Musical instruments - beats phenomenon

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Discussion Overview

The discussion centers on the beats phenomenon associated with musical instruments and waves in general. Participants explore the mathematical representation of beats and seek clarification on the concept.

Discussion Character

  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant requests a description of the beats phenomenon, expressing difficulty in understanding the concept despite consulting textbooks.
  • Another participant provides a mathematical explanation of beats, describing the interference of two sinusoidal signals with slightly different frequencies and how this leads to a modulation of amplitude perceived as beats.
  • A third participant offers a correction regarding the mathematical representation of the waveform, specifying that the correct form includes the factor of 2π.
  • A fourth participant shares a link to a plot that visually represents the concept discussed, illustrating the interference of two sine waves with slightly different frequencies.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the understanding of the beats phenomenon, as one participant expresses confusion while others provide mathematical insights and corrections.

Contextual Notes

The discussion includes varying levels of mathematical detail and assumes familiarity with wave functions and sinusoidal signals, which may not be universally understood among all participants.

CAF123
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Can anyone give me a description of the beats phenomenon associated with musical instruments (or in general, any waves with which beats are associated).
I have looked at numerous textbooks however I feel I don't completely understand the concept still.
Thanks
 
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Suppose you are hearing two sinusoid signals of slightly different frequencies f and f'. Say the phases are such that you are hearing the time-dependent waveform f(t) = sin(f*t)+sin(f'*t).

Mathematically, this is the same as

f(t) = 2 * sin(0.5*(f+f')*t) * cos(0.5*(f-f')*t)

Since f-f' is small, the cos() will be a slowly varying function of time, while the sin() will vary quickly in time. The cos() will be heard as beats, i.e. a slowly varying amplitude of the quickly oscillating signal with frequency 0.5*(f+f') which is the average of f and f'.

It will be very similar for real signals cantaining harmonics.
 
Just a small correction, if frequency is f, the waveform is sin(2π f t).
 

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