The G string on a guitar is 59 cm long and has a fundamental frequency of 196 Hz. A guitarist ca

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The G string on a guitar is 59 cm long and has a fundamental...

The G string on a guitar is $59 \mathrm{cm}$ long and has a fundamental frequency of 196 Hz. A guitarist can play different notes by pushing the string against various frets, which changes the string's length. The first fret from the neck gives $A b(207.65 \mathrm{Hz})$ the second fret gives $\mathrm{A}(220 \mathrm{Hz}) .$ How far apart are the first and second frets?

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Key Concepts

Fundamental Frequency and Vibrating String Length

The frequency at which a string vibrates—its fundamental frequency—is inversely proportional to its effective vibrating length when tension and linear density remain constant. Shortening the string by pressing it against a fret raises its frequency, forming the basis for note variation on string instruments.

Equal Temperament Scale

In equal temperament, the octave is divided into equal frequency ratios, specifically the twelfth root of two for each semitone. This consistent multiplicative step between notes ensures that each fret change corresponds to a uniform pitch increase across the instrument.

Fret Placement on String Instruments

Fret placement is determined by combining the physical properties of vibrating strings with the mathematical structure of the equal temperament scale. The positions of the frets are calculated so that shortening the string by the appropriate ratios produces the desired frequencies according to the musical scale.

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