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How do frequency ratios such as 2:1 and 3:1 occur in the harmonic series?

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musical sound The universality and innate nature of music, and the pentatonic scale, as demonstrated by singer Bobby McFerrin.

In the context of musical sound, frequency ratios such as 2:1 and 3:1 arise from the physical properties of standing waves on a vibrating medium, such as a stretched string.

When a string vibrates, it produces a fundamental frequency ( $f_{1}$ ), which is the lowest frequency of the vibration. Additional standing waves, known as harmonics, occur as integral multiples of this fundamental frequency ( $f_{n} = n \times f_{1}$ ). These harmonics are created when the string vibrates in equal segments called loops:

2:1 Ratio: The second harmonic ( $n = 2$ ) occurs when the string vibrates in two sections. Because the wavelength is halved, the frequency is exactly twice that of the fundamental, creating an octave.
3:1 Ratio: The third harmonic ( $n = 3$ ) occurs when the string vibrates in three sections. Its frequency is three times that of the fundamental.

These relationships demonstrate that the harmonic series is composed of frequencies that are integer multiples of the fundamental, providing the mathematical basis for musical intervals and harmony.1