Dubhe: Difference between revisions
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'''Dubhe''' is a [[temperament]] in the 3.5.7.17 [[subgroup]] generated by a sharp [[~]][[17/15]] (or equivalently a flat ~[[45/17]]). As a weak extension of [[BPS]], it [[tempering out|tempers out]] the sensamagic comma, [[245/243]], while splitting the BPS generator of ~[[9/7]] in two (by tempering out [[2025/2023]]) so that a stack of two generators represents 9/7, and a stack of four generators represents [[5/3]] in addition to 81/49, which generates | '''Dubhe''' is a [[temperament]] in the 3.5.7.17 [[subgroup]] generated by a sharp [[~]][[17/15]] (or equivalently a flat ~[[45/17]]). As a weak extension of [[BPS]], it [[tempering out|tempers out]] the sensamagic comma, [[245/243]], while splitting the BPS generator of ~[[9/7]] in two (by tempering out [[2025/2023]]) so that a stack of two generators represents 9/7, and a stack of four generators represents [[5/3]] in addition to 81/49, which generates [[MOS scale]]s of [[8L 1s (3/1-equivalent)|8L 1s]] and [[9L 8s (3/1-equivalent)|9L 8s]] against the tritave. The "canonical" tuning for the generator is [[26edt|3\26]]edt, representing the double [[Bohlen-Pierce]] scale, but a range of other tunings are valid, including [[17edt|2\17]]edt and [[43edt|5\43]]edt. | ||
For technical data, see ''[[No-twos subgroup temperaments#Dubhe]]''. | For technical data, see ''[[No-twos subgroup temperaments#Dubhe]]''. | ||
Revision as of 05:00, 3 October 2024
Dubhe is a temperament in the 3.5.7.17 subgroup generated by a sharp ~17/15 (or equivalently a flat ~45/17). As a weak extension of BPS, it tempers out the sensamagic comma, 245/243, while splitting the BPS generator of ~9/7 in two (by tempering out 2025/2023) so that a stack of two generators represents 9/7, and a stack of four generators represents 5/3 in addition to 81/49, which generates MOS scales of 8L 1s and 9L 8s against the tritave. The "canonical" tuning for the generator is 3\26edt, representing the double Bohlen-Pierce scale, but a range of other tunings are valid, including 2\17edt and 5\43edt.
For technical data, see No-twos subgroup temperaments#Dubhe.
Interval chains
In the below, tritave-reduced harmonics below 243 are indicated in bold. All ratios are considered to be in the 287-throdd limit.
| # | Cents* | Approximate Ratios |
|---|---|---|
| 3.5.7.17 subgroup | ||
| -12 | 1160.5 | 49/25, 243/125 |
| -11 | 1380.8 | 189/85 |
| -10 | 1601.1 | 63/25 |
| -9 | 1821.4 | 49/17, 243/85, 357/125 |
| -8 | 139.7 | 27/25, 49/45 |
| -7 | 360.0 | 21/17, 153/125 |
| -6 | 580.3 | 7/5, 243/175 |
| -5 | 800.6 | 27/17, 119/75, 245/153 |
| -4 | 1020.8 | 9/5, 49/27 |
| -3 | 1241.1 | 35/17, 51/25, 243/119 |
| -2 | 1461.4 | 7/3, 81/35 |
| -1 | 1681.7 | 45/17, 119/45, 459/175 |
| 0 | 0.0 | 1/1, 245/243 |
| 1 | 220.3 | 17/15, 135/119, 175/153 |
| 2 | 440.6 | 9/7, 35/27 |
| 3 | 660.9 | 25/17, 51/35, 119/81 |
| 4 | 881.1 | 5/3, 81/49 |
| 5 | 1101.4 | 17/9, 225/119, 459/245 |
| 6 | 1321.7 | 15/7, 175/81 |
| 7 | 1542.0 | 17/7, 125/51 |
| 8 | 1762.2 | 25/9, 135/49 |
| 9 | 80.5 | 51/49, 85/81, 125/119 |
| 10 | 300.8 | 25/21 |
| 11 | 521.1 | 85/63 |
| 12 | 741.4 | 75/49, 125/81 |
| 13 | 961.7 | 85/49 |
| 14 | 1181.9 | 125/63 |
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