(source unknown)
(click to enlarge)
1) The science which treats of the nature and laws of sound.
2) The sensation of sound consists in the communication of a vibratory motion to the tympanic membrane of the ear, through slight and rapid changes in the pressure of the air on its outer surface.
3) The mode of propagation of sound in air may be explained in the following manner. Suppose a small particle of fulminating silver to be exploded in free air; the air particles immediately contiguous are driven outwards in all directions by the explosion, their motion is almost instantaneously communicated to the adjacent ones, those first agitated coming at the same time to rest; the adjacent ones pass on the impulse in the same way to those at a greater distance, and so on; thus the explosion gives rise to what may be looked on as a rapidly expanding shell or constant thickness, containing at any instant between its exterior and interior surfaces a stratum of agitated air particles each one of which performs a single vibration to and fro during the passage of the shell over it; in other words the exterior and interior surfaces of the shell are at any time the loci of all those points at which the particles at that instant come under the influence of the impulse, and are left at rest by it respectively, so that its thickness depends both on the rapidity of their vibration and the rate at which they pass on the impulse, one to another. Let us suppose now that immediately after the first explosion a second were to take place; then, in exactly the same way there would be a second pulse propagated in all directions. If a series of explosions at regular intervals were to take place, there would be a regular series of expanding shells; and if the intervals were sufficiently small, the alternate changes of pressure, due to the successive collisions of the air particles against the tympanic membrane of an ear in the neighborhood of the explosions would convey to the brain a sensation of a continuous note. Exactly the same thing occurs if, for a series of explosions, are substituted the vibrations of an elastic body; and it is, in general, by the latter means that all sounds, and especially musical ones, are produced. The motion of a sound wave must not be confounded with the motion of the particles which transmit the wave. In the passage of a single wave each particle over which it passes makes only a small excursion to and fro, the semi-length of which is called the amplitude of the vibration, the time occupied during one vibration being called its period.
4) The intensity of a sound is proportional to the square of the maximum velocity of the vibrating particle. It also approximately varies inversely as the square of the distance from the origin of the sound; for, supposing the latter to be produced at a uniform loudness, the same amount of energy has to be communicated to the particles contained within the external and internal surfaces of shells of the same thickness but of different radii. For example, if we take a shell of air whose internal radius is one foot, one of the same thickness whose radius is two feet will contain four times the quantity of matter; one whose radius is three feet, nine times the quantity, and so on. Thus the amount of matter over which a given quantity of energy has to be distribute augments as the square of the distance from the origin of sound, and therefore the amount of energy, or, what comes to the same thing, the intensity of the sound, diminishes in the same ratio. [See Square Law]
5) At a temperature of zero Centigrade sound is propagated at the rate of about 1090 feet per second, and this speed augments about two feet per second for every additional degree of temperature; thus at 15° C. the rate of propagation would be about 1120 feet per second. The velocity of sound in air depends on the elasticity of the air in relation to its density. It is also directly proportional to the square root of the elasticity, and inversely proportional to the square root of the density. Now for a constant temperature the elasticity varies as the density, hence in this case they neutralise one another, and the velocity of the sound is independent of the density of the air.
6) One sound differs from another not only in quantity, but also in quality and pitch. The pitch of a sound depends on the number of vibrations per second by which it is caused; the greater this number is the higher is the sound, and vice verse; thus pitch is a more or less relative term, and it is therefore necessary to have some standard to which different sounds may be referred. This standard is so chosen that the middle C of the pianoforte shall be produced by 264 vibrations per second. (That is, according to German musical pitch, Western pitch is set at 244 cps.)
7) Knowing the velocity of sound in air we can estimate the different wave lengths corresponding to notes of different pitch in the following manner. The wavelength is the distance through which the sound travels while any particle over which it passes describes a complete vibration; hence, if we know the number of vibrations the particle performs per second, the required wavelength will be found by dividing the number of feet over which the sound travels per second, by that number. Now, by means of an instrument invented by Cagniard de la Tour, and by him named syren, the number of vibrations corresponding to a note of any given pitch can be determined very exactly. For a detailed account of this instrument and of its improvements by Helmholtz, the reader is referred to Tyndalls Lectures on Sound, p. 64; but to describe it shortly may be said in its original form to consist of two equal disks, one forming the top of a hollow fixed cylinder, into which air can be driven, the other capable of revolving concentrically upon it with the smallest possible amount of friction. A circle of small holes equidistant from each other is bored upon each disk and concentric with it; those in the upper disk being inclined slantwise to its plane, those in the lower being slantwise also but in the opposite direction; there are also arrangements both for driving a constant supply of air into the hollow cylinder, and for registering the number of revolutions the upper disk performs in a minute; thus, when the upper disk is so turned that its holes coincide with those of the lower, and air is forced into the cylinder, it will pass out through the perforations, and by reason of their obliquity will cause the movable disk to revolve with a rapidity corresponding to the pressure; and each time that the holes of the former coincide with those of the latter a number of little puffs of air get through simultaneously, giving rise to an agitation in the surrounding atmosphere which spreads round in all directions in the way before described, and if the pressure of the air in the cylinder is sufficient, the series of impulses thus given will link themselves together, forming a continuous note. (It should be remarked that the pitch of the sound would be exactly the same if there were only one perforation in the revolving disk, the number of holes merely serving to increase its intensity; if the number of holes in the revolving disk is less than the number in the lower one, those of the former must be situated so as all to coincide simultaneously with an equal number of the latter. (It should also be remarked that this syren is an old invention. Today there are any number of electronic and computer instruments that are far more accurate and precise than this mechanical device.)
