This video, by Eric Archer, explores the harmonic series visually, using an oscilloscope. 

via EA78751:

An electronic filter dissects a sawtooth waveform (82 Hz, pitch E) into the first twenty-four partials of the harmonic series. The waveform is displayed on an oscilloscope with annotations to indicate the harmonic relationships. One can see that by counting the number of crests in the waveform per fundamental cycle, the harmonic number is easily obtained.

Have you heard a melody played with the notes of the harmonic series? It can be a pleasant alternative to the 12-tone equal tempered octave. Which instruments are capable of this?

You can approximate the harmonic series on a chromatic instrument by playing the series of pitches listed in the right hand column over a drone note at the fundamental frequency.

You can also hear the harmonic series by whirling an open-ended corrugated tube around in a circle. These are found in some toy stores and educational science shops. Try this and you’ll notice it takes more and more effort to get to the next highest overtone, despite the fact that the pitch increments are getting smaller.

A photograph is included of the voltage-controlled diode ladder filter that was used for this video. Text scanned from THE EMI WALL CHART For Musical Instrument Making. Filmed from a Tektronix 2430A digital oscilloscope.

Original content by Eric Archer 2011