Irrational frequency ratios make Lissajous curves rotate through all phase shapes

Evidence

If the ratio cannot be expressed so simply, the curve traced out appears to be slowly rotating through different shapes it attains with various relative phases of a nearby integer ratio. For example, if the ratio is 3.073317:4, the curve will very nearly trace out all the possible phases of the exact 3:4 case. These considerations lead us down the path of number theory, rational approximations to real numbers, and so on, which is a beautiful but very deep subject. — Eric J. Heller, Why You Hear What You Hear- An…, 318

Categorization

• QUESTION: The offhand mention that Lissajous curves lead into number theory and rational approximation of irrationals is a tangent worth chasing — how exactly does the near-miss ratio trace nearly-all phases?
• SHARE: Nerdy enough that only someone into acoustics or mathematical visualization would enjoy it; a math-minded friend might appreciate the rational-approximation connection.