Misty: Difference between revisions
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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 parts of [[512/405]] each. This temperament, supported by [[12edo|12et]], is notably in the [[ | '''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}}. | ||
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]]. | ||
| Line 7: | Line 7: | ||
== Interval chain == | == Interval chain == | ||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
|- | |||
! rowspan="2" | # | ! rowspan="2" | # | ||
! colspan="2" | Period 0 | ! colspan="2" | Period 0 | ||
| Line 59: | Line 60: | ||
| 125/63, 448/225 | | 125/63, 448/225 | ||
|} | |} | ||
<nowiki>* | <nowiki />* In 7-limit CTE tuning | ||
== Tunings == | == Tunings == | ||
| Line 68: | Line 69: | ||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|- | |||
! Edo<br>Generator | ! Edo<br>Generator | ||
! [[Eigenmonzo|Eigenmonzo<br>( | ! [[Eigenmonzo|Eigenmonzo<br>(rnchanged-interval)]] | ||
! Generator<br>(¢) | ! Generator<br>(¢) | ||
! Comments | ! Comments | ||
Revision as of 16:17, 10 December 2024
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 |