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Music & Acoustics

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

Equal-Temperament Note Frequency

NEWIntermediate
fn=f0×2n/12f_n = f_0 \times 2^{n/12}

Frequency of a note n semitones from a reference pitch.

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Octave Frequency Ratio

NEWBasic
f2=2f1f_2 = 2 f_1

An octave above a note has exactly double its frequency.

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Sound Intensity Level (dB)

NEWIntermediate
L=10log10 ⁣(II0)L = 10 \log_{10}\!\left(\frac{I}{I_0}\right)

Sound intensity expressed in decibels relative to the hearing threshold.

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Combining Two Sound Levels

NEWAdvanced
L=10log10(10L1/10+10L2/10)L = 10 \log_{10}(10^{L_1/10} + 10^{L_2/10})

Total level when two sound sources play together.

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Wave Speed

NEWBasic
v=fλv = f \lambda

Sound speed equals frequency times wavelength.

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Vibrating String Frequency

NEWAdvanced
f=12LTμf = \frac{1}{2L}\sqrt{\frac{T}{\mu}}

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

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Harmonic Series

NEWIntermediate
fn=nf1f_n = n f_1

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

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Pitch Interval in Cents

NEWAdvanced
c=1200log2 ⁣(f2f1)c = 1200 \log_2\!\left(\frac{f_2}{f_1}\right)

Measures the size of a musical interval in cents.

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Beat Duration from Tempo

NEWBasic
t=60BPMt = \frac{60}{BPM}

Length of one beat in seconds from the tempo.

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Delay Time for Tempo

NEWIntermediate
tms=60000BPMt_{ms} = \frac{60000}{BPM}

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

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Doppler Pitch Shift

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

Observed pitch changes as a source moves toward or away.

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Room Resonance (Axial Mode)

NEWAdvanced
f=c2Lf = \frac{c}{2L}

Lowest standing-wave frequency between two parallel walls.

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Frequency and Period

NEWBasic
f=1Tf = \frac{1}{T}

Frequency is the reciprocal of the wave's period.

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Open Pipe Fundamental

NEWIntermediate
f=v2Lf = \frac{v}{2L}

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

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Perceived Loudness Doubling

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

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

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Sabine Reverberation Time

NEWAdvanced
RT60=0.161VART_{60} = 0.161\frac{V}{A}

Time for sound to decay 60 dB in a room.

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Mel Scale (Pitch Perception)

NEWAdvanced
m=2595log10 ⁣(1+f700)m = 2595 \log_{10}\!\left(1 + \frac{f}{700}\right)

Converts frequency to the perceptual mel pitch scale.

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