Beating & Roughness

Beating is a physical interference pattern: two nearby frequencies sum into a fast oscillation whose loudness rises and falls. Roughness is the perceptual consequence when those fluctuations become too fast to hear as separate pulses and are instead heard as a harsh, buzzing texture.

🎯 Simple version: If two tones are very close in pitch, they make a “wah-wah” pulse. As that pulse speeds up, it stops sounding like separate beats and starts sounding rough.

Physical Beating

For two pure tones at f1 and f2, the summed waveform is:

cos(2πf₁t) + cos(2πf₂t) = 2cos(2π·(f₁-f₂)/2·t)·cos(2π·(f₁+f₂)/2·t)

• The fast factor ((f₁+f₂)/2) is the carrier: the average pitch region.
• The slow factor ((f₁-f₂)/2) is the envelope term.
• Envelope peaks repeat at |f1 - f2| per second, so beat rate is |f1 - f2| Hz.

From Beating to Roughness

At very small separations, the envelope is slow and you hear distinct beats. Around ~15-20 Hz, the auditory system stops tracking each pulse as a separate event. The sensation shifts from rhythmic fluctuation to roughness: a continuous rasping quality.

For pure tones, peak roughness appears near ~25% of the critical bandwidth (Plomp-Levelt region). As separation approaches and then exceeds the full critical bandwidth, roughness drops because the cochlea resolves the tones into separate channels. Past that point, you hear two tones rather than one rough fused event.

The Critical Bandwidth Connection

Critical bandwidth is the cochlear filter width for a local frequency region (see ear-cochlea.md). A useful rule in this register is ~10-20% of center frequency:

• Center frequency: fc = (f1 + f2) / 2
• Approximate critical bandwidth: CB ≈ 0.15 × fc

Because CB scales with frequency, the same Hz separation can be rough in one register and cleanly separated in another. This is why interval color depends on register, not just interval name.

The Interactive

Use the explorer to move from slow beating to roughness and then to two separate tones.

Implications

• Close intervals can sound rough not because of abstract interval labels, but because many component pairs land inside the same cochlear filter.
• Register matters: lower tones have narrower relative spacing demands, so nearby components interact more strongly.
• Timbre matters: bright spectra add more nearby partial pairs, which can increase roughness at the same pitch interval.
• Beating is physical and measurable; whether a listener treats the result as desirable texture or unwanted harshness is a separate aesthetic layer.

Translation Table

Connections

• Consonance & Dissonance — places roughness inside a broader multi-factor perception model
• The Ear & Cochlea — source of critical bandwidth and frequency-channel limits
• Fourier Analysis — explains why harmonic component spacing predicts roughness behavior
• Timbre — shows how harmonic density changes roughness at fixed pitch spacing