Hence to determine the number of vibrations per second, corresponding to a sound of a given pitch, we have only to maintain such a pressure of air in the syren as will cause it to produce the same sound for the space of a minute, and note the number of revolutions registered in that time. Now, for every revolution of the upper disk, the same number of sound waves are propagated around as there are perforations, hence the whole number propagated in a second will be the product of the number of holes and number of revolutions per minute divided by 60; and this result will evidently be the required number of vibrations per second caused by the given sound.
To apply this to find the wavelength corresponding to the note given by the open C string of the violoncello, we should adjust the supply of air to the syren until it gives a note of the same pitch. Supposing the number of holes in each disk to be 18, the number of revolutions per minute would be found to be 220. Hence the number of vibrations per second of the string, and therefore of the surrounding particles of air, would be 220 x 18/ 60 = 66. Supposing the temperature were 16° C the velocity of sound would be about 1122 feet per second, and the number by 66 gives the wave length corresponding to that number of vibrations per second; that is, just 17 feet; the sound then will travel through this distance during the time the string takes to perform one complete vibration.
8) If the number of vibrations per second be increased, the pitch of the sound caused by them is raised, and vice versa, as can easily be illustrated by driving more or less air into the syren, and observing the sound it produces. Dr. Wollaston has shown (Phil. Trans. 1820, p. 336) that if the number be increased, beyond a certain limit the sound becomes inaudible, although this limit is not the same for all ears, some persons being perfectly sensible to sounds inaudible to others. In general it is probable that no sound is heard when the number of vibrations per second exceeds 40,000; while on the other hand the perception of pitch appears to begin when the number of vibrations is somewhere between 8 and 32, the wave length being in the former case about 0.03 of an inch - in the latter ranging from 140 feet to 35 feet.
9) Sounds are primarily divided into two classes, musical and unmusical; the former being defined as those produced by regular or periodic vibrations, the latter by such as are irregular or non-periodic. These definitions require some explanation, since, by sounding together a sufficient number of notes sufficiently near in pitch, it is plain that we could produce as unmusical (noise) a sound as we pleased, although the components would be themselves due to periodic vibrations, and would therefore musical. The answer to this is found in the fact that when two or more sets of sound waves impinge on the ear at the same instant, since each one cannot impress its own particular vibration on the tympanum contemporaneously with those of the others, the motion of the latter membrane must be in some way (be) the sum of all the different motions which the different sets of waves would have separately caused it to follow; and this is what in fact does happen, i.e., the vibrations due to each set combine and throw the tympanum into a complicated state of vibration, causing the sensation of the consonance or combination of the different sounds from which the sets of sound waves proceed.
Now the unassisted ear is only able to distinguish the separate notes out of a number sounded at once up to a certain point; beyond this it fails to distinguish them individually, and is conscious only of a confused mixture of sounds which approaches the more nearly to the character of noise the more components there are, or the nearer they lie to one another. A noise, then, may be defined as a sound so complicated that the ear is unable to resolve or analyze it into its original constituents.
10) As the character of a sound depends upon that of the vibrations by which it is caused, it is important to know of what kind the latter must be in order that they may give the sensation of a perfectly simple tone, i.e., one which the ear cannot resolve into any others. Such a vibration is perhaps best realized by comparison with that of the pendulum of a clock when it is swinging only a little to and fro. Under these circumstances it is performing what are called harmonic vibrations, and when the air particles in the neighborhood of the ear are caused by any means to vibrate according to the same law as that which the pendulum follows, and also with sufficient rapidity, a perfect simple tone is the result. Such a tone is, however, rarely heard except when produced by means specially contrived for the purpose. If a note on the pianoforte is struck, the impact of the hammer on the string throws it into a state of vibration, which, though periodic, is not really harmonic; consequently we do not hear a perfectly simple tone, but one which is in reality a mixture of several higher simple tones with that one which corresponds to the actual length of the string. The former are, however, generally faint, and become associated by habit with the latter, appearing to form with it a single note of determinate pitch. These higher tones are harmonics of the string, and are produced by vibrations whose numbers per second are respectively twice, three times, four times, etc., as great as those of the fundamental tone of the string. The same may be said of the notes of all instruments, including the human voice, which are usually employed for the production of musical sounds.
11) Since the consonance of two or more such simple tones always gives a more or less musical sound, and since also the ear is always more or less capable of resolving the latter into its components, the question naturally arises whether all sounds are not, theoretically at least, resolvable into simple tones. The answer to this is contained in a celebrated theorem due to the French mathematician Fourier. He has shown that any periodic vibration is the result of combining together a certain number of simple harmonic vibrations whose periods are aliquot parts of that of the former; and we have conclusive reasons for supposing that, in the same way as a compound periodic vibration gives rise to a compound sound, so the simple tones into which the ear resolves the latter are respectively due to the simple harmonic vibrations which, as the above mentioned theorem proves, make up the former.
