Orbital resonances are found throughout the solar system
(and there are even some examples found between exo-planets and their stars).
There are two main types of orbital resonances: unstable resonances, which clear objects at specific radii from the
parent body (think gaps between the rings of Saturn caused by its inner moons);
and mean-motion resonances which
maintain and self-correct the orbits of the bodies involved.
For the purposes of this exploration, I want to focus on the
stable orbits between planets or moons in the solar system. Three of the moons
of Jupiter – Ganymede, Europa, and Io – are in a stable ratio of orbits of
1:2:4 (specifically, for every single orbit that Ganymede completes, Europa
will complete two orbits and Io; four). These types of orbital resonances are
somewhat rare in our solar system, but do include the following notable
examples: Pluto – Neptune, 2:3; Tethys – Mimas (moons of Saturn), 2:4; Dione –
Enceladus (other moons of Saturn), 1:2; Hyperion – Titan (further moons of
Saturn), 3:4; Haumea – Neptune (suspected), 7:12; 225088 Gonggong – Neptune,
3:10; and Pluto’s moons, Styx – Nix – Hydra, have a ratio of 11:9:6. We will
also take a look at orbital resonances found in distant exo-planet systems
later.
The term resonance
is also a common feature of another interest of mine: music. The exploration of
some of the mathematical formalism behind harmonious sounds was done by
Pythagoras, who found that sounds based frequencies that were small number
ratios of each other sounded consonant. This work laid the foundation for
myriad western tuning systems and western harmonies. Pythagoras found that
frequencies of ratios 1:2, 2:3, and 4:3 sounded best together. To try this out
for yourself, find an online tone generator and open up two copies in different
tabs. For a 2:1 ratio, try 440
Hz (A4 on the piano; I’ll use this as a reference tone for all the other
harmonies we’ll explore later) and 880 Hz. This is known in
western music as the octave. For 2:3, use 440 Hz and 660 Hz. This is called an
interval of a fifth. For 4:3, use 440 Hz and 586.67 Hz. This
interval is known as a fourth. These Pythagorean intervals can be used to
construct the twelve notes in the Pythagorean scale. This scale, and many other
scales that came after it, suffers from the circle of fifths not being closed
and from certain terrible-sounding intervals called wolf intervals.
Multiple variations on western tuning systems have been developed
since Pythagoras’ day including, but not limited to: Just Intonation tuning, which created better sounding thirds at a
ratio of 4:5 (try 440 Hz
and 550 Hz); Meantone scales, which preserved nice
thirds but had slightly worse fifths (compare 440 Hz and 657.95 Hz to the
previous 660 Hz); Well Temperament, favoured by Bach and granting each key
signature its own distinct character; and 12-Tone
Equal Temperament (12-TET), which spreads all 12 notes in an octave logarithmic-evenly across the octave. I encourage you to look these scales
up in more detail; each one has strengths and weaknesses. I’ll reference these
scales later when looking for the closest match to intervals created from
orbital resonances.
We’ve established that scales and tuning systems can be
created using ratios of frequencies, and that orbital resonances occur within
our solar system that are based on ratios of orbits. I want to explore what
some of these ratios sound like. From previous; Ganymede, Europa, and Io are in
stable orbit ratios of 1:2:4 respectively. This is exactly the same as
Pythagoras’ octave; try it out using 440 Hz, 880 Hz and 1760 Hz. Similar to this,
Tethys and Mimas have a ratio of orbits of 2:4 (880 Hz and 1760 Hz); and Dione and
Enceladus have a ratio of 1:2 (back to 440 Hz and 880 Hz). Pluto and Neptune
have a stable ratio of orbits of 2:3, the same as Pythagoras’ interval of a
fifth (440 Hz and 660 Hz). Hyperion and
Titan have a ratio of orbits of 3:4, giving us the same interval as a
Pythagorean fourth (440 Hz
586.67 Hz).
Haumea and Neptune have a supposed ratio of orbits of 7:12. To
hear this, use 440 Hz,
and 440 * (12/7) = 754.28
Hz. This harmonic interval is closest a 12-TET 6th interval (use
n = 9 in the formula 440 * 2n/9 = 739.99 Hz; to hear it, use 440 Hz and 739.99 Hz). Does one
interval sound better to you?
