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Problem: You know that certain musical notes sound good together -harmonious - whereas others do not. This harmony is related to the various harmonics of the notes played. The musical notes C (262 Hz) and G (392 Hz) make a pleasant sound when played together; we call this consonance. As the Figure shows, the harmonics of the two notes are either far from each other or very close to each other (within a few Hz). This is the key to consonance: harmonics that are spaced either far apart or very close. The close harmonics have a beat frequency of a few Hz that is perceived as pleasant. If the harmonics of two notes are close, but not too close, the rather high beat frequency between the two is quite unpleasant. This is what we hear as dissonance. Exactly how much a difference is maximally dissonant is a matter of opinion, but harmonic separations of 30 - 40 Hz seem to be quite unpleasant for most people.A. An organ pipe open at both ends is tuned so that its fundamental frequency is G. How long is the pipe? Show your work. B. If the C were played on an organ pipe that was open at one end and closed at the other, which of the harmonic frequencies in the Figure would be present? Show your work.

FREE Expert Solution

The fundamental frequency of an open pipe:

n(G)=v2l, where v is the velocity of sound in air (v = 343 m/s is assumed if not otherwise given) and l is the length of the air column.

A.

Solving for l:

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Problem Details

You know that certain musical notes sound good together -harmonious - whereas others do not. This harmony is related to the various harmonics of the notes played. 

The musical notes C (262 Hz) and G (392 Hz) make a pleasant sound when played together; we call this consonance. As the Figure shows, the harmonics of the two notes are either far from each other or very close to each other (within a few Hz). This is the key to consonance: harmonics that are spaced either far apart or very close. The close harmonics have a beat frequency of a few Hz that is perceived as pleasant. 

If the harmonics of two notes are close, but not too close, the rather high beat frequency between the two is quite unpleasant. This is what we hear as dissonance. Exactly how much a difference is maximally dissonant is a matter of opinion, but harmonic separations of 30 - 40 Hz seem to be quite unpleasant for most people.

A. An organ pipe open at both ends is tuned so that its fundamental frequency is G. How long is the pipe? Show your work. 

B. If the C were played on an organ pipe that was open at one end and closed at the other, which of the harmonic frequencies in the Figure would be present? Show your work.

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