The Music of the Spheres
This is sand on a vibrating square plate, gathering into the lines that stay still. Each planet's figure is tuned to the musical interval Johannes Kepler heard in its orbit: Mars, whose speed around the Sun swings between two notes a fifth apart, plays the 2 and 3 of that fifth. Watch the sand run off the moving parts and settle into the pattern.
| Mars, a fifth | Kepler 2 : 3 (1.50), real 1.45 |
|---|---|
| Mercury | Kepler 5 : 12 (2.40), real 2.30 |
| Kepler's error | Within 5% for every planet |
| Grains of sand | 14,000 to 40,000 |
What you are looking at
In 1787 Ernst Chladni scattered sand on metal plates, held them at the centre and drew a violin bow along the edge. Each note set the plate ringing in its own pattern, and the sand ran off the parts that moved and gathered on the lines that did not. Those lines are the plate's nodal lines, and the figures are named after him.
Here the sand starts scattered evenly and gathers over a few seconds, still shivering a little as the plate rings. The label under the plate names the planet, Kepler's ratio for it and the musical interval it makes. Mars draws a bold cross with arches on every side. Saturn and Jupiter draw stepped diamonds. Mercury draws a field of rings. The Earth and Venus, whose intervals are almost a unison, draw a fine even weave.
The math
The plate follows the standard approximation for a square plate's modes, f(x, y) = cos(nπx) cos(mπy) minus cos(mπx) cos(nπy), on a square running from minus 1 to 1 each way. The sand lies where f is zero. The diagonals are always among those lines, and the pattern has four-fold symmetry, like a plate held at its centre. It is a good likeness of a real plate, not an exact one: a steel plate's true modes bend these lines a little.
In his Harmonices Mundi of 1619, Kepler took each planet's slowest and fastest daily motion around the Sun and heard their ratio as a musical interval: Saturn 4 to 5, a major third; Jupiter 5 to 6, a minor third; Mars 2 to 3, a fifth; the Earth 15 to 16, a semitone, which he wrote as mi, fa; Venus 24 to 25, a diesis; and Mercury 5 to 12, an octave and a minor third. The real ratio follows from each orbit's shape, ((1 + e) / (1 minus e)) squared, where e is the eccentricity. It gives 1.45 for Mars against Kepler's 1.50, 2.30 for Mercury against 2.40 and 1.07 for the Earth against 1.07: his numbers are within 5 percent every time.
How it is computed
Turning an interval into a plate figure is our choice, not Kepler's: the two numbers of each planet's ratio become the plate's two mode numbers, n and m. The ratios are his. The closer the two numbers, the finer and more even the weave, which is why the near-unisons of the Earth and Venus look like cloth.
The sand is 14,000 grains for Mars and up to 40,000 for the finest plates, scattered the same way every time so a picture can be shared exactly. Before anything is drawn, each grain is walked onto its nearest still line by a few steps of Newton's method, each step kept shorter than a fraction of a cell so a grain cannot jump to the wrong line. Each frame then places every grain partway between where it started and where it settles, with a shiver that dies away, so the slowest grain is within 1 percent of its line after nine and a half seconds.
Questions
What are Chladni figures?
The patterns sand makes on a vibrating plate. The sand is thrown off the parts of the plate that move and collects on the lines that stay still. Ernst Chladni published them in 1787, and Sophie Germain won the French Academy's prize in 1816 for the first mathematical theory of how such plates vibrate.
What did Kepler mean by the music of the spheres?
That the planets' motions follow musical ratios. In Harmonices Mundi he showed that each planet's fastest and slowest speed around the Sun stand in a simple musical interval, and that the planets together sound chords. The same book contains his third law of planetary motion.
Is the link between the intervals and the plates real?
The intervals are Kepler's and the plates are real physics, but joining them is our artistic choice: we use the two numbers of each interval as the plate's two mode numbers. A real plate does not know which planet it is playing.