Kleismic: Difference between revisions
| Line 107: | Line 107: | ||
== List of patent vals == | == List of patent vals == | ||
The following patent vals support 2.3.5.13 Kleismic. Contorted vals are not included. | |||
{| class="wikitable" | |||
!|Edo !! Generator tuning !! Fifth tuning | |||
|- | |||
|| 15 || 320.000 || 720.000 | |||
|- | |||
||34 || 317.647 || 705.882 | |||
|- | |||
||155 || 317.419 || 704.516 | |||
|- | |||
||121 || 317.355 || 704.132 | |||
|- | |||
||208 || 317.308 || 703.846 | |||
|- | |||
||295 || 317.288 || 703.729 | |||
|- | |||
||87 || 317.241 || 703.448 | |||
|- | |||
||401 || 317.207 || 703.242 | |||
|- | |||
||314 || 317.197 || 703.185 | |||
|- | |||
||227 || 317.181 || 703.084 | |||
|- | |||
||367 || 317.166 || 702.997 | |||
|- | |||
||507 || 317.160 || 702.959 | |||
|- | |||
||140 || 317.143 || 702.857 | |||
|- | |||
||613 || 317.129 || 702.773 | |||
|- | |||
||473 || 317.125 || 702.748 | |||
|- | |||
||333 || 317.117 || 702.703 | |||
|- | |||
||526 || 317.110 || 702.662 | |||
|- | |||
||193 || 317.098 || 702.591 | |||
|- | |||
||439 || 317.084 || 702.506 | |||
|- | |||
||246 || 317.073 || 702.439 | |||
|- | |||
||299 || 317.057 || 702.341 | |||
|- | |||
||53 || 316.981 || 701.887 | |||
|- | |||
||72 || 316.667 || 700.000 | |||
|- | |||
||19 || 315.789 || 694.737 | |||
|} | |||
{{Navbox regtemp}} | {{Navbox regtemp}} | ||
{{cat|Temperaments}} | {{cat|Temperaments}} | ||
Revision as of 00:40, 14 March 2026
| Kleismic |
325/324, 625/624 (2.3.5.13)
2.3.5.13 15-odd-limit: 2.35¢
2.3.5.13 15-odd-limit: 15 notes
Kleismic, [15 & 19], is a high-accuracy temperament (usually seen in its basic form as a 2.3.5 temperament) that equates a stack of six 6/5 minor thirds to one 3/1. Via a structurally induced extension to 2.3.5.13, it equates three 6/5's to one semitwelfth 26/15 and equates 25/24 to 26/25 and 27/26.
Kleismic harmony is naturally based on splitting 5/3 into 4/3 and 5/4 thus making 3:4:5 or 12:15:18 triads. Indeed, Kleismic[15] (4L11s) can be constructed from the generator sequence GS(3:4:5)[19] by tempering out four kleismas.
Kleismic, despite generating a heptatonic scale, is not particularly usefully a 7-form temperament; this is because 3/2 is an imperfect sixth rather than a perfect fifth. In the 11-form, however, it is much better as 5/4 and 4/3 are mapped to the same degree, resulting in a dichotomy of [0 4 8]/11 triads similar to the [0 2 4]/7 ones found in fifth-centric temperaments.
Interval chain
In the following table, odd harmonics 1–15 are labeled in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 317.1 | 6/5 |
| 2 | 634.2 | 13/9, 36/25 |
| 3 | 951.3 | 26/15 |
| 4 | 68.4 | 25/24, 26/25, 27/26 |
| 5 | 385.5 | 5/4 |
| 6 | 702.6 | 3/2 |
| 7 | 1019.6 | 9/5 |
| 8 | 136.7 | 13/12, 27/25 |
| 9 | 453.8 | 13/10 |
| 10 | 770.9 | 25/16, 39/25 |
| 11 | 1088.0 | 15/8 |
| 12 | 205.1 | 9/8 |
| 13 | 522.2 | 27/20 |
| 14 | 839.3 | 13/8 |
| 15 | 1156.4 | 39/20 |
| 16 | 273.5 | 75/64 |
| 17 | 590.6 | 45/32 |
| 18 | 907.7 | 27/16 |
| 19 | 24.7 | 65/64, 81/80 |
* In 2.3.5.13-subgroup CWE tuning, octave reduced
List of patent vals
The following patent vals support 2.3.5.13 Kleismic. Contorted vals are not included.
| Edo | Generator tuning | Fifth tuning |
|---|---|---|
| 15 | 320.000 | 720.000 |
| 34 | 317.647 | 705.882 |
| 155 | 317.419 | 704.516 |
| 121 | 317.355 | 704.132 |
| 208 | 317.308 | 703.846 |
| 295 | 317.288 | 703.729 |
| 87 | 317.241 | 703.448 |
| 401 | 317.207 | 703.242 |
| 314 | 317.197 | 703.185 |
| 227 | 317.181 | 703.084 |
| 367 | 317.166 | 702.997 |
| 507 | 317.160 | 702.959 |
| 140 | 317.143 | 702.857 |
| 613 | 317.129 | 702.773 |
| 473 | 317.125 | 702.748 |
| 333 | 317.117 | 702.703 |
| 526 | 317.110 | 702.662 |
| 193 | 317.098 | 702.591 |
| 439 | 317.084 | 702.506 |
| 246 | 317.073 | 702.439 |
| 299 | 317.057 | 702.341 |
| 53 | 316.981 | 701.887 |
| 72 | 316.667 | 700.000 |
| 19 | 315.789 | 694.737 |
