Monday, August 3, 2020

Orbital Resonances and Musical Harmonies

This week, our summer undergraduate researcher, Simon Friesen, takes a step back from pondering the dusty skies of Mars to consider the Music of the Spheres. Music has long been associated with mathematics and the physical sciences. I know more than a few researchers who started out studying music only to later find themselves drawn to the harmonies of nature. This includes past PVL researchers as well as a few people with whom I went to graduate school. No talent or study ever goes to waste and there are surprising patterns that arise again and again in different fields. One such variation on a theme in planetary science takes place when considering orbital resonances and how this data can be represented musically.

by Simon Friesen

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