In stringed instruments we are concerned only with the transverse vibrations of a string--that is, its movements in a direction at right angles to the axis of the string. A string can also vibrate longitudinally--that is, in the direction of its axis--as may be proved by drawing a piece of resined leather along a violin string. In this case the harmonics "step up" at the same rate as when the movements were transverse. Let us substitute for a wire a stout bar of metal fixed at one end only. The longitudinal vibrations of this rod contain overtones of a different ratio. The first harmonic is not an octave, but a twelfth. While a tensioned string is divided by nodes into two, three, four, five, six, etc., parts, a rod fixed at one end only is capable of producing only those harmonics which correspond to division into three, five, seven, nine, etc., parts. Therefore a free-end rod and a wire of the same fundamental note would not have the same _timbre_, or quality, owing to the difference in the harmonics. COLUMNS OF AIR. In wind instruments we employ, instead of rods or wires, columns of air as the vibrating medium. The note of the column depends on its length. In the "penny whistle," flute, clarionet, and piccolo the length of the column is altered by closing or opening apertures in the substance encircling the column. RESONANCE OF COLUMNS OF AIR. Why does a tube closed at one end, such as the shank of a key, emit a note when we blow across the open end? The act of blowing drives a thin sheet of air against the edge of the tube and causes it to vibrate. The vibrations are confused, some "pulses" occurring more frequently than others. If we blew against the edge of a knife or a piece of wood, we should hear nothing but a hiss. But when, as in the case which we are considering, there is a partly-enclosed column of air close to the pulses, this selects those pulses which correspond to its natural period of vibration, and augments them to a sustained and very audible musical sound. [Illustration: FIG 136.--Showing how the harmonics of a "stopped" pipe are formed.]