The portion of the string of a certain musical instrument...

The portion of the string of a certain musical instrument between the bridge and upper end of the finger board (that part of the string that is free to vibrate) is $60.0 \mathrm{~cm}$ long, and this length of the string has mass $2.00 \mathrm{~g}$. The string sounds an $\mathrm{A}_{4}$ note $(440 \mathrm{~Hz})$ when played. (a) Where must the player put a finger (what distance $x$ from the bridge) to play a $D_{5}$ note $(587 \mathrm{~Hz}) ?$ (See Fig. E15.43.) For both the $\mathrm{A}_{4}$ and $\mathrm{D}_{5}$ notes, the string vibrates in its fundamental mode. (b) Without retuning, is it possible to play a $\mathrm{G}_{4}$ note $(392 \mathrm{~Hz})$ on this string? Why or why not?

Key Concepts

Fundamental Mode of Vibration

This mode is the simplest standing wave pattern on a string, where the string vibrates in a single half-wavelength between its fixed endpoints. The frequency in the fundamental mode is determined by the length of the string, its tension, and its mass per unit length according to the relationship f = (1/2L) ?(T/?).

Frequency-Length Relationship

For a vibrating string with constant tension and mass per unit length, the frequency is inversely proportional to the effective vibrating length. This means that shortening the string increases the pitch because a shorter string supports a higher frequency for the fundamental vibration.

Effective Vibrating Length and Fingering Technique

In stringed instruments, when a player presses the string against the fingerboard, they effectively shorten the length of the vibrating portion of the string. This change in length results in a change in the produced pitch. Understanding how far to press along the string to achieve a desired frequency involves applying the inverse relationship between length and frequency.

Instrumental Pitch Constraints

The range of pitches that can be produced on a string without retuning is limited by the physical length of the string and the fixed parameters like tension and mass distribution. If the calculated effective length necessary for a particular note falls outside the available range of the instrument, that note cannot be played without altering the tension (i.e., retuning) or using alternate techniques.

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