12) The theorem of Fourier referred to in the preceding article is of such great importance in all questions connected with acoustics that a few words illustrative of it may not be out of place.
If a peg is fixed into the rim of a wheel capable of revolving about a fixed center, and at right angles to the plane of the wheel, and if the latter is caused to rotate uniformly and is looked at edge ways the peg will appear to move up and down in a straight line, its velocity being the greatest at the middle of its course, and diminishing as it approaches each end. Under these circumstance the peg appears to perform harmonic vibrations.
Now suppose a second wheel, also furnished with a peg in its rim, is made to revolve about the peg of the first as an axis. If the latter is at rest the peg of the second will appear, looked at as above, to perform harmonic vibrations; but if the former is also caused to revolve these vibrations are no longer harmonic, but are the result of adding together the separate harmonic vibrations of the two pegs, in other words of superposing the harmonic vibrations which the second peg performs if the first wheel is at rest, upon those which the first peg performs when it is itself in motion. Now it is evident that by continuing this process indefinitely, and by giving the wheels different radii, and different uniform velocities of rotation, the final motion of the last peg looked at sideways as before, would be an exceedingly complicated one, and that an indefinite number of different vibrations could be produced by varying the number, position at starting, radii, and velocities of the wheels, though it could not be assumed without proof that every possible variety could be so produced. This however is what Fouriers theorem asserts, provided that the velocities of rotation of the several wheels of the series are in the proportion of 1, 2, 3, 4, 5, etc. In other words, every periodic vibration is the resultant of a certain number of harmonic vibrations whose periods are one-half, one-third, one-fourth, etc., that of the former.
13) A harmonic scale is formed by taking a series of notes produced by vibration whose numbers in a given time are respectively as 1, 2, 3, 4, et.
If we take as fundamental tone the open C string of the violoncello, the series of tones which with it forms a harmonic scale will be as pictured:
The notes marked with an asterisk do not exactly represent the corresponding tones; but are the nearest representatives which the modern notation supplies. All the notes of the harmonic scale can theoretically be produced by either a single string, or by a simple tube used as a trumpet. If we lightly touch the string of a violin, without causing it to come in contact with the finger board, at any one of a series of points dividing it into a number of a equal parts, and excite it by means of a bow, it no longer vibrates as a whole, but separates into the number of equal vibrating segments which is the least possible consistent with that point forming one of their points of diversion; the latter remain stationary, or very nearly so, and are called nodes, their number being evidently just one less than that of the segments. It is plain that if the point of application of the bow be one of a series of nodes, no sound will be produced, provided, of course, the finger remains on any other of the same series, and this may serve to explain why it is sometimes difficult to bring out the higher harmonics of a violin, as the bow may, unconsciously to the performer, be passing exactly over one of the corresponding nodes. The first harmonic, as it is called, of the open string is produced by touching it while on a state of vibration at its middle point, and thereby dividing it into two equal portions, both of which vibrate twice as fast as the whole, and accordingly give the octave. The second harmonic, or the twelfth of the fundamental, corresponds to a division of the string into three equal portions, and so on. And generally, in order to produce the nth harmonic the finger should touch the string at any one of the series of points which divide it into n equal portions. In practice, however, the finger should always touch the string at the point of division adjacent to either end.
14) The harmonics of a simple tube used as a trumpet are the same as those of a vibrating string, viz., the octave, twelfth, fifteenth, etc., and are produced by modifications of the breath and lips; but there is a great difference between the nature of the vibrations which produce sound, in the case of strings and pipes. In the former case the vibrations are executed at right angles to the length of the string, that is, are lateral or transverse, while in the latter they are in the direction of the pipe, or longitudinal, and are the vibrations of the air itself with it. Actually the string has all three primary modes of vibration with the lateral or transverse predominating.
15) When an open organ pipe is sounding its fundamental tone, the particles of the column of air within it are all, more or less, in a state of vibration parallel to the length of the pipe, of which the intensity is at its maximum at the two ends, growing less and less towards the middle, where there is a node, that is, a point of no disturbance. The harmonics of an open organ pipe follow the same law as those of a simple trumpet, or vibrating string.
The fundamental note of a stopped organ pipe is an octave below the fundamental note of an open organ pipe of the same length. When it is sounding this note there is no node, and the first harmonic is a fifth above the octave, the second a major sixth above the first, the third a diminished fifth above the second, and so on. Or, more simply, the successive tones of the harmonic scale of an open pipe are produced by vibrations which are as 1, 2, 3, 4, etc., those of a stopped pipe by vibrations which are as 1, 3, 5, 7, etc.
