When two bodies share a star, their orbital periods tend to settle into small whole-number ratios. Those same ratios are the consonant intervals of music — a fact needs no tuning fork, because a ratio carries no arbitrary reference pitch. Here each one is drawn as the curve it traces and played as the chord it is. Click any panel to hear it.
Featured trace — the golden ratio, 1.618 : 1 — the curve that never closes
Feed one frequency to the horizontal axis and a second to the vertical, and the beam draws a Lissajous figure. Whole-number ratios close into a steady loop. Irrational ratios — like φ — never repeat, so the trace keeps wandering and filling the field forever.
That endlessly-wandering curve is the visual twin of an orbit that never falls into resonance: no repeating alignment, no periodic kick. In celestial mechanics that turns out to be the most stable arrangement of all.
click the curves below ↓Real pairs of moons, planets, and minor bodies that lock into each ratio. The interval name and its size in cents are exact — derived from the ratio alone, the same numbers a piano tuner uses.
The golden ratio is the most irrational number — the hardest of all to approximate with a fraction. Sounded as a dyad it never settles into a beat pattern; drawn as a curve it never closes.
In orbital dynamics this is the punchline (the KAM theorem): the orbit whose period ratio is golden is the last one to break apart under gravitational tugging, because it never lines up the same way twice and so never gets a repeated push. The most dissonant ratio is the most durable orbit.