
|-
| 3
| 290.6
| 32/27
| 690.6
| 112/75
| 1090.6
| 15/8
|-
| 4
| 387.4
| '''5/4'''
| 787.4
| 63/40
| 1187.4
| 125/63, 448/225
|}
<nowiki />* In 7-limit CTE tuning
 
== Tunings ==
* 7-limit POTE tuning: ~3/2 = 703.0212
* 7-limit CTE tuning: ~3/2 = 703.1448
 
=== Tuning spectrum ===
 
{| class="wikitable center-all left-4"
|-
! Edo<br>Generator
! [[Eigenmonzo|Eigenmonzo<br>(rnchanged-interval)]]
! Generator<br>(¢)
! 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
|}
 
[[Category:Misty| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Misty family]]
[[Category:Hemimean clan]]
[[Category:Hemifamity temperaments]]

Latest revision as of 14:25, 28 April 2025

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 more technical data.

Interval chain

# Period 0 Period 1 Period 2
Cents* Approximate Ratios Cents* Approximate Ratios Cents* Approximate Ratios
0 0.0 1/1 400.0 63/50 800.0 100/63
1 96.9 135/128 496.9 4/3 896.9 42/25
2 193.7 28/25 593.7 45/32 993.7 16/9
3 290.6 32/27 690.6 112/75 1090.6 15/8
4 387.4 5/4 787.4 63/40 1187.4 125/63, 448/225

* In 7-limit CTE tuning

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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