Final answer:
A compound inequality representing the range of a basic acoustic guitar tuned to concert pitch over three octaves is 82.4 Hz ≤ f ≤ 659.2 Hz. The strings of a guitar are designed to be tuned to specific frequencies for standard notes, influencing the pitch of the sounds produced. The perception of frequency as pitch is important in music and is modified on string and wind instruments by changing string length or air column length, respectively.
Step-by-step explanation:
A compound inequality that describes the range of frequencies for a typical acoustic guitar when tuned to concert pitch, spanning three octaves, and inclusive of the endpoints, is 82.4 Hz ≤ f ≤ 659.2 Hz, where f represents the frequency in Hertz (Hz). This inequality indicates that the guitar can play frequencies as low as 82.4 Hz and as high as 659.2 Hz. The standard tuning of a guitar's strings, which are the high E, B, G, D, A, and low E, corresponds to fundamental frequencies of 329.63 Hz, 246.94 Hz, 196.00 Hz, 146.83 Hz, 110.00 Hz, and 82.41 Hz, respectively.
The significance of these frequencies lies in how they are perceived by humans as pitch. People can usually discern a frequency difference as slight as 0.3%, which means two sounds, such as 500.0 Hz and 501.5 Hz, are discernibly different. The perception of frequency, particularly in music, allows for the identification and distinction of various musical notes and the creation of melodies and harmonies when these notes are played in combination.
Understanding the physics behind musical instruments, such as the vibration of strings or the changes in resonating air columns in wind instruments, allows for a deeper appreciation of music. For instance, altering the length of a guitar string changes the frequency, hence the pitch, of the note it produces. This principle also applies to other instruments, such as flutes, where changing the length of the resonating air column by covering finger holes adjusts the pitch.