Misty
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 |
3136/3125, 5120/5103 (7-limit);
256/255, 324/323, 400/399, 476/475
(2.3.5.7.17.19)
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 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
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 703.2481 ¢ | CWE: ~3/2 = 703.1489 ¢ | POTE: ~3/2 = 703.1114 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 703.1448 ¢ | CWE: ~3/2 = 703.0551 ¢ | POTE: ~3/2 = 703.0212 ¢ |
| 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 |