SOUND. - I. PHYSICAL GENESIS OF ACOUSTIC STIMULI. - The energy adequate to stimulate the auditory apparatus is a particular form of mechanical energy, that is, vibratory energy. A conservative mechanical system, in which elastic potential energy is contained, tends to give it back in the form of kinetic energy, and this, if certain conditions of symmetry are satisfied, can transform into potential energy according to a closed cycle of energy exchange, giving rise to a periodic vibratory movement of the material masses of the system around their position of rest. The parasitic dissipations of the energy contained in the system mean that this cycle can repeat itself only a limited number of times, so that if the vibrating system is not replenished with energy the vibrations die out in a longer or shorter time.
The amplitude of the vibrations and the number of vibrations that the system itself performs in the unit of time (frequency) depend on the mechanical quantities in play, in particular on the masses and on the restoring forces that continually urge them toward the position of rest. One complete vibration per second, that is, one period per second, is the unit of frequency and is called Hertz (Hz).
If the vibrating system is located in an elastic medium (solid, liquid or gaseous), it is also a "source of vibrations" insofar as its alternating movements are communicated to the medium and are transmitted in it by pressure waves. It is in this way that the vibrations reach the auditory apparatus, through the air, directly, setting the tympanic membrane vibrating and then successively the structures of the middle and inner ear; a small part of the vibratory energy reaches it by direct conduction through solids (e.g., the ground), arriving at the inner ear directly by bone conduction, as likewise happens also for a modest fraction of that conducted by the gaseous medium.
The tympanic membrane is sensitive to the "pressure" of the incoming waves, and the vibratory stimulus proves adequate only if this pressure is comprised within certain limits ("threshold of audibility" and "threshold of pain"), while its frequency is likewise comprised within other defined limits (lower limit and upper limit of the tonal audibility range). The increase in the pressure of the stimulus is perceived as an increase in the (subjective) intensity of the sound, while the rising of its frequency is perceived as a rise in the so-called (subjective) pitch of the sound.
Through the outer and middle ear the mechanical vibrations at acoustic frequency reach the cochlea, where the premises for the excitation of nerve VIII are created; the nerve impulses travel, being modified in various ways, through a chain of 4 neurons and their synapses and reach the auditory radiations and the acoustic cortical area. Then the conscious subjective movement begins, during which the sound arises following the higher associative, perceptual and apperceptive processes.
There is no occasion to speak of sound as present in the environment that surrounds us. There the vibratory phenomenon has physical reality. And it is to be noted that the sensation of sound can arise even in the total absence of vibrations in the external environment, e.g., through the application of electric currents to the auditory apparatus (electrophonic phenomenon); the electric stimulus here is an inadequate but sufficient stimulus.
II. VIBRATING BODIES
Stretched strings, metal plates, rods, bells and almost all the objects that surround us are mechanical systems capable of vibrating at acoustic frequency; even air can become a vibrating body if it is confined within cavities, as in organ pipes and sound boxes.
A remarkable vibrating system is the one constituted by the "vocal cords," more properly called "vocal lips."
III. SPEED OF VIBRATION
The amplitude and the frequency of the vibratory motion determine the speed with which the particles of the vibrating body follow their periodic alternating motion around the position of rest, toward which they are urged at every instant, as was said, by the elastic restoring forces.
In a tuning fork having a frequency of 100 Hz, excited in such a way that the amplitude of vibration of the tip of the prongs is one millimeter, each particle of that tip travels in one second 100 times, in journeys of 1 millimeter each, equal to 200 mm. of path, with an average speed therefore of 20 cm. per second, equal to 0.72 km./hour.
The membrane of the eardrum, which at the frequency of 3000 Hz can be set in vibration by an energy of barely 1 billionth of a microwatt per square centimeter, has, according to Wilska, an amplitude of vibration equal to 5.2 hundred-billionths of a centimeter; under such conditions it has, as is easy to calculate, an average speed of 3.12 millionths of a millimeter per second, that is, about 12 microns per hour.
IV. SPEED OF PROPAGATION
Calculations of this kind can also be repeated for the particles of the medium (gaseous, liquid or solid) in which the vibratory motion propagates. In air, for example, the velocity of the particles reaches at most fractions of a millimetre per second, since even though the movement is rapid, their displacement around the position of rest is always extremely small.
The velocity, on the other hand, with which the vibratory motion propagates (commonly indicated by the name "velocity of sound") depends on the physical characteristics of the medium, more precisely on its density and its modulus of elasticity, quantities which are in turn bound to various factors such as, for example, temperature and pressure. In air the velocity of propagation is 331.45 metres per second at zero degrees centigrade under the normal atmospheric pressure of 760 mm. of Hg., and it increases by 0.607 m. per second for each degree of rise in temperature, so that one may assume for it the average value of about 340 m. per second in relation to the average ambient temperature at our latitudes. In liquid media the velocity is greater (in distilled water, for example, it is 1430 m./second at 5° C.), and in solids it is several times greater still.
