It can be fun and enlightening for piano students to learn about deeper musical concepts such as the overtone series, more correctly known as the “harmonic series.” The harmonic series can help piano students understand such intriguing subjects as the origins of tonality, why a piano sounds the way it does, and how music relates to math. I recommend giving a live demonstration of the harmonic series on an acoustic piano. If you don’t know how to do that, be sure to watch Leonard’s Bernstein’s brief but inimitable talk (consisting of two short videos) on the subject.
When you play a note (called the “fundamental”) on the piano (or any instrument) you are not just hearing that note, but also a series of successively higher frequencies called “harmonics” or “overtones.” These frequencies are known as the harmonic series.
How strongly the frequencies in the harmonic series resonate on a given instrument is the primary determinant of that instrument’s characteristic sound quality or timbre (pronounced “tamber”). In other words, the way the harmonic series resonates on the piano is why the piano sounds like a piano and not a trombone!
In the above harmonic series, middle C (C4) is the first harmonic.
As you can see, the intervals between the pitches become smaller as the harmonic series progresses.
The harmonic series is identical for any pitch. The first harmonic is always one octave higher than the fundamental, the second harmonic is a twelfth (octave + fifth) higher, etc.
Interestingly, the first harmonic in the harmonic series is exactly twice the frequency of the fundamental. Another way of saying this is the ratio of the frequencies of notes a perfect octave apart is 2:1. The ratio of the next interval in the harmonic series, a perfect fifth, is 3:2, and the perfect fourth is 4:3. This is the mathematical reason why the intervals of the octave, fifth and fourth were called “perfect” intervals – the human brain hears intervals with lower-numbered ratios as more consonant or pleasing! By contrast, the ratio of the much more dissonant minor second (half step) is 16:15. (Because pianos are tuned using the equal temperament system – which means that a perfectly-tuned piano is somewhat out of tune according to these ratios! – a minor second is actually slightly narrower than 16:15.)
The fact that the first harmonic of the series is the same note as the fundamental (just one octave higher, i.e. the tonic), and that it is followed by the dominant on the second overtone, reveals the significance of the harmonic series to tonality itself.
Notice that the first four frequencies in the harmonic series together comprise the tones of a major chord, which is almost certainly why major chords sound so pleasing to the human ear and are used so often in most musical styles.