This individual character of tones depends in part on certain superficial characteristics. The note of the piano comes impulsively, suddenly strong, and then rapidly dying out, while the tones of an organ do not come instantly to full strength, but are then sustained and steady. But tones equally sustained and steady may yet differ greatly in quality, as, for example, the tones of tuning-fork, organ-pipe, and violin. To investigate the cause of this difference we shall need resonators which vibrate in sympathy with the tones studied.

307. Sympathetic Vibration.-A bell ringer by timing his pulls on the rope to correspond to the swing of the bell is able to set a heavy bell strongly swinging, while mere random pulls would accomplish very little; so it is that sound waves or other comparatively slight impulses may set up strong vibrations in a body if they are exactly timed to correspond to its natural period of vibration. This fact of sympathetic vibration may be illustrated by tuning two strings on a sonometer to the same pitch, and then sounding one strongly; the other will be set in vibration by the impulses communicated to it through the supporting bridges.

Again, if the dampers are raised from the strings of a piano and a clear strong note is sung near the instrument, the corresponding string will be heard sounding after the singer's voice is silent.

A very interesting case of sympathetic resonance is that in which a tuning-fork is set in vibration by the sound waves from a similar fork placed twenty or thirty feet away. The two forks are mounted on suitable resonance boxes and must be of exactly the same pitch; if they are thrown out of unison even very slightly, as may be done by affixing a bit of beeswax to a prong of one of them, they will no longer respond to each other.

308. Resonators. When water is poured into a tall cylindrical jar the noise produced has a noticeable pitch which grows higher as the water level rises in the jar. This pitch is due to the air column in the jar, which has a natural time of vibration. of its own and responds to any component vibration of the same pitch which may exist in a noise produced in its vicinity. On blowing sharply across the mouth of the jar the same pitch is

noticed, the confused rustling noise having some component to which the jar can respond. The roaring heard in sea shells is explained in the same way.

If an ordinary tuning-fork, not mounted on a resonance box, is held by its stem and struck, it will scarcely be heard a few feet away, but if it is held, as shown in the figure, over the mouth of a jar tuned to respond, a strong tone will be given out. The arrangement shown in the figure permits the tuning to be easily effected by raising or lowering the connected

water reservoir, thus changing the level of the water in the resonance tube until the response is most powerful.

The pitch of the air column may be lowered also by partially closing the opening of the jar.

Such an air column being easily set in vibration by the proper tone is known as a resonator and may be useful in detecting in a mass of tones the presence of the particular one to which it is tuned.

309. Helmholtz Resonators.-Helmholtz, in the analysis of composite tones, made use of spherical resonators, each having a large opening and also a small one adapted to the ear.

FIG. 173. Resonance.

A resonator of this form is particularly useful because it responds easily to vibrations of one pitch only and so is well suited to the analysis of sounds.

310. Complex aud Simple Tones.-The following experiment will now give us a clue to the cause of the difference in quality of tones. Take a series of tuning-forks mounted on resonating boxes, the frequencies of the forks being in the order of the series of whole numbers, 1, 2, 3, 4, 5, etc., which is known as the harmonic series. If the deepest toned fork in the series makes 250 vibrations per second, the next will make 500, and the next 750, etc. Provide also a set of Helmholtz resonators, one adapted to each fork.

Now, on sounding the lowest pitched fork alone, a deep tone is obtained to which the corresponding resonator alone will respond. If the next lowest is now sounded at the same time with the other, the tones blend and come to the ear as a single tone of the

same pitch as before but of a different quality. And so by sounding along with the deepest or fundamental fork any or all of the others, making some of the component tones strong and some weak, great variety of tones may be obtained differing in quality but all of the same pitch.

But if any of these tones is tested by the resonators it is found that all those resonators respond which correspond to the forks used in producing the tone. Such a tone is called complex, while a tone to which only one resonator will respond is called a simple tone.

311. Analysis of Sounds. In the case just considered it is evident from the way in which the sounds of various qualities were produced that they were complex and consisted of sounds of different pitches blended together. But if we now sound an open organ pipe of the same pitch as the deepest toned resonator we find that not only does that resonator respond, but so also to a greater or less degree do the whole series of resonators, showing that though the sound comes from a single pipe it is just as truly complex as though originating in a series of tuningforks.

The component simple tones which unite to form a complex tone are known as its partial tones, the lowest of these in pitch is the fundamental, and the others are the upper partial tones or upper harmonics. The latter term is especially applicable when the upper partial tones are members of the harmonic series which starts with the fundamental.

From the laws of dynamics as well as from experiment there is reason to believe that a simple tone, to which a resonator of only one certain pitch will respond is one in which the vibrations of the air are simple harmonic (§120).

The ear seems to hear the simple harmonic components of a complex tone as separate simple tones, for persons with ears trained to the analysis of sound can often detect the different harmonics in a tone without the aid of resonators.

312. Synthesis of Sounds.-Helmholtz devised an interesting apparatus by which complex sounds might be built up from their simple components. This consisted of a set of ten tuningforks, corresponding to the first ten terms of a harmonic series, which were kept continuously vibrating by means of electro

magnets, each fork being mounted in front of an appropriate resonator as shown in figure 174.

The resonators were cylindrical brass boxes, each mounted with its opening close to the prongs of the corresponding fork, the openings being closed by covers which could be drawn back by pressing the keys of the key-board. When the resonators were closed scarcely any sound came from the forks, but drawing back the cover from any resonator by depressing its key brought

FIG. 174.-Helmholtz apparatus for the synthesis of sound.

out the corresponding tone with an intensity which depended upon the amount that the key was depressed. By means of such an apparatus the sound of an open or closed organ pipe, a violin, or reed instrument can be closely imitated.

An interesting modern instance of the synthesis of sounds is found in the ingenious "telharmonium" of Mr. Cahill in which the separate harmonics are transmitted by means of alternating currents of electricity of different frequencies which combine to form a single resultant current which acts on the telephone receiver at the end of the line. By combining the proper simple harmonic components all the instruments of an orchestra are imitated.

313. Quality of a Musical Tone.-The quality of a musical tone may then be said to be determined by the pitch and intensity of the different simple tones or harmonics into which it may be resolved.

314. Fourier's Analysis.-It was shown by the distinguished French mathematician Fourier that any regular periodic vibration, such as can take place in a sound wave, may be resolved into a sum of simple harmonic components all of which belong to a harmonic series, in which the fundamental has the same period as the vibration analyzed. Thus, according to this theorem of Fourier, it is possible to analyze any sound wave into its simple harmonic components, and it is these simple harmonic components which are the simple partial tones detected by resonators.

D

FIG. 175.-Vibration curves.

FIG. 176. Combination of vibrations.

For example, the upper curve in figure 175 represents a simple sine wave. The three lower curves represent waves having the same wave length and therefore the same periodicity as the upper curve, but they represent entirely different modes of vibration.

Now, according to Fourier's theorem, each of these curves can be resolved into simple harmonic components. In figure 176, for example, the wave forms expressed by the heavy lines are the resultants of the simple harmonic waves represented by the dotted sine curves. The resultant curve in each case is