Hanson and cata
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Hanson is a rank-2 temperament of the kleismic family, characterized by the vanishing of the kleisma. It is generated by a classical minor third (6/5), six of which make a twelfth (3/1). This naturally gives us hemitwelfths at only 3 generator steps, which can be interpreted as 26/15 (and thus hemifourths as 15/13), resulting in a low-complexity but high-accuracy extension to the 2.3.5.13 subgroup, sometimes known as cata.
7-limit extensions include keemun, catalan, catakleismic, countercata, and metakleismic.
For technical data, see Kleismic family #Hanson.
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 |
| 3 | 950.3 | 26/15 |
| 4 | 68.4 | 25/24, 26/25, 27/26 |
| 5 | 385.6 | 5/4 |
| 6 | 702.7 | 3/2 |
| 7 | 1019.8 | 9/5 |
| 8 | 136.9 | 13/12, 14/13, 27/25 |
| 9 | 454.0 | 13/10 |
| 10 | 771.1 | 25/16 |
| 11 | 1088.2 | 15/8 |
| 12 | 205.3 | 9/8 |
| 13 | 522.4 | 27/20 |
| 14 | 839.6 | 13/8, 21/13 |
| 15 | 1156.7 | 39/20 |
| 16 | 273.8 | 75/64 |
| 17 | 590.9 | 45/32 |
| 18 | 908.0 | 27/16 |
| 19 | 25.1 | 65/64, 81/80 |
* in 2.3.5.13-subgroup CTE tuning