Marvel–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 marvel–sensamagic equivalence continuum is a continuum of 7-limit temperament families which equate a number of marvel commas (225/224) with a sensamagic comma (245/243). This continuum is theoretically interesting in that these are all 7-limit temperament families supported by septimal magic temperament.
All temperaments in the continuum satisfy (225/224)n ~ 245/243. Varying n results in different temperament families listed in the table below. It converges to marvel 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 magic (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 1.84017119, and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament family | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| −4 | 19 & 41 & 141 | 52734375/51380224 | [-20 3 9 -2⟩ |
| −3 | 19 & 41 & 69 | 234375/229376 | [-15 1 7 -1⟩ |
| −2 | 5-limit magic | 3125/3072 | [-10 -1 5⟩ |
| −1 | Keemic | 875/864 | [-5 -3 3 1⟩ |
| 0 | Sensamagic | 245/243 | [0 -5 1 2⟩ |
| 1 | Hemimage | 10976/10935 | [5 -7 -1 3⟩ |
| 2 | Breeze | 2460375/2458624 | [-10 9 3 -4⟩ |
| 3 | 19 & 41 & 142 | 553584375/550731776 | [-15 11 5 -5⟩ |
| 4 | 19 & 41 & 214 | [-20 13 7 -6⟩ | |
| … | … | … | … |
| ∞ | Marvel | 225/224 | [-5 2 2 -1⟩ |
Examples of temperaments with fractional values of n:
- 19 & 41 & 126bc (n = −5⁄2 = −2.5)
- 19 & 41 & 42bc (n = −3⁄2 = −1.5)
- 19 & 41 & 42 (n = −1⁄2 = −0.5)
- septimagic (n = 1⁄2 = 0.5)
- acromagic (n = 3⁄2 = 1.5)
- 19 & 41 & 359 (n = 5⁄3 = 1.666…)
- 19 & 41 & 511 (n = 7⁄4 = 1.75)
19 & 41 & 69
Comma list: 234375/229376
Mapping: [⟨1 0 0 -15], ⟨0 1 0 1], ⟨0 0 1 7]]
- WE: ~2 = 1201.118¢, ~3 = 1901.764¢, ~5 = 2783.489¢
- CWE: ~2 = 1200.000¢, ~3 = 1900.566¢, ~5 = 2780.998¢
Optimal ET sequence: 3d, 6b, 9d, 13, 16d, 19, 22, 41
Badness (Sintel): 16.808