16) It was stated (§10) that the sound of a vibrating string was in general compounded of a number of simple tones, and a well trained ear can detect a considerable number of them. If it were not for these harmonic components the tones of strings, pipes, of the human voice, or in short, of every instrument most generally used for the production of sound, would be flat and uninteresting like pure water. Each harmonic component is by itself a simple tone, and is due to the vibration of the corresponding segment of the string superposed upon that of the whole. The same statement applies, mutatis mutandis to pipes, whether open or stopped. That the harmonics of different instruments greatly influence their several characters is observable in the difference of the tones of a flute, and clarinet. A flute is an open pipe, a clarinet a stopped one end; in the former, therefore, the harmonics follow the order of the natural numbers 1, 2, 3, 4, and in the latter the order 1, 3, 5, 7; the intermediate notes being supplied by opening the lateral orifices of the instrument.
17) When two simple tones, that is (as explained above), notes deprived of all the harmonic components which under ordinary circumstances accompany them, are sounded together very nearly in unison, there are heard what are called beats succeeding one another at regular intervals, their rapidity depending inversely on the smallness of the interval between the two tones. Their origin may be explained thus: Suppose the tones to be produced by vibrations numbering 500 and 501 per second respectively, then every 500th sound wave of the former will strike on the tympanum at exactly the same instant as every 501st of the latter and will reinforce it; while at the 250th of the first the corresponding wave of the other will be just half a period in front of it. Now a sound wave consists of a condensed and rarefied stratum of air particles, and therefore the condensed portion of one wave here coincides with the rarefied portion of the other and neutralizes it. Thus there will be an alternate reinforcement and diminution of sound, every second, from the maximum intensity when both waves impinge on the tympanum at the same instant to the minimum when they counteract each other as much as possible and vice versa.
In the above case it was supposed that the number of vibrations of one tone were only one more per second than those of the other; but if the difference of the numbers had been two, for instance, then in one second the first tone would have gained two vibrations on the other, and there would have been two beats; and in general the number of beats per second is always equal to the difference between the two rates of vibrations per second.
18) In the preceding section, the cause of beats due to two simple tones of nearly the same pitch was explained, and it was seen that the number of beats per second was always equal to the difference of the numbers of vibrations per second of each tone; so that as the interval between them increased so would the number of beats increase in a given time. Hence it is obvious that if the interval became sufficiently large, the beats would succeed each other so rapidly as to become undistinguished. For instance, in the case of the fifth whose lower and upper tones are produced by vibrations numbering 264 and 396 per second respectively, the number of beats per second would be 132 and would therefore be undistinguished; and still more so supposing the upper tone to have 397 or more vibrations per second; but, on the other hand it is a well-known fact, that if an imperfect fifth, octave, or any other tolerably simple interval is played on a violin or violincello, the beats are most distinctly heard succeeding each other at perceptible intervals; whereas according to what was said above they should occur so rapidly as not to be heard at all. Two explanations of this phenomenon have been given, of which by far the most simple is due to Helmholtz; and which here follows. It appears that when the tones are simple and at a sufficiently large interval the beats should occur too rapidly to be heard, whereas when the interval is played on a violin they are easily distinguishable. The reason of this fact is that in the latter case the tones are no longer simple but compound; and the beats which are heard are not due to the fundamental tones themselves but arise from two of their harmonic components which are nearly unison. Suppose the ratio of the interval between the fundamental tones to be m/n, that is, let m/n, be the fraction, reduced to its lowest terms, which is formed by putting in the numerator the number of vibrations per second of the upper tone, and in the denominator those of the lower. Then it is plain that the nth harmonic component of the tone m, will be of the same pitch as the mth harmonic component of the tone n; for they will each have exactly mn vibrations per second. Now let M/N be the ratio, expressed in the same way, of another interval, nearly, but not quite, equal to m/n; then the nth harmonic component of M will have Nm vibrations per second, while the mth component of N will have Nm. Now since M/N is nearly equal to m/n, the difference between Mn and Nm will be a small number; and when the two notes are sounded together the number of beats per second will be equal to that difference.
For example, let m/n, be the ratio of a fifth, that is the fraction 3/2, and let M/N represent very nearly the same interval, say 397/264; then the difference between Mn and Nm, or 794 and 792, is 2; hence if two strings tuned apart at an interval represented by 397/264 are sounded simultaneously there will be two beats heard per second.
19) When the vibrations of the air due to a number of different sounds which co-exist at the same time are infinitely small, they are merely superposed one on another, so that each separate sound passes through the air as if it alone were present; and this Law of Supposition holds, though only approximately, until the vibrations have increased up to a certain limit, beyond which it is no longer true. Vibrations which give rise to a large amount of disturbance produce secondary waves; and it is to these that the phenomena of resultant tones are due.
Thus if two notes a fifth apart, for instance, are forcibly sounded together, a third tone is heard an octave below the lower of the two, and this ceases to be perceptible when the loudness of the concord diminishes. In general the resultant tone of any combination of two notes is produced by a number of vibrations per second equal to the difference of the numbers per second of the notes. This fact formerly led to the supposition that the resultant tone was produced by the beats due to the consonance, which, when they occurred with sufficient rapidity, linked themselves together so as to form a continuous musical note. If this were so it is clear that the resultant ought to be heard when the original notes are sounded gently as well as forcibly; and it was the failure of this condition that led Helmholtz to the re-investigation of their origin. These resultant tones have been named by him difference tones; he has also discovered the existence of resultant tones formed by the sum of the numbers of vibrations of the primaries. These summation tones as they are called cannot be explained on the old theory.
