Six Seven of the TRAPPIST-1 exoplanets form the longest known chain of near-resonant exoplanets. Each planet makes some number of complete orbits for every 24 of b's orbits.
| Planet | Orbital period in days | Orbits for every 24 of b's orbits |
|---|---|---|
| b | 1.5109 | 24 |
| c | 2.4218 | 15 |
| d | 4.0496 | 9 |
| e | 6.0996 | 6 |
| f | 9.2067 | 4 |
| g | 12.3529 | 3 |
| h | 18.7640 | 2 |
The ratios of the orbital frequencies for neighboring pairs of planets form a series of perfect intervals.
| Planets | Ratio | Interval |
|---|---|---|
| b : c | 8 : 5 | minor sixth |
| c : d | 5 : 3 | major sixth |
| d : e | 3 : 2 | fifth |
| e : f | 3 : 2 | fifth |
| f : g | 4 : 3 | fourth |
| g : h | 3 : 2 | fifth |
These are also rhythms: f makes 4 orbits in the same time g makes 3, or a 4 against 3 polyrhythm.
Taking the root as C, the intervals produce this series, spread out over almost five octaves.
| Octave | Interval | Note |
|---|---|---|
| +2 | unison | C₅ |
| +1 | major third | E₄ |
| fifth | G₃ | |
| unison | C₃ | |
| -1 | fourth | F₂ |
| -1 | unison | C₂ |
| -2 | fourth | F₁ |
The orbits are slightly off from the perfect ratios, though close enough to be considered resonant. The exact ratios can be computed musically in cents.
| Exact Ratio in Cents | Error from Just Intonation | Error from Equal Temperment | ||
|---|---|---|---|---|
| 2,400 | + | 16.01 | 16.01 | 16.01 |
| 1,200 | + | 399.15 | 12.84 | -0.85 |
| 0 | + | 709.12 | 7.17 | 9.12 |
| 0 | + | 0.00 | 0.00 | 0.00 |
| -1,200 | + | 487.24 | -10.80 | -12.76 |
| -1,200 | + | -21.68 | -21.68 | -21.68 |
| -2,400 | + | 454.59 | -43.46 | -45.41 |
The top five orbits are very close to exact. Tuned to orbit e, orbit g is appreciably off by almost exactly a syntonic comma, 81/80, the smallest interval in 5-limit just intonation. Orbit h is flat by almost double that, almost half a semitone.
Fixing one orbit to some frequency, the exact frequencies of the others can be calculated.
| Orbit e as C₃ | Orbit e as A₂ |
|---|---|
| 528.111 | 444.087 |
| 329.466 | 277.046 |
| 197.033 | 165.685 |
| 130.813 | 110.000 |
| 86.666 | 72.877 |
| 64.593 | 54.316 |
| 42.523 | 35.758 |
Compared to instruments tuned in equal temperment, these frequencies are slightly spread: Upper tones a bit sharp, lower tones a little flat, the lowest by a fair bit.
This page was created as a resource for the March 16, 2017, music composition prompt in the Disquiet Junto project series. More information at disquiet.com/0272.
Update March 18, 2017: We learned from astronmer Michaël Gillon that the orbital period of the 7th planet, h has now been determined. It's resonant, too, and adds a very low note to our series!
Disquiet Junto: Marc Weidenbaum
Page & computations: Mark Lentczner
image credit: NASA/JPL-Caltech/R. Hurt, T. Pyle (IPAC)
The TRAPPIST-1 system: http://www.trappist.one/
NASA: https://exoplanets.nasa.gov/trappist1/
Wikipedia: https://en.wikipedia.org/wiki/TRAPPIST-1#Orbital_near-resonance