The harmonic series
The Harmonic Series
The harmonic series is a sequence of frequencies or pitches that are integer multiples of a fundamental frequency. It is a natural phenomenon that forms the basis of how we perceive musical tones and harmonies. The harmonic series explains why instruments and voices have unique timbres and why certain combinations of notes sound pleasing together.
How It Works
- Fundamental Frequency
The lowest frequency in the series is called the fundamental. It defines the pitch we perceive.
Example: If the fundamental is 100 Hz, all other frequencies in the series are derived from it.
- Overtones (Harmonics)
These are the higher frequencies in the series, which are multiples of the fundamental.
For a fundamental frequency f, the harmonic series includes:
f, 2f, 3f, 4f, 5f, …
If f = 100 Hz, then:
- 1st harmonic (fundamental): 100 Hz
- 2nd harmonic: 200 Hz
- 3rd harmonic: 300 Hz
- 4th harmonic: 400 Hz
- ... and so on.
- Relative Strengths
In real sounds, the harmonics are not equally loud. Their relative strengths determine the timbre of an instrument or voice.
Musical Implications
- Intervals in the Harmonic Series
The harmonics correspond to musical intervals:
- 1st harmonic: Unison (same note)
- 2nd harmonic: Octave
- 3rd harmonic: Perfect fifth above the octave
- 4th harmonic: Another octave
- 5th harmonic: Major third above the second octave
- ...
This pattern of intervals gives rise to the tonal relationships we use in music.
- Timbre
Different instruments emphasize different harmonics, which is why a violin sounds different from a trumpet, even when playing the same note.
- Tuning and Scales
The harmonic series forms the foundation for just intonation, a tuning system based on simple ratios derived from the series. However, modern music often uses equal temperament, which slightly adjusts these intervals for consistency across all keys.
Example: Harmonic Series for A = 440 Hz
- 1st harmonic: 440 Hz (A)
- 2nd harmonic: 880 Hz (A, 1 octave higher)
- 3rd harmonic: 1320 Hz (E, perfect fifth above 880 Hz)
- 4th harmonic: 1760 Hz (A, 2 octaves higher)
- 5th harmonic: 2200 Hz (C♯, major third above 1760 Hz)
- 6th harmonic: 2640 Hz (E, perfect fifth above 1760 Hz)
In Summary
The harmonic series is a naturally occurring sequence of frequencies that shapes our understanding of pitch, tone, harmony, and musical structure. It underpins the science of sound and forms the foundation for much of Western music theory.