20) The theory of beats explains the law that the smaller the two numbers are, which express the ratio of their vibrations, the smoother is the combination of any two tones. When two simple tones are sounded together whose rates of vibration per second differ by more than 32, the beats, according to Helmholtz, totally disappear. As the difference grows less the beats become more and more audible, the interval meanwhile growing proportionately dissonant, till they number 33 per second, at which point the dissonance of the interval is at its maximum.
This, however, depends upon the position of the interval as regards its pitch. For it should be remembered that though the ratio of any given interval remains the same whatever the absolute pitch of its tones may be, yet the difference of the actual numbers of their vibrations, and therefore the number of beats due to their consonance, alters with it. And vice verse, if the difference of the number of vibrations remains constant, the interval must diminish as its pitch rises. For instance, either of the following combinations would give rise to 33 beats per second, since the numbers of vibrations of their tones per second, are 99-66, and 528-495, respectively. Now it is obvious that in the latter case the dissonance would be far greater than in the former.
The above explanation of the cause of dissonance is also due to Helmholtz, and completely solves a question which had remained unanswered since the time of Pythagoras, although that philosopher made the important discovery that the simpler the ratio of the two parts into which a vibrating string was divided, the more perfect was the consonance of the two sounds.
21) The sound of the piano, violin, etc., is only in a small measure due to the actual vibration of the strings themselves. The latter communicate their own motion to the sound board of the piano, and to the front, back, and enclosed air of the violin. In the latter instrument communication is made to the surrounding air from that within it by means of the holes.
If a string were merely stretched between two pegs firmly fixed in a stone wall and caused to vibrate, scarcely any sound would be heard at all, owing to the mass and rigidity of the wall, which would refuse to be thrown into vibration by so small an amount of energy as that which the string would possess. On the other hand, the sound board of a piano readily answers to the vibrations imposed on it when the string is struck, and having a large surface in contact with the air, every point of which originates a system of waves, it causes a full and powerful sound.
22) The vibrations of straight rods may be either longitudinal or transversal. The former have not been generally employed for the production of musical sounds; the latter are such as take place when a tuning fork is struck, or when a musical box or triangle is played. In the case of a curved rod the vibrations are more complicated, but there is one interesting case, namely, that in which the curved rod takes the form of a circular ring. In this case the fundamental tone is obtained by suspending it horizontally by four strings attached at equidistant points in the circumference, and by lightly tapping it midway between any two. If the number of vibrations then given be 2n per second, those of the successive harmonics are proportional to 3nsqrt6, 4nsqrt13, 5nsqrt22, etc.
23) The nature of the vibrations of a bell may be partly inferred from those of a ring, as the bell may be considered as consisting of a connected series of rings of different diameters all vibrating simultaneously; thus the fundamental tone of a bell would cause it to divide itself longitudinally into four equal segments, corresponding to the four quadrants into which the suspended ring divides. The period of its vibrations could not, however, be similarly inferred.
24) The vibration of plates is not, musically speaking, a subject of much interest, as the only instruments which depend upon it directly for the production of their sounds, are gongs and cymbals, and the same may be said of membranes. Chladni was the first to show the positions of the lines of nodes on a plate, by clamping it horizontally in a vice, and causing it to vibrate by passing a violin bow over one edge, having previously sprinkled it with a little sand. The lines of nodes being those parts of the plate which, like the nodes of a string (§13), are not thrown into vibration, remain covered with the sand which collects there from the vibrating portions, and in this way very curious and interesting figures are produced.
ChatGPT analyzes the above text [2/28/25]:
Here is an analysis and rewrite of each numbered point to enhance clarity and understanding:
1) Definition of Acoustics
Original: The science which treats of the nature and laws of sound. Rewritten: Acoustics is the branch of physics that studies sound, its generation, transmission, reception, and interaction with various materials.
2) Sensation of Sound
Original: The sensation of sound consists in the communication of a vibratory motion to the tympanic membrane of the ear, through slight and rapid changes in the pressure of the air on its outer surface. Rewritten: Sound perception occurs when air pressure variations cause vibrations in the eardrum, transmitting these vibrations to the brain via the auditory system.
3) Propagation of Sound in Air
Original: A detailed explanation using an explosion analogy, describing sound waves as expanding pressure shells. Rewritten: Sound travels through air in waves, where air particles pass along vibratory motion in a chain reaction. If a series of sound-producing vibrations occur at regular intervals, they create a continuous sound, as in musical tones. The motion of air particles is distinct from the movement of the sound wave itself.
4) Sound Intensity and the Inverse Square Law
Original: Sound intensity is proportional to the square of the vibrating particle’s velocity and decreases with the square of the distance from the source. Rewritten: The loudness of sound depends on the speed of vibrating particles. As sound spreads outward, its energy is distributed over an increasing area, decreasing intensity by the square of the distance.
5) Velocity of Sound
Original: Sound travels at 1,090 feet per second at 0°C, increasing by 2 feet per second for every additional °C. Rewritten: The speed of sound in air is approximately 1,090 feet per second (332 m/s) at 0°C, increasing slightly with temperature. The velocity depends on air’s elasticity and density, but in normal conditions, air’s elasticity and density cancel out, making sound speed independent of air density.
