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