How are harmonics related to exact multiples of a fundamental frequency?

In the context of musical sound, harmonics are defined by their relationship to the fundamental frequency, which is the lowest frequency produced by a vibrating source such as a string or air column.12

A harmonic is any frequency that is an exact integer multiple of this fundamental frequency. The fundamental frequency itself is considered the first harmonic. Subsequent harmonics are produced when the vibrating medium divides into equal sections, resulting in higher frequencies that are whole multiples of the fundamental (e.g., the second harmonic is twice the fundamental frequency, the third is three times, and so on).12

These harmonics form simple ratios with the fundamental frequency, such as 2:1 or 3:1, which contributes to the perceived quality or timbre of a musical tone. While many instruments produce these harmonic series, some sources—such as drums, cymbals, or certain bars—produce nonharmonic overtones, where the frequencies are not exact multiples of the fundamental.1