The velocity of propagation does not depend, within broad limits of approximation, on the frequency of the mechanical vibrations (at least within the limits of the acoustic range); a strict dependence between frequency and velocity would make it almost impossible to hear words and music at a distance, since tones of different pitch would reach the listener's ear with different delays. The velocity of propagation is not perceived as such by the ear, but we are accustomed to appreciate it as an indirect consequence in the fact, for example, of hearing the sound with greater or lesser delay (in relation to distance) from the instant in which one sees the act that generated the mechanical vibrations. Furthermore, this velocity is of primary importance in determining the direction from which a vibration of acoustic frequency comes, which is bound to the differences in time and phase of perception at the two ears.
The vibratory displacement of the particles takes place in the very direction of propagation; in this sense there do not exist in the medium waves as such, that is, according to what one is led to imagine by taking as a comparison the model offered by waves on the surface of water, in which the motion of each particle takes place along a direction perpendicular to the direction of propagation of the disturbance on the surface.
In the propagation of elastic waves in a homogeneous medium there therefore exist only compressed and rarefied zones which follow one another in orderly fashion, spaced at equal intervals in the direction of propagation of the elastic disturbance. The length of the interval takes the name "wavelength" and is inversely proportional to the frequency. At 100 periods per second with a velocity of 340 m. per second, each wave is 3.4 m. long; at 1000 periods per second, with the same velocity each wave is 34 cm. long.
V. SIMPLE MOTIONS AND COMPOUND MOTIONS: TIMBRE
The mechanical vibrations of bodies at acoustic frequency are periodic movements similar to those of a pendulum, but they do not attain its regularity and simplicity except in the rare cases in which precisely particular conditions of simplicity and symmetry of structure are realized (as, for example, in a tuning fork). The course of the vibratory motion over time is in such cases representable by a sinusoidal function of the type y = sin x.
An elastic body of any shape, on the other hand, vibrates by dividing itself into various vibrating portions (compound vibratory motion), so that vibrations having various frequencies and amplitudes are simultaneously produced. And there may or may not exist among such frequencies simple relations of multiplicity (harmonic relations). The component with a frequency double that of the fundamental frequency takes the name of second harmonic, that with triple frequency of third, etc. Any compound motion is always decomposable into a series of simple vibratory motions, that is, sinusoidal ones, which, coexisting, determine the compound motion itself. The search for such components commonly takes the name "analysis of sound."
It is to the "composition" of the vibratory motion that the subjective appreciation of timbre corresponds. Timbre is found to vary as a consequence of the variation in the number, frequency and amplitude of the vibrations that make up the complex vibration. Differences of phase are of lesser importance.
When the component vibrations are not in harmonic relations, but completely independent of any law of mathematical relation, especially when the vibratory motion dies away rapidly, the listener defines his auditory sensation more as noise than as sound. A typical example is that offered by the "Savart wheel," that is, a toothed wheel on which a small card may be rested. When the wheel turns, at the passage of each tooth the card performs an oscillation which results in a well-determined noise; but if the successive impacts on the teeth follow one another rapidly, the auditory sensation may be better defined as a sound of pitch depending on the number of impacts per second, that is, on the velocity and the number of teeth of the wheel.
Hearing is endowed with notable powers of analysis and can distinguish in a complex sound the individual components, appreciating their pitch and intensity. This analysis takes place as if the ear were constituted by a complete and very numerous series of resonators and should be, according to the acoustic law of Ohm, independent of the phase relations of the component tones. This is true within limits of first approximation, since strictly speaking the ear is not a perfect analyser, in the sense that adjacent zones of the basilar membrane, to which the most recent researches of Bekesy credit this task, are found to respond to a lesser degree also to stimulations of a frequency close to that for which they are specifically tuned. This, while guaranteeing on the one hand the response to stimuli of any frequency (within the acoustic range), gives on the other hand rise to the possibility that, through simultaneous stimulation with two stimuli of slightly different frequency, there occur effects of periodic reinforcement and weakening of the sound on the rhythm of the difference between the two frequencies (subjective beats).
Modifications of timbre due to phase shifts of the individual components of a complex tone may occur through phenomena of interferential combination with the auricular harmonics; that is, the stimulus effectively applied to the cochlear organ is modified and consequently the subjective appreciation turns out different. Serious modifications of timbre are a consequence of notable reductions in the amplitude of the tonal range of audibility due to affections of the auditory organ in the numerous forms of deafness.
