• Linear Superposition
• Beats
• Combination Tones
• Other Nonlinear Effects
• Phase and Timbre

Combination Tones

This is a ``self-study'' section. I'll just provide a brief outline here, emphasizing the topics that I think you should most carefully consider.

Linear Superposition

• The response of a linear system to multiple inputs is just the sum of the individual responses.
• N different frequency inputs will produce a response given by the same N frequencies (with the possibility of total cancellation of same-frequency components).
• Two simple harmonic waveforms of the same frequency will add to produce a single harmonic waveform of the same frequency. The resulting amplitude and phase will depend on the individual amplitudes and phases of the two waveforms.

Beats

• When two pure tones of slightly different frequency are superposed, our ears perceive audible beats at a rate given by the difference of the two frequencies.
• Beats are heard as a pulsation in the loudness of a tone at the ``average'' frequency $f = 0.5 (f_{1} + f_{2})$.
• When the difference frequency exceeds about 15 Hz, the beat sensation disappears and a characteristic roughness of sound appears.
• As the difference frequency is increased further, a point is reached at which the ``fused'' tone at the average frequency gives way to two tones, though still with roughness. At this point, the resonance regions on the basilar membrane are sufficiently separated to produce two distinct pitch signals, but these regions still overlap to produce roughness.
• Finally, when the difference frequency exceeds the critical bandwidth, the roughness disappears completely.

Combination Tones

• When two tones are sounded together, a tone of lower frequency is frequently heard. Such a tone is called a Tartini tone or combination tone.
• The most commonly heard combination tone occurs at a frequency $f_{2} - f_{1}$ (or $f_{1} - f_{2}$).
• Other combination tones can occur at frequencies given by $2 f_{1} - f_{2}$, $3 f_{1} - 2 f_{2}$, ..., $f_{1} - k(f_{2} - f_{1})$, though these are much less common.
• Combination tones expose nonlinearities in our auditory system.

Other Nonlinear Effects

• A single, loud tone of frequency $f$ should produce additional pitch sensations at $2f, 3f, 4f$, etc. These are called aural harmonics.
• A nonlinear mechanism in the ear should result in summation, as well as difference, tones. Perception of such higher frequency tones, however, has yet to be adequately demonstrated.

Phase and Timbre

• The human auditory system is a relatively poor detector of phase.
• Sounds with equal harmonic amplitude but varying phase, however, can be distinguished under certain circumstances.
• Cyclical variations in phase, which occur in the case of mistuned consonances, can be heard as beats.