Harmonic Partials and Timbre
Each harmonic partial in a vibrating system has a frequency that is a whole-number multiple of the fundamental. As the harmonic number increases, the frequency rises and the wavelength shortens: 4 2 9
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The 2nd harmonic has twice the frequency and half the wavelength of the fundamental (1st harmonic).
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The 3rd harmonic has three times the frequency and one-third the wavelength, and so on.
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For example, the 4th harmonic is double the frequency of the 2nd, and the 6th is double the frequency of the 3rd.
However, not all pitches contain every harmonic partial, and the relative strength of each harmonic varies. If all pitches had identical harmonic content, they would all sound the same, a world with no difference in tone color. Fortunately, this is not the case. The unique blend of harmonics in a sound is what gives it its timbre, its distinct personality or tone quality.
Additive Synthesis: Building Complex Tones
When multiple sine waves (each at a harmonic frequency) are added together, they form a composite waveform. This process is called additive synthesis, and it’s a powerful way to understand how tone color is shaped by harmonic content.
Figure 1c: Square Wave Construction
The square wave is a classic example of a sound that includes only the odd-numbered harmonics (1st, 3rd, 5th, etc.). The more odd harmonics we add, the closer the waveform approximates an ideal square shape.
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(a) Only the fundamental — results in a pure sine wave.
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(b) Fundamental + 3rd harmonic — creates a richer but still smooth waveform.
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(c) Fundamental + 3rd + 5th + 7th + 9th — begins to resemble a square wave.
The square wave has the same fundamental frequency (i.e., pitch) as a sine wave, but sounds very different due to its richer harmonic content.
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
Audio Examples: Hear the Difference
Below is an audio sample of a completed square wave. It contains the same fundamental frequency as a sine wave but has a completely different timbre. This demonstrates that pitch remains constant, while tone color changes with harmonic content.
Try it: Listen to the difference between a sine wave (pure tone) and a square wave (rich tone). Notice how they have the same pitch, but evoke completely different sound qualities.
The pitch or frequency is the same but, the timbre or tone color is different.
Figure 1e is a short video courtesy of SynthSchool which discusses additive synthesis, and shows how sounds can be constructed by adding together pure sine waves.
Fig. 1e