VI. PSYCHOLOGICAL ATTRIBUTES OF SOUND
The principal ones are intensity and pitch, respectively rendered in the subject's judgment by indications of "loud" and "soft" and by indications of "high" and "low". The (subjective) intensity corresponds, for a given vibratory stimulus, to the acoustic pressure at the level of the tympanic membrane and hence to the amplitude of vibration, in general to the number of ergs transmitted per second through one square centimeter. Pitch, on the other hand, corresponds to the frequency of the vibratory stimulus. These two correspondences are, however, not as strict and above all not as univocal as was believed until some time ago and as is still found in many treatises; both pitch and intensity are each a function of the two variables, acoustic pressure and frequency; so that it is, for example, possible to induce variations in pitch by modifying the intensity of a sound. Such variations are percentage-wise small, but they account for many phenomena of physiological acoustics.
The level of acoustic pressure of a vibratory stimulus can be expressed by the number of decibels (dB), which indicates by how much it exceeds the value of the reference threshold, that is, of the minimum acoustic pressure capable of generating an acoustic sensation, a threshold which by convention is taken as relative to the frequency of 1000 Hz. On this scale:
| zero dB | 10⁻¹⁶ watt cm⁻² |
|---|---|
| " " | 0.000204 dyne cm⁻² |
| " " | 73.8 dB below 1 dyne per cm². |
These reference levels represent the auditory threshold for a normal ear, but they do not constitute starting values for a (subjective) "scale of intensity" strictly understood. The field of audibility extends between 16 Hz and 19,000 Hz on average for normal hearing; that is, between wavelengths of 21.3 m and 1.8 cm respectively.
VII. OTHER PSYCHOLOGICAL ATTRIBUTES OF SOUND
Besides the timbre already mentioned, there exist other attributes; it is noted that, to define a new one, it is enough that the reaction on the part of the organism to variations of the stimulus be systematic and that the corresponding values be plottable on a graph. Rich in 1916 stated in key terms the fact that sounds have a subjective character of "breadth" or "extension" (volume) depending, according to his findings, on pitch. Halverson in 1924 emphasized the dependence of volume on intensity. Stevens in 1934 showed that it is possible to make two sounds appear equal in volume while being different in pitch and in intensity precisely because, if one wants the volume to remain constant, the higher sound must have greater intensity. Thus, for the volume to remain constant, starting from a vibratory stimulus of frequency 900 Hz and pressure at the tympanic membrane 55 dB above threshold, an increase of 100 Hz in frequency must be followed by an increase of 10 dB in pressure.
Many subjects then have a tendency to attribute to pure sounds the subjective evaluation of "compactness" or "density," and this has been the object of detailed research on the part of Stevens (1934). Density has a course inverse to that of volume, in the sense that an increase in frequency is compensated by a decrease in pressure. Thus, for density to remain constant, the increase in frequency from 500 to 600 Hz in a vibratory stimulus that generates at the tympanic membrane a pressure of 60 dB above threshold must be compensated by a decrease of ca. 4 dB in the aforesaid pressure.
It is not easy to indicate through what mechanisms one arrives at the perception of volume and density. It is obvious that the fact that the responses are differentiated implies a different neural and cortical excitation. Thus one may think that density is linked to the density of nervous excitation in the cortex, while volume (Boring, 1926) would depend on the "breadth" of the cortical area involved.
The possibility of judging sounds as "bright" and "dull" has led some to define the brightness of sound as another attribute. It does not seem, however, that this attribute can enjoy a life of its own, inasmuch as Troland already in 1929 indicated it as closely connected to pitch, while for Stevens it is to be considered, on the basis of more rigorous analysis, identical to density. The latter has in fact ascertained that producing tones of varying brightness is not possible except by means of a variation of timbre: so that brightness is more a subjective appreciation of timbre than a new attribute of sound. Other very important attributes of sounds are the "musical" ones, consisting in the fact that two or more sounds of different frequency can be judged consonant or dissonant, whether they are produced simultaneously or successively, and that the relative interval can be judged equal to, greater than, or less than another, all of which leads to the constitution of musical scales, and, in union with the appreciations of the intervals of time elapsing between one sound and another, to harmony, counterpoint, orchestration, in a word to music in general.
VIII. REFLECTION. ECHO
If, in its propagation, the vibratory motion of the particles of air encounters an elastic wall, such as, for example, a slab of marble, a smooth wall, etc., it is reflected according to laws similar to those that govern the reflection of light by a mirror. The reflection is translated into echo under particular conditions of the environment.
