In Search of the Missing Fundamental: by Richard K. Jones

# Harmonic Partials

The second harmonic partial has half the wavelength and twice the frequency of the first. The third harmonic partial has one-third the wavelength and three times the frequency of the first. The fourth harmonic partial has one-quarter the wavelength and four times the frequency of the first, and so on. The fourth harmonic partial has twice the frequency of the second harmonic, and the sixth harmonic partial has twice the frequency of the third harmonic partial.

Not every pitch contains every harmonic partial. If that were the case then every pitch would have the same timbre or tone color, which wouldn’t very interesting. Timbre is what gives pitch its personality.  When these harmonic partials or individual sine waves are added together they create a composite harmonic waveform.

Figure 1c shows how a specific type of composite harmonic wave form with a specific timbre, the square wave, results if a sine wave and several odd numbered harmonic partials are added together. (a) just the fundamental note (so the sum is just a sine wave, 1p), (b) the fundamental and third harmonic (1p +3p), (c) the fundamental and odd harmonics up to the 9th (1p + 3p + 5p + 7p +9p). The square wave happens to contain only odd harmonics. The more odd harmonics that are included, the closer the result approximates to a square wave.

Fig. 1c

Figure 1d is an animation showing the additive synthesis of a square wave with an increasing number of odd harmonics up to the 25 partial.

Fig. 1d

The audio file below is an example of what the square wave sounds like when it is complete.

The pitch or frequency is the same but, the timbre or tone color is different.

Figure 1e is a short video courtesy of which discusses additive synthesis, and shows how sounds can be constructed by adding  together pure sine waves.

Fig. 1e

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