Hydra, Nix, and Styx have a ratio of orbits of 6:9:11. 6:9
is the same as 2:3, so the interval is a Pythagorean fifth. To find the
frequency of the highest pitch, we use: 440 * 11/6 = 806.67 Hz. This is closest
to a Just major seventh interval, but is close to a quarter tone flatter.
Putting all three tones together (440 Hz, 660 Hz, and 806.67 Hz) creates
something that you might describe as an A major 7th suspension
chord, but is really outside of western tonality.
Orbital resonances have also been found in exo-planets and
extra-solar systems. Though it is still a rare phenomenon, some of these
extra-solar systems also exhibit longer chains of orbital resonances than are
found in our solar system.
On the simple end of things, Kepler-29 has an observed 7:9
resonance between a pair of planets. Translating this to our 440 Hz reference pitch
gives us 440 * (9/7) = 565.71
Hz. This pitch is close to both a Pythagorean and a 12-TET major third.
Kepler-37 has three planets in ratios of 8:15:24. Referencing 440 Hz we find the upper
two pitches to be 440 * (15/8) = 825 Hz, and 440 * (24/8) =
1320 Hz (I will omit
calculations from here on; all further frequencies will be created from ratios
in a similar manner). The middle tone is a Just major seventh, and the high
pitch is an octave plus a fifth, giving us another A major 7th
suspended chord. Kepler-233 has a series of planets in a 3:4:6:8 ratio of
orbits (frequencies 440 Hz,
586.67 Hz, 880 Hz, and 1173.33 Hz). This
chord is only made of octaves and fourths, so we might consider it an A-suspended 4th chord.
Kepler-80 has six planets in a 4:6:9:12:18 ratio (440 Hz, 660 Hz, 990 Hz, 1320 Hz, and 1980 Hz). This chord is
made of a fifth, a second (up an octave, in tune with the Pythagorean and Just
scales), and two further fifths in upper octaves. This chord would then be a
stellar example of an A major suspended second. Kepler-90 has six planets in a
ratio close to (but not exactly) 2:3:4:7:11 (440 Hz, 660 Hz, 880 Hz, 1540
Hz, and 2420 Hz).
This chord consists of a fifth, an octave, a high sharp major second, and a
high sharp minor seventh, making this some strange variation on an A 9th
suspended 2nd, but is not really part of any scale system. Finally,
TRAPPIST-1 has an astounding 7 planets in a ratio close to 2:3:4:6:9:15:24 (440 Hz, 660 Hz, 880 Hz, 1320 Hz, 1980 Hz, 3300 Hz, and 5280 Hz (feel free to
skip this one)). This chord consists of a fifth, an octave, a high fifth, a
high Pythagorean second, and an extremely high fifth, giving us an A major
suspended 2nd chord.
And so, we now have a wider understanding of both western
harmonies and resonances between orbiting objects in and beyond our solar
system. I had expected to find many more notes that did not fit neatly into
scales, but surprisingly only Kepler-90; pluto’s moons Hydra, Nix, and Styx;
Haumea and Neptune; and Kepler 29 produced notes not included from our
reference scales. The universe seems to doubling or 1.5-factor periods in
orbits. Some of the intervals outlined here could inform new tuning systems or
harmonies to explore musically.
Via Wikipedia: "Sequence of conjunctions of
Hydra (blue), Nix (red) and Styx (black) over one third of their
resonance cycle. Movements are counterclockwise and orbits completed are
tallied at upper right of diagrams (click on image to see the whole
cycle)." Photo: By WolfmanSF - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=4175932