Frequency & Pitch

Sound is a mechanical wave — a travelling disturbance that compresses and rarefies the air (or any medium) as it moves. One of the most important properties of any wave is its frequency: the number of complete oscillation cycles that pass a fixed point every second. For sound waves, frequency has a direct perceptual consequence — it determines pitch. A high-frequency sound wave vibrates the air rapidly and is heard as a high-pitched note, while a low-frequency wave produces a deep, bass sound. This relationship is fundamental to music, speech, and hearing science. The human ear can detect sound frequencies roughly between 20 Hz and 20,000 Hz (20 kHz), a range called the audible spectrum. Below 20 Hz lies infrasound (felt rather than heard — think of a large earthquake); above 20 kHz lies ultrasound (used in medical imaging and sonar). Within the audible range, pitch perception is approximately logarithmic: doubling the frequency raises the perceived pitch by exactly one musical octave. This is why the piano key A4 vibrates at 440 Hz and A5 vibrates at 880 Hz — twice the frequency, one octave higher. Frequency is measured in hertz (Hz), defined as one cycle per second. It is related to the period T of the wave — the time for one complete cycle — by the simple reciprocal relationship f = 1/T. Understanding frequency and pitch connects physics to everyday experiences: the difference between a piccolo and a tuba, a soprano and a bass singer, or a dog whistle and a foghorn.

Key takeaways
  • What frequency actually measures
  • Period: the flip side of frequency
  • The wave equation: linking frequency, wavelength, and speed
  • Pitch: the perceptual result of frequency
  • Logarithmic perception and musical octaves

What frequency actually measures

Frequency f counts how many complete pressure cycles (compressions followed by rarefactions) pass a given point per second. If you watch a speaker cone, it pushes forward and then pulls back once per cycle. A speaker vibrating at 440 Hz completes 440 full back-and-forth motions every second. The unit hertz (Hz) is simply s⁻¹ — cycles per second, where 'cycle' is dimensionless.

Period: the flip side of frequency

The period T is the time for one complete oscillation. Frequency and period are exact reciprocals: f = 1/T and T = 1/f. A 440 Hz sound has a period of T = 1/440 ≈ 0.00227 s ≈ 2.27 ms. A 55 Hz bass note has T = 1/55 ≈ 18.2 ms — the low-pitched wave takes much longer per cycle. Period and frequency always move in opposite directions: increasing frequency means decreasing period.

The wave equation: linking frequency, wavelength, and speed

The speed of sound in air (approximately 343 m/s at 20 °C) is fixed by the medium's properties, not by the source. Once the frequency is set by the vibrating source, the wavelength λ adjusts automatically via v = fλ. A 440 Hz note therefore has wavelength λ = 343/440 ≈ 0.78 m. A 110 Hz note has λ = 343/110 ≈ 3.12 m — four times longer, matching the four-times lower frequency.

Pitch: the perceptual result of frequency

Pitch is a subjective, psychological quantity — it is what the brain infers from the physical stimulus of frequency. For pure tones in the mid-frequency range, pitch tracks frequency very closely. However, at extreme volumes or frequencies, or with complex tones, the perception can shift slightly. Importantly, pitch depends almost entirely on frequency, NOT on amplitude (loudness) or waveshape. This is why changing the volume of a note does not change its pitch.

Logarithmic perception and musical octaves

Human pitch perception is logarithmic in frequency. Equal ratios of frequency produce equal musical intervals. An octave always corresponds to a factor of 2 in frequency, regardless of the starting pitch. The twelve equal-tempered semitones in an octave each involve a frequency ratio of 2^(1/12) ≈ 1.0595. This means going from A4 (440 Hz) up one semitone to A♯4 gives 440 × 1.0595 ≈ 466 Hz. Physics provides the numbers; perception provides the music.

The Key Equation

v=fλv = f\lambda
  • v — Speed of sound (m s⁻¹): The speed at which the pressure disturbance propagates through the medium. In dry air at 20 °C, v ≈ 343 m s⁻¹.
  • f — Frequency (Hz (s⁻¹)): The number of complete wave cycles passing a fixed point per second. Determines the pitch of the sound.
  • \lambda — Wavelength (m): The distance between successive points of identical phase (e.g. compression to compression). Longer wavelength means lower pitch for a given wave speed.

Worked Example

Problem. A guitar string vibrates at 330 Hz (the musical note E4). The speed of sound in air is 343 m s⁻¹. (a) Calculate the wavelength of this sound wave in air. (b) Calculate the period of the oscillation.

λ≈1.04 m;T≈3.03×10−3 s\lambda \approx 1.04\text{ m};\quad T \approx 3.03 \times 10^{-3}\text{ s}

Answer. Wavelength ≈ 1.04 m; Period ≈ 3.03 × 10⁻³ s

Figure
Diagram
Frequency and Wavelength of Sound WavesHigh frequency(high pitch)λ shortLow frequency(low pitch)λ longf highf lowBoth waves travel at the same speed v in air.Higher frequency → shorter wavelength (v = fλ)Direction of propagation →
Origins · 1 minHermann von Helmholtz and how this was discovered
HV
Hermann von Helmholtz
1821–1894 · German

Helmholtz published his landmark work 'On the Sensations of Tone' (Lehre von den Tonempfindungen, 1863), which systematically established the physical and physiological basis of pitch perception. He demonstrated that the ear performs a form of frequency analysis — different regions of the basilar membrane in the cochlea respond to different frequencies — and rigorously connected the physics of vibrating systems to the psychology of musical pitch. His resonator devices, now called Helmholtz resonators, could isolate individual frequency components from complex sounds. Helmholtz built a set of brass resonating spheres (Helmholtz resonators) of carefully chosen sizes to identify pure frequency components in vowel sounds — effectively an analogue frequency analyser decades before electronic technology existed.

  • Amplitude & Loudness — Amplitude and frequency are the two fundamental properties of a sound wave; understanding both together gives a complete picture of how sound is perceived.
  • Wave Speed in Media — The wave equation v = fλ links frequency directly to wave speed; knowing how speed varies with medium explains why pitch is unchanged when sound travels between air and water.
  • The Doppler Effect — The Doppler effect describes how relative motion between source and observer changes the observed frequency — and therefore the perceived pitch — of a sound.
  • Period & Frequency of Oscillations — Sound frequency originates from the oscillating source; the period–frequency relationship T = 1/f is introduced in the study of simple harmonic motion and directly applies here.