If you were to remove all the harmonics from a sound,

Question

BlackTom AI · Accepted Answer

A pure tone is a single-frequency sinusoidal sound (one frequency with no additional overtones), so removing all harmonics leaves only that single frequency.

1. Relevant knowledge points and concepts
- Any periodic sound can be represented as a sum of sinusoidal components (Fourier series): a fundamental frequency plus harmonics (integer multiples of the fundamental).
- Harmonics (often called overtones) are the additional frequency components that determine timbre; the fundamental sets perceived pitch.
- A pure tone is a single sinusoid (one frequency component) and therefore has no harmonics or overtones.

2. Step-by-step reasoning
- Start with a complex sound: it contains a fundamental frequency f0 and harmonic components at 2f0, 3f0, 4f0, etc.
- “Removing all the harmonics” means eliminating those additional frequency components while leaving the fundamental (the phrasing in common usage treats harmonics/overtones as the extra components).
- After removal, only the single remaining component is the fundamental frequency, so the waveform is a single sinusoid.

3. Calculations / intermediate steps
- Represent sound as S(t) = A0 sin(2πf0t) + Σ An sin(2πn f0 t) for n ≥ 2.
- Removing harmonics sets An = 0 for n ≥ 2, leaving S'(t) = A0 sin(2πf0t), which is a pure tone (single frequency).

4. Why the correct answer is correct
- The resulting waveform S'(t) has only one frequency component; by definition this is a pure tone. Timbre is removed, leaving only pitch.

5. Why the other options are incorrect
- Option b (“you would hear no sound at all”): false unless you also remove the fundamental; removing only harmonics does not eliminate the fundamental, so a tone remains.
- Option c (“you would be left with a tone of multiple frequencies”): contradictory to the premise—removing harmonics removes additional frequencies, so you do not have multiple frequencies left.

6. Conclusion
- Therefore, removing all harmonics leaves a single-frequency sinusoid — a pure tone. This matches the provided correct answer: a. you would be left with a pure tone.

If you were to remove all the harmonics from a sound,单项选择题

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