Music & Acoustics

The physics of sound: note frequencies, decibels, harmonics, tempo and the Doppler pitch shift

Music & Acoustics brings together the core formulas behind how sound behaves, from note frequencies, harmonics and tuning systems to decibels, wave speed, tempo, beats and Doppler pitch shifts. It’s useful for students, exam practice and anyone working with audio, music, acoustics or engineering, because these formulas help you calculate, compare and predict sound in real-world situations.

Equal-Temperament Note Frequency

NEWIntermediate

f_n = f_0 \times 2^{n/12}

Frequency of a note n semitones from a reference pitch.

Octave Frequency Ratio

f_2 = 2 f_1

An octave above a note has exactly double its frequency.

Sound Intensity Level (dB)

NEWIntermediate

L = 10 \log_{10}\!\left(\frac{I}{I_0}\right)

Sound intensity expressed in decibels relative to the hearing threshold.

Combining Two Sound Levels

L = 10 \log_{10}(10^{L_1/10} + 10^{L_2/10})

Total level when two sound sources play together.

Wave Speed

v = f \lambda

Sound speed equals frequency times wavelength.

Vibrating String Frequency

f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}

Fundamental pitch of a string from length, tension and mass.

Harmonic Series

NEWIntermediate

f_n = n f_1

Overtones of a note are whole-number multiples of the fundamental.

Pitch Interval in Cents

c = 1200 \log_2\!\left(\frac{f_2}{f_1}\right)

Measures the size of a musical interval in cents.

Beat Duration from Tempo

t = \frac{60}{BPM}

Length of one beat in seconds from the tempo.

Delay Time for Tempo

NEWIntermediate

t_{ms} = \frac{60000}{BPM}

Milliseconds per beat, used to sync echo/delay effects to a song.

Doppler Pitch Shift

f' = f\,\frac{v + v_o}{v - v_s}

Observed pitch changes as a source moves toward or away.

Room Resonance (Axial Mode)

f = \frac{c}{2L}

Lowest standing-wave frequency between two parallel walls.

Frequency and Period

f = \frac{1}{T}

Frequency is the reciprocal of the wave's period.

Open Pipe Fundamental

NEWIntermediate

f = \frac{v}{2L}

Lowest note of a pipe open at both ends (like a flute).

Perceived Loudness Doubling

\Delta L = 10\,\text{dB}

A 10 dB increase is heard as roughly twice as loud.

Sabine Reverberation Time

RT_{60} = 0.161\frac{V}{A}

Time for sound to decay 60 dB in a room.

Mel Scale (Pitch Perception)

m = 2595 \log_{10}\!\left(1 + \frac{f}{700}\right)

Converts frequency to the perceptual mel pitch scale.