Overtones are frequencies that exist above the fundamental frequency and play a crucial role in enriching and complicating the sound profile of musical notes.

To elaborate, the fundamental frequency is the lowest frequency of a periodic waveform. In musical terms, it represents the lowest pitch that can be perceived and is generated by the entire object vibrating as a whole. In contrast, overtones are higher frequencies that result from the vibration of specific parts of the object.

For instance, when a guitar string is plucked, it vibrates as a whole, generating the fundamental frequency. However, various sections of the string also vibrate at higher frequencies, producing overtones. These overtones are typically integer multiples of the fundamental frequency, occurring at frequencies of $2$, $3$, $4$, and so on times the fundamental frequency. These specific overtones are referred to as harmonic overtones and are essential in defining the timbre or tone color of the sound, thereby making it richer and more intricate.

The relationship between the fundamental frequency and the overtones is direct; if the fundamental frequency changes, the frequencies of the overtones also change proportionally. This proportionality arises because the overtones are simply multiples of the fundamental frequency. For example, if the fundamental frequency doubles, the frequencies of the overtones will also double. This explains why adjusting the tension in a guitar string alters the pitch of the note: both the fundamental frequency and the frequencies of the overtones are affected.

In conclusion, overtones are frequencies higher than the fundamental frequency, produced by the vibration of specific parts of the object. They significantly contribute to the richness and complexity of sound. The relationship between the fundamental frequency and the overtones is such that any change in the fundamental frequency results in a proportional change in the overtones’ frequencies.