6) Pitch, Frequency, and Standardization
Original: Sound differs in quantity, quality, and pitch. Pitch is determined by frequency, with middle C defined as 264 Hz (German pitch) or 244 Hz (Western pitch). Rewritten: Pitch is the perceived highness or lowness of a sound, determined by the number of vibrations per second (Hertz, Hz). Standard tuning assigns Middle C to approximately 264 Hz (historical) or 244 Hz (Western tuning at the time).
7) Wavelength and Frequency Relationship
Original: Wavelength is the distance sound travels while one particle completes a vibration cycle. An early device (syren) was used to measure frequency. Rewritten: Wavelength is the physical distance a sound wave travels in the time it takes for one full cycle of vibration. It is calculated by dividing sound speed by frequency. Early experiments used the syren device to determine frequency by measuring airflow interruptions.
8) Upper and Lower Limits of Audibility
Original: Sound pitch perception varies by individual; frequencies above 40,000 Hz are usually inaudible, while perception starts between 8 Hz and 32 Hz. Rewritten: The human hearing range is typically 20 Hz to 20,000 Hz, though some individuals can hear beyond these limits. Frequencies above 40 kHz are generally inaudible, while the lowest perceivable frequencies start around 8–32 Hz.
9) Musical vs. Unmusical Sounds (Noise)
Original: Musical sounds have regular vibrations; noise is irregular. Too many simultaneous frequencies cause noise rather than music. Rewritten: Music consists of periodic (regular) vibrations, while noise is a chaotic mix of irregular frequencies. When too many tones overlap, the human ear perceives a blurred sound rather than distinct notes.
10) Simple Tones and Harmonics
Original: A pure tone consists of a single frequency. Most musical sounds contain additional harmonics that define timbre. Rewritten: A simple tone is a sound wave containing only one frequency (like a tuning fork). Musical sounds typically consist of multiple harmonics, which give instruments their unique tone.
11) Fourier’s Theorem and Sound Decomposition
Original: Any periodic vibration can be broken into a sum of harmonic vibrations (Fourier’s Theorem). Rewritten: Fourier’s Theorem states that any complex sound wave can be broken down into a series of simpler sine waves (harmonic components). This explains how different tones combine to form complex sounds.
12) Illustration of Fourier’s Theorem
Original: A peg on a rotating wheel represents harmonic motion. Adding more wheels with pegs creates complex vibrations. Rewritten: Fourier analysis is demonstrated by imagining a peg on a rotating wheel, moving up and down in a simple harmonic motion. When multiple wheels rotate at different speeds, their combined motion represents complex sound waves formed by multiple harmonics.
13) Harmonic Scale and String Vibrations
Original: The harmonic series is generated when a vibrating string is divided into equal sections. Rewritten: The harmonic series consists of tones with frequencies in integer ratios (e.g., 1:2:3:4). A string, when lightly touched at certain points, vibrates in sections, producing harmonics that define musical timbre.
14) Harmonics in Wind Instruments
Original: Harmonics of a trumpet-like tube are similar to string harmonics but involve longitudinal vibrations. Rewritten: In wind instruments, air vibrations form harmonic series similar to vibrating strings, but with longitudinal rather than transverse waves.
15) Open vs. Stopped Pipes
Original: Open organ pipes follow the harmonic series, while stopped pipes produce only odd harmonics. Rewritten: Open organ pipes vibrate in full harmonic series (1, 2, 3, 4…), while stopped pipes allow only odd harmonics (1, 3, 5, 7…), resulting in a different timbre.
16) Harmonics Define Instrument Timbre
Original: Harmonics distinguish instrument sounds, e.g., flute vs. clarinet. Rewritten: The unique tone color (timbre) of instruments depends on their harmonic structure. A flute (open pipe) follows a full harmonic sequence, while a clarinet (stopped pipe) emphasizes odd harmonics.
17) Beats and Interference
Original: When two frequencies are slightly different, they create beats due to alternating reinforcement and cancellation. Rewritten: Beats occur when two close frequencies interfere, causing alternating loud and soft cycles. The beat frequency equals the difference between the two frequencies.
18) Beats and Tuning
Original: Imperfect fifths and octaves create beats, making tuning adjustments necessary. Rewritten: Small deviations in musical intervals (like a fifth or octave) create beats, helping musicians fine-tune instruments by listening for periodic pulsations.
19) Resultant Tones
Original: Loud sounds create difference tones and summation tones, not just primary notes. Rewritten: Resultant tones arise from two interacting tones, generating new frequencies at the sum or difference of their frequencies.
20) Dissonance and Beating
Original: Dissonance increases when beats become audible. Rewritten: Dissonance occurs when beats between frequencies become rapid and noticeable, interfering with smooth sound perception.
21) Soundboards and Amplification
Original: A string alone produces little sound, but a resonating body (e.g., a violin’s wooden body) amplifies it. Rewritten: Soundboards (e.g., piano and violin bodies) amplify vibrations, making strings louder by enhancing resonance.