The vibratory motion that reaches the ear is always different from that produced by the source, as a result of the various and successive reflections on the walls of the environment and of the objects contained in it. Thus there is a persistence of acoustic energy in the environment: the phenomenon of reverberation, always present unless one operates in a free field, that is, in a theoretical environment in which (reflecting) walls do not exist and are infinitely distant; or in which such walls, present and at a finite distance, are completely absorbent, that is, reflect nothing of the energy incident upon them. The existence of "absorbent" materials such as, for example, glass wool, mineral wool, etc., makes it possible to construct rooms called precisely "absorbent," in which the energy reflected by the walls can be only a small percentage of the incident energy. No wall, moreover, is so reflecting as to reflect all the energy incident upon it; there do exist, however, "reflecting" materials (e.g., marble is typical) which make it possible to construct the rooms called "reverberant," for purposes of acoustic investigation. The proportioning, within the environment, of reflecting and absorbent materials is very important for the acoustics of auditoriums and theaters.
IX. RESONANCE
A material system can be set into oscillation under the action of periodic external forces (forced vibration). In that case the period and the amplitude of vibration are not those proper to the system when it vibrates on its own, but those proper to the external forces, though partly modified by the very presence of the material system on which they act.
The vibrations of the latter can become very large when its own period coincides with that of the external forces; in that case the system enters into resonance, and even a small force can generate conspicuous oscillations. Thus two pendulums oscillating with the same period mutually impel each other to oscillate if placed near one another; and if one of them is at rest and the other is set oscillating, after a little while the first is seen to oscillate as well, as a consequence of the rhythmic impulses it receives through the air. A similar experiment can be carried out with two tuning forks having the same frequency; by exciting one and placing it near the other, it will be observed that the latter begins to vibrate: and again with two musical instruments, e.g., two violins, whose strings have been brought into unison, placed in the same room even at a distance of some metres, it can be observed that by exciting one by one the strings of the first and then stopping them by placing the fingers on them, those of the other respond in order; it is understood that they are excited by "plucking" and not with a bow or other means, which can induce frequency variations in the rubbed string.
Even a mass of air delimited by a cavity can enter into resonance; that contained in a tube, e.g., resonates at a frequency depending on the length of the tube; in a complex manner the air contained in the cavities called "sound boxes" in musical instruments resonates. These cavities, a typical example being those of violins, have been studied empirically so that the air contained in them resonates as uniformly as possible over the whole vast range of frequencies supplied by the instrument, since, given the complex shape, there are different lengths of air column and different masses of air in different directions. Noteworthy for the uses made of them in acoustics are the "Helmholtz resonators," hollow spheres provided with two diametrically opposed holes through which the internal air communicates with the surrounding air and with the listener's ear. The internal air resonates at a frequency determined by the diameter of the cavity.
X. INTERFERENCE
Two sinusoidal vibratory motions of the same frequency and phase give rise to a motion still of the same frequency and phase with an amplitude that is the sum of the amplitudes of the two motions; whereas if the phase is different, the resulting motion has a smaller amplitude (interference). For opposite phase, that is, for a phase angle equal to 180° (a displacement of ½ wavelength) and equal amplitude, the sum is zero, that is, the two motions cancel each other out. This case is called by some "total interference." It corresponds to the silence obtained by "adding sound to sound," according to an ancient saying.
XI. BEATS
In the case where the vibrations have a different frequency, e.g., 200 Hz and 201 Hz respectively, and at a given instant are in phase, it is clear that as time passes the phase difference becomes more pronounced, and that, precisely after half a second, one will have completed 100 full periods and the other 100 and ½. Thus they will be in phase opposition. After another half second, however, they will again be in phase and the amplitudes will add together. A listener will periodically perceive reinforcements and weakenings of the sound, called "beats." If the difference between the frequencies of the two vibrations is 2, 3, 4, etc. Hz, then 2, 3, 4, etc. are the beats per second, as is easy to calculate; so that the number of beats turns out to be equal to the difference between the two frequencies; and if such a difference reaches the range of acoustic frequencies, one hears a "third sound," called precisely "difference sound" or "Tartini's sound," after the name of the famous violinist who was the first to make use of it. This "third sound" is very important in musical composition and in orchestration, since one must take into account its presence and the fact that it forms consonance or dissonance with the other "sounds" present.
XII. RECORDING OF MECHANICAL VIBRATIONS AT ACOUSTIC FREQUENCY
Edison's early first device consisted of a stretched membrane that collected the vibrations of the air and transmitted them to a stylus, capable of cutting into a rotating wax cylinder a groove of variable depth, whose profile came to reproduce the form of the vibrations that had acted upon the membrane itself. Passing the stylus a second time through the groove, the membrane connected to the stylus was made to repeat the recorded vibrations and thus to communicate to the air a vibratory motion similar to that which had been recorded. In other words, the recording and reproduction of mechanical vibrations at acoustic frequency was achieved. Subsequently, wax cylinders were replaced by discs. Today recording and reproduction are carried out through the transformation of acoustic vibrations into alternating currents of equal frequency and form, whether on disc or following the many other recording processes devised, such as those on photographic track and on magnetic tape or wire.