TRAPPIST-1 Planetary Intervals

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 1.510924
c 2.421815
d 4.04969
e 6.09966
f 9.20674

The ratios of the orbital frequencies for neighboring pairs of planets form a series of perfect intervals.

Planets Ratio Interval
b : c8 : 5minor sixth
c : d5 : 3major sixth
d : e3 : 2fifth
e : f3 : 2fifth
f : g4 : 3fourth
g : h3 : 2fifth

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
+1major thirdE₄

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

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)