22) Vibrations of Rods and Rings
Original: Straight and curved rods vibrate differently. Rewritten: Rods can vibrate longitudinally or transversely, while rings exhibit complex vibratory patterns.
23) Bell Vibrations
Original: Bells vibrate as multiple connected rings. Rewritten: Bells vibrate as concentric rings, each producing unique harmonics.
24) Vibrations of Plates and Membranes
Original: Chladni figures show vibration patterns using sand on a vibrating plate. Rewritten: Chladni patterns reveal how plates and membranes vibrate, forming distinct nodal lines where no motion occurs.
Grok Analyzes the above original text [2/28/25]:
Below is an analysis and rewritten version or definition of each numbered point from the provided text. The goal is to clarify and simplify the concepts for better understanding while preserving the original meaning.
1) Original: The science which treats of the nature and laws of sound.
Rewritten: Acoustics is the branch of science that studies the properties, behavior, and principles governing sound.
2) Original: The sensation of sound consists in the communication of a vibratory motion to the tympanic membrane of the ear, through slight and rapid changes in the pressure of the air on its outer surface.
Rewritten: Sound is perceived when rapid, small changes in air pressure cause the eardrum (tympanic membrane) to vibrate, transmitting these vibrations to the brain as the sensation of sound.
3) Original: The mode of propagation of sound in air may be explained in the following manner...
Rewritten: Sound travels through air as a wave, created by a disturbance (like an explosion) that pushes air particles outward in all directions. These particles bump into neighboring ones, passing the motion along like a ripple. This forms an expanding "shell" of vibrating air, where each particle briefly oscillates once as the wave passes. Multiple disturbances, like a series of explosions or vibrations from an object (e.g., a string), create repeated waves. If these occur quickly enough, they produce a continuous sound, such as a musical note. Importantly, the wave’s movement is distinct from the particles’ motion—each particle only shifts slightly back and forth, not traveling with the wave.
Key Definitions:
Amplitude: The distance a particle moves from its resting position during vibration (half the total back-and-forth distance). Period: The time it takes for one complete vibration.
4) Original: The intensity of a sound is proportional to the square of the maximum velocity of the vibrating particle...
Rewritten: The loudness (intensity) of a sound depends on how fast the air particles vibrate—specifically, it’s proportional to the square of their maximum speed. Additionally, sound gets quieter as you move farther from its source because the same energy spreads over a larger area. For example, at twice the distance, the energy is spread over four times the area (due to the square of the distance), so the intensity drops to a quarter of what it was. This is known as the "inverse square law."
Key Concept:
Inverse Square Law: Sound intensity decreases with the square of the distance from its source (e.g., at 3 feet, it’s 1/9th as intense as at 1 foot).
5) Original: At a temperature of zero Centigrade sound is propagated at the rate of about 1090 feet per second...
Rewritten: The speed of sound in air is about 1,090 feet per second at 0°C and increases by roughly 2 feet per second for each degree Celsius warmer. For example, at 15°C, it’s about 1,120 feet per second. This speed depends on air’s elasticity (how easily it compresses and rebounds) and density (how heavy it is). Elasticity increases with temperature, speeding up sound, while density’s effect cancels out at a constant temperature, leaving speed independent of density alone.
Key Formula:
Speed ≈ √(elasticity / density).
6) Original: One sound differs from another not only in quantity, but also in quality and pitch...
Rewritten: Sounds vary in three ways: loudness (quantity), tone color (quality), and pitch (highness or lowness). Pitch is determined by how many vibrations occur per second—more vibrations mean a higher pitch, fewer mean a lower one. To compare pitches, we use a standard: middle C on a piano is set at 264 vibrations per second (German pitch; Western pitch uses 244).
Key Terms:
Pitch: The perceived highness or lowness of a sound, tied to vibration frequency. Quality: The unique "timbre" or character of a sound, distinguishing, say, a flute from a violin.
7) Original: Knowing the velocity of sound in air we can estimate the different wave lengths corresponding to notes of different pitch...
Rewritten: Wavelength—the distance a sound wave travels during one vibration—can be calculated by dividing the speed of sound (e.g., 1,122 feet per second at 16°C) by the number of vibrations per second (frequency). For example, a cello’s C string vibrating 66 times per second has a wavelength of 1,122 ÷ 66 ≈ 17 feet. An old device called a syren, with spinning disks and air puffs, helps measure frequency by matching a sound’s pitch and counting vibrations.
Key Formula:
Wavelength = Speed of sound ÷ Frequency.
8) Original: If the number of vibrations per second be increased, the pitch of the sound caused by them is raised...
Rewritten: Increasing the vibration frequency raises a sound’s pitch; decreasing it lowers it. This can be tested with a syren by adjusting airflow. Humans hear sounds between about 8 and 40,000 vibrations per second—below 8, it’s too low to hear; above 40,000, it’s ultrasonic (inaudible), though hearing limits vary by person.
Key Range:
Audible pitch: ~8 to ~40,000 vibrations per second (wavelengths from 140 feet to 0.03 inches).
9) Original: Sounds are primarily divided into two classes, musical and unmusical...
Rewritten: Sounds are either musical (from regular, repeating vibrations, like a note) or unmusical (from irregular vibrations, like noise). However, combining many musical notes close in pitch can create a noise-like effect because the ear can’t separate them—it hears a jumbled mix instead. Noise is thus a complex sound the ear can’t break into distinct parts.
Key Terms:
Musical Sound: Periodic, predictable vibrations. Noise: Irregular, chaotic vibrations or unresolved combinations.
10) Original: As the character of a sound depends upon that of the vibrations by which it is caused...
Rewritten: A sound’s nature reflects its vibrations. A “simple tone” (pure, single pitch) comes from smooth, regular vibrations, like a pendulum’s swing (harmonic motion). Most real sounds, like a piano note, are complex—vibrations mix a main tone with fainter “harmonics” (multiples of the base frequency: 2x, 3x, etc.). These harmonics shape the sound’s unique character, though we often hear them as one note.
Key Term:
Harmonics: Higher-pitched tones accompanying the main (fundamental) tone.
11) Original: Since the consonance of two or more such simple tones always gives a more or less musical sound...
Rewritten: When simple tones combine pleasantly (consonance), the ear can often pick out individual pitches. This raises a question: can all sounds be broken into simple tones? Fourier’s theorem says yes—for periodic vibrations, any complex wave is a sum of simple harmonic waves with frequencies that are multiples (1x, 2x, 3x, etc.) of a base frequency.
Key Concept:
Fourier’s Theorem: Every repeating vibration is a blend of simple harmonic vibrations.
12) Original: The theorem of Fourier referred to in the preceding article is of such great importance...
Rewritten: Fourier’s theorem is crucial in sound science. Imagine a peg on a spinning wheel—it moves up and down smoothly (harmonic motion). Add another wheel spinning around that peg, and the motion gets complex, combining both wheels’ harmonic patterns. With more wheels at speeds of 1x, 2x, 3x, etc., any periodic motion can be built, showing how complex sounds break into simple tones.
Key Idea: Complex vibrations = sum of simpler harmonic parts.
13) Original: A harmonic scale is formed by taking a series of notes produced by vibration whose numbers in a given time are respectively as 1, 2, 3, 4, etc.
Rewritten: A harmonic scale is a series of notes with vibration frequencies in ratios like 1, 2, 3, 4, etc. For a cello’s C (66 vibrations/second), harmonics are 66, 132, 198, etc. On a string, touch it lightly at points (e.g., halfway) to divide it into vibrating segments, producing these higher tones (octave, twelfth, etc.). Nodes are still points between segments.
Key Term:
Nodes: Points of no vibration on a string.
14) Original: The harmonics of a simple tube used as a trumpet are the same as those of a vibrating string...
Rewritten: A trumpet (simple tube) produces harmonics like a string (octave, twelfth, etc.) using breath and lip adjustments. String vibrations are sideways (transverse); tube vibrations are lengthwise (longitudinal) air movements. Strings have both types, but transverse dominates.
Key Difference: String = transverse; Tube = longitudinal.
15) Original: When an open organ pipe is sounding its fundamental tone...
Rewritten: An open organ pipe’s base tone comes from air vibrating strongest at both ends, weakest at a middle node. Its harmonics match a string’s (1, 2, 3, etc.). A stopped pipe (closed at one end) has a lower base tone (octave below open) and odd harmonics only (1, 3, 5, etc.), with no middle node for the base tone.
Key Pattern:
Open pipe: 1, 2, 3, 4… Stopped pipe: 1, 3, 5, 7…
16) Original: It was stated (§10) that the sound of a vibrating string was in general compounded of a number of simple tones...
Rewritten: String and pipe sounds mix a main tone with harmonics, adding richness. A trained ear hears these. A flute (open pipe) has all harmonics (1, 2, 3…); a clarinet (stopped) has odd ones (1, 3, 5…), shaping their distinct tones.
Key Idea: Harmonics define an instrument’s sound.
17) Original: When two simple tones... are sounded together very nearly in unison, there are heard what are called beats...
Rewritten: Two close simple tones (e.g., 500 and 501 vibrations/second) create “beats”—a pulsing sound. When waves align, they amplify; when misaligned, they cancel. Beats per second equal the frequency difference (e.g., 501 - 500 = 1 beat/second).
Key Formula: Beats = |Frequency1 - Frequency2|.
18) Original: In the preceding section, the cause of beats due to two simple tones of nearly the same pitch was explained...
Rewritten: Beats increase with larger frequency gaps, blending into a hum if too fast (e.g., 132 beats/second for 264 vs. 396). On violins, audible beats in intervals like a fifth come from harmonics, not base tones. For a 3:2 ratio (fifth), harmonics align (e.g., 3x vs. 2x), but slight mistuning (e.g., 397/264) causes slow beats between harmonics (e.g., 2 beats/second).
Key Insight: Beats from harmonics, not fundamentals, explain real-world effects.
19) Original: When the vibrations of the air due to a number of different sounds... are infinitely small...
Rewritten: Small vibrations from multiple sounds overlap cleanly, each acting independently. Loud sounds create “difference tones” (frequency = |f1 - f2|) and “summation tones” (f1 + f2), audible when strong. For a fifth