Pairs of tones with irrational frequency ratios never exactly repeat their combined shape.

Evidence

Suppose we have two frequencies, 77 Hz and 109 Hz (figure 3.20). Technically, both of these are different multiples of the same fundamental frequency—namely, 1 Hz. … The two frequencies are related by the irrational number ; the shape of this combination of only two sinusoids never exactly repeats itself! The GCD of 77 and does not exist. Irrational combinations of frequencies are not available in Fourier, but MAX Partials1 will give any combination quite accurately. The combination 77 Hz + 109 Hz is periodic, but not audio-periodic. To the ear, it is indistinguishable from 77 Hz + 108.89444 … Hz, which is not periodic at all. — Eric J. Heller, Why You Hear What You Hear- An…, 354

Categorization

• QUESTION: Does the ear genuinely hear two incommensurate tones as non-periodic, or does the auditory system forcibly harmonicize them? This connects to psychoacoustic beating and roughness research.
• XREF: Irrational frequency combinations producing never-repeating waveforms connects to the mathematical idea of quasiperiodicity and to what composers like Xenakis exploited.