Kleismic
| 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.
Kleismic can be seen as a counterpart to Diaschismic, which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - Diaschismic splits 9/8 into two; Kleismic splits it into three. The intersection of both temperaments in the 5-limit is 34edo, which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.
Kleismic contains interordinal intervals, because 3/1 is split into an even number of parts. Because it equates 9/8 to three 25/24s, the inframinor and ultramajor third in alpha-dicot tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.
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
- Main article: Kleismic/Patent vals
