Sympathetic resonance or sympathetic vibration is a harmonic phenomenon wherein a formerly passive string or vibratory body responds to external vibrations to which it has a harmonic likeness. The classic example is demonstrated with two similar tuning-forks of which one is mounted on a wooden box. If the other one is struck and then placed on the box, then muted, the un-struck mounted fork will be heard. In similar fashion, strings will respond to the external vibrations of a tuning-fork when sufficient harmonic relations exist between the respective vibratory modes. A unison or octave will provoke the largest response as there is maximum likeness in vibratory motion. Other links through shared resonances occur at the fifth and, though with much less effect, at the major third. The principle of sympathetic resonance has been applied in musical instruments from many cultures and times. Apart from the basic principle at work on instruments with many undamped strings, such as harps, flutes, guitars and pianos with the dampers raised, other instruments are fitted with extra choirs of sympathetic strings, which respond with a silvery halo to the tones played on the main strings.
Lewcock et al.(2006) states that:
The property of sympathetic vibration is encountered in its direct form in room acoustics in the rattling of window panes, light shades and movable panels in the presence of very loud sounds, such as may occasionally be produced by a full organ. As these things rattle (or even if they do not audibly rattle) sound energy is being converted into mechanical energy, and so the sound is absorbed. Wood paneling and anything else that is lightweight and relatively unrestrained have the same effect. Absorptivity is at its highest at the resonance frequency, usually near or below 100 Hz.
An example of proper sympathetic resonance is a windowpane rattling steadily at the very low powerful sound of a bus or truck engine going stationary. The rattling will usually occur at a higher harmonic of the sound made by the engine. As soon as the driver changes into gear the rattling will stop, often changing its rhythm before it stops altogether. Powerful sopranos bursting wineglasses fits into the same category - sympathetic resonance at a distance.
String resonance in music instruments 
String resonance occurs on string instruments. Strings or parts of strings may resonate at their fundamental or overtone frequencies when other strings are sounded. For example, an A string at 440 Hz will cause an E string at 330 Hz to resonate, because they share an overtone of 1320 Hz (3rd harmonic of A and 4th harmonic of E).
According to Grove Music Online (2007) article on Duplex Scaling, Steinway progressed a system of aliquot scaling to provide sympathetic resonance with the intention of enriching the treble register of the pianoforte. In the 'octave duplex' piano by Hoerr of Toronto, each note had four strings, of which two, three or four could potentially be struck by the hammer depending on the depression of one of four pedals. Steinway’s duplex scale was precipitated a half century earlier by an experiment conducted by the German piano maker Wilhelm Leberecht Petzoldt, in which a small bridge was placed behind the standard larger one with the intention of maximizing the potential additional resonance of a sympathetically vibrating additional length of string.
- Lewcock, Ronald, Rijn Pirn (with Jürgen Meyer), et al. (2006). Acoustics: Resonance, reverberation and absorption. Oxford University Press. Grove Music Online. Source: http://www.grovemusic.com/shared/views/article.html?section=music.00134.1.3#music.00134.1.3 (accessed: Wednesday January 17, 2007)