Starling–sensamagic equivalence continuum
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
The starling–sensamagic equivalence continuum is a continuum of 7-limit temperament families which equate a number of starling commas (126/125) with a sensamagic comma (245/243). This continuum is theoretically interesting in that these are all 7-limit temperament families supported by septimal sensi temperament.
All temperaments in the continuum satisfy (126/125)n ~ 245/243. Varying n results in different temperament families listed in the table below. It converges to starling as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 7-limit temperament families supported by septimal sensi (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of n is approximately 1.02868884, and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament family | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| −4 | 19 & 46 & 110c | 50824368/48828125 | [4 3 −11 6⟩ |
| −3 | 19 & 46 & 33 | 403368/390625 | [3 1 −8 5⟩ |
| −2 | 19 & 46 & 21 | 9604/9375 | [2 −1 −5 4⟩ |
| −1 | Sengic | 686/675 | [1 −3 −2 3⟩ |
| 0 | Sensamagic | 245/243 | [0 −5 1 2⟩ |
| 1 | Ragismic | 4375/4374 | [−1 −7 4 1⟩ |
| 2 | Sensipent | 78732/78125 | [2 9 −7⟩ |
| 3 | 19 & 46 & 161 | 9920232/9765625 | [3 11 −10 1⟩ |
| 4 | 19 & 46 & 173 | 1249949232/1220703125 | [4 13 −13 2⟩ |
| … | … | … | … |
| ∞ | Starling | 126/125 | [1 2 −3 1⟩ |
Examples of temperaments with fractional values of n:
- Sensibeta (n = 1⁄2 = 0.5)
- 19 & 46 & 193 (n = 2⁄3 = 0.666…)
- 19 & 46 & 265 (n = 3⁄4 = 0.75)
- 19 & 46 & 354 (n = 5⁄4 = 1.25)
- 19 & 46 & 255 (n = 4⁄3 = 1.333…)
- 19 & 46 & 183 (n = 3⁄2 = 1.5)