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The Harmonic Series
Fundamental Frequency and Overtone Structure
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The harmonic series explains how one vibrating string or air column can produce many related frequencies at the same time. It is a core idea in both physics and music because it connects wave behavior to pitch, tone color, and resonance. When an instrument plays a note, the sound is usually not just one pure frequency but a blend of the fundamental and higher harmonics. This pattern helps explain why different instruments can play the same note yet sound different.
A standing wave forms when reflections in a string or air column fit the length of the system in specific ways. The lowest allowed frequency is the fundamental, and higher allowed frequencies are harmonics with frequencies that are whole-number multiples of the fundamental. For a string fixed at both ends, each harmonic adds more nodes and antinodes while keeping the same boundary conditions. In music, these harmonics shape timbre and also relate to intervals such as the octave, fifth, and major third.
Key Facts
- Harmonic frequencies follow fn = n f1, where n = 1, 2, 3, ...
- For a string fixed at both ends, lambda_n = 2L/n
- For a string, v = f lambda
- For a string under tension, v = sqrt(T/mu)
- The 2nd harmonic has frequency 2f1 and sounds one octave above the fundamental.
- The 3rd and 4th harmonics have frequencies 3f1 and 4f1, which help produce musical intervals above the fundamental.
Vocabulary
- Fundamental frequency
- The lowest natural frequency of a vibrating system and the first harmonic.
- Harmonic
- A frequency that is a whole-number multiple of the fundamental frequency.
- Standing wave
- A wave pattern that stays in place with fixed nodes and oscillating antinodes.
- Node
- A point on a standing wave that does not move.
- Timbre
- The characteristic sound quality of an instrument that depends on the mix of harmonics present.
Common Mistakes to Avoid
- Thinking harmonics are random extra notes, which is wrong because harmonic frequencies occur at whole-number multiples of the fundamental.
- Confusing pitch with timbre, which is wrong because pitch mainly depends on the fundamental frequency while timbre depends on the relative strengths of many harmonics.
- Using lambda = L/n for a string fixed at both ends, which is wrong because the correct standing-wave condition is lambda_n = 2L/n.
- Assuming a louder sound always has more harmonics, which is wrong because loudness is related to amplitude and not automatically to the number or pattern of harmonics.
Practice Questions
- 1 A guitar string has a fundamental frequency of 110 Hz. Find the frequencies of the 2nd, 3rd, and 4th harmonics.
- 2 A string fixed at both ends has length 0.80 m. Find the wavelength of the 1st harmonic and the 4th harmonic.
- 3 Two instruments play the same fundamental frequency, but one sounds bright and the other sounds mellow. Explain how differences in their harmonic content can cause this.