Harmonics are very important for understanding how good a sound is from acoustic instruments. To grasp this better, let's look at some basic ideas.

Fundamental Frequency

The fundamental frequency is the lowest sound frequency an instrument makes, which we call $f_0$. It’s the main tone we hear when we listen. For example, the note A above middle C has a fundamental frequency of about 440 Hz.

Overtones

Overtones are the higher sounds that come along with the fundamental frequency. These higher sounds are related to the fundamental frequency by whole numbers. We usually write them as $f_n = n \cdot f_0$, where $n = 2, 3, 4, ...$ This means:

• The first overtone (or second harmonic) is $2 \cdot f_0$.
• The second overtone (or third harmonic) is $3 \cdot f_0$, and so on.

Harmonic Series

The harmonic series shows how these frequencies relate to the fundamental frequency:

$$f, 2f, 3f, 4f, \ldots$$

For example, if the fundamental frequency is 100 Hz, the first five harmonics are:

1. $100 \text{ Hz}$
2. $200 \text{ Hz}$
3. $300 \text{ Hz}$
4. $400 \text{ Hz}$
5. $500 \text{ Hz}$

Effect on Sound Quality

When the fundamental frequency combines with the overtones, it makes a complex sound wave. This combination creates the timbre, which is the unique color or quality of the instrument's sound. Different instruments have different relative strengths of these harmonics, which changes how they sound. For example:

• A piano and a violin can play the same note and have the same fundamental frequency, but they sound different because their overtones are not the same.
• Some instruments highlight certain harmonics to make the sound warmer or brighter. For instance, brass instruments focus on lower harmonics, giving them a rich and bold sound.

Statistical Insights

Studies show that harmonics can account for more than 80% of how we perceive sound quality. The first few harmonics are really important too, as about 60% of what we hear comes from just the first five harmonics.

In summary, harmonics are key to how we experience sound in acoustic instruments. They help mix the fundamental frequencies and overtones, allowing us to tell different musical timbres apart.