One example of an independent variable related to how the pitch of a sound affects the frequency is the tension of a vibrating string or the length of a vibrating column of air.
For example, in a stringed instrument like a guitar or violin, the pitch of the sound produced can be altered by changing the tension of the string. Increasing the tension of the string makes it vibrate at a higher frequency, resulting in a higher pitch. Conversely, decreasing the tension lowers the frequency and produces a lower pitch.
In the case of wind instruments like flutes or clarinets, the pitch can be adjusted by changing the length of the vibrating column of air. By opening or closing certain finger holes or keys, the effective length of the vibrating air column is altered. A longer column produces a lower frequency and a lower pitch, while a shorter column produces a higher frequency and a higher pitch.
In both cases, the tension of the string or the length of the vibrating column of air serve as independent variables that can be controlled to influence the frequency and, consequently, the pitch of the sound produced.