The shape of a wave is directly related to its spectral content, or the particular frequencies, amplitudes and phases of its components. Spectral content is the primary factor in our perception of timbre or tone color. We are familiar with the fact that white light, when properly refracted, can be broken down into component colors, as in the rainbow. So too with a complex sound wave, which is the composite shape of multiple frequencies.
So far, we have made several references to sine waves, so called because they follow the plotted shape of the mathematical sine function. A perfect sine wave or its cosine cousin will produce a single frequency known as the fundamental. Once any deviation is introduced into the sinus shape (but not its basic period), additional frequencies, known as harmonic partials are produced.
Partials refers to frequencies that are generated by a simple or complex waveform. In real world sound, these are not necessarily harmonic partials. Harmonics or harmonic partials are integer (whole number) multiples of the fundamental frequency(ƒ) (so 1ƒ, 2ƒ, 3ƒ, 4ƒ ). Overtones are the harmonics above the fundamental. For conventions sake, we usually refer to the fundamental as partial #1, since it is 1ƒ. The first few harmonic partials are the fundamental frequency, the 8ve above, a perfect fifth, two 8ves above, two 8ves + major 3rd, two 8ves + major 5th as pictured below for the pitch 'A.' After the 8th partial, the pitches begin to grow ever closer and do not necessarily correspond to equal-tempered pitches, as shown in the chart. In fact, even the fifths and thirds are slightly off their equal-tempered frequencies. You may note that the first few pitches correspond to the harmonic nodes of a violin (or any vibrating) string.
Partials that deviate in frequency from the ideal arithmetic multiples of the fundamental are referred to as inharmonic, and a general characterization of the deviation is called inharmonicity. To artificially generate inharmonic partials requires the addition of instability to a periodic wave, creating aperiodicity. Inharmonicity is common in the real world—piano strings, particularly towards either end of the keyboard, generate a small amount of inharmonicity, whereas bells generate a much greater amount.