Kleismic: Difference between revisions
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readded some changes, added formulas, etc. Chords of 13 are some of the simplest chords of cata in terms of generator complexity. |
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| | | 3''g''<sup>3</sup> + 4''g'' - 10 = 0 | ||
| 317.0010 | | 317.0010 | ||
| [[Delta-rational chord|DR]] 13:15:18 | | [[Delta-rational chord|DR]] 13:15:18 | ||
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| Line 197: | Line 197: | ||
| 317.1429 | | 317.1429 | ||
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| ''g''<sup>6</sup> + 2''g''<sup>5</sup> - 8 = 0 | |||
| 317.1496 | |||
| 1–3–5 equal-beating tuning, [[Delta-rational chord|DR]] 3:4:5 | |||
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| Line 239: | Line 244: | ||
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| | | 3''g''<sup>3</sup> - ''g'' - 4 = 0 | ||
| 317.5679 | | 317.5679 | ||
| [[Delta-rational chord|DR]] 9:13:15 | | [[Delta-rational chord|DR]] 9:13:15 | ||
|- | |- | ||
| [[34edo|9\34]] | | [[34edo|9\34]] | ||
| Line 254: | Line 259: | ||
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| | | ''g''<sup>6</sup> - 2''g''<sup>5</sup> + 2 = 0 | ||
| 317.9593 | | 317.9593 | ||
| [[Delta-rational chord|DR]] 4:5:6, close to 2/7-kleisma | | 1–3–5 equal-beating tuning, [[Delta-rational chord|DR]] 4:5:6, close to 2/7-kleisma | ||
|- | |- | ||
| [[83edo|22\83]] | | [[83edo|22\83]] | ||
Revision as of 12:43, 17 September 2024
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, 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 |
| 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
Tunings
Tuning spectrum
| Edo Generator |
Eigenmonzo (Unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 6/5 | 315.6413 | Untempered tuning, lower bound of 5-odd-limit diamond tradeoff | |
| 5\19 | 315.7895 | Lower bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone | |
| 27/26 | 316.3343 | 1/4-tunbarsma | |
| 29\110 | 316.3636 | 110ff val | |
| 24\91 | 316.4835 | 91f val | |
| 27/25 | 316.6547 | 1/8-kleisma | |
| 19\72 | 316.6667 | ||
| 9/5 | 316.7995 | 1/7-kleisma | |
| 33\125 | 316.8000 | 125f val | |
| 26/25 | 316.9750 | 1/4-marveltwin comma | |
| 14\53 | 316.9811 | ||
| 3/2 | 316.9925 | 1/6-kleisma | |
| 3g3 + 4g - 10 = 0 | 317.0010 | DR 13:15:18 | |
| 75/52 | 317.0274 | 1/2-tunbarsma | |
| 51\193 | 317.0984 | ||
| 15/8 | 317.1153 | 2/11-kleisma | |
| 88\333 | 317.1171 | ||
| 13/10 | 317.1349 | ||
| 37\140 | 317.1429 | ||
| g6 + 2g5 - 8 = 0 | 317.1496 | 1–3–5 equal-beating tuning, DR 3:4:5 | |
| 13/8 | 317.1805 | ||
| 60\227 | 317.1807 | ||
| 23\87 | 317.2414 | ||
| 5/4 | 317.2627 | 1/5-kleisma, upper bound of 5-odd-limit diamond tradeoff | |
| 13/12 | 317.3216 | ||
| 32\121 | 317.3554 | ||
| 41\155 | 317.4194 | ||
| 26/15 | 317.4197 | 1/3-marveltwin comma | |
| 3g3 - g - 4 = 0 | 317.5679 | DR 9:13:15 | |
| 9\34 | 317.6471 | ||
| 25/24 | 317.6681 | 1/4-kleisma, virtually DR 10:12:15 | |
| g6 - 2g5 + 2 = 0 | 317.9593 | 1–3–5 equal-beating tuning, DR 4:5:6, close to 2/7-kleisma | |
| 22\83 | 318.0723 | 83f val | |
| 13/9 | 318.3088 | 1/2-marveltwin comma, upper bound of 2.3.5.13-subgroup 15-odd-limit diamond tradeoff | |
| 125/72 | 318.3437 | 1/3-kleisma | |
| 13\49 | 318.3673 | 49f val | |
| 125/104 | 318.4135 | Full tunbarsma | |
| 625/432 | 319.6949 | 1/2-kleisma | |
| 4\15 | 320.0000 | Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone | |
| 65/54 | 320.9764 | Full marveltwin comma |
* besides the octave
Other tunings
- DKW (2.3.5): ~2 = 1\1, ~6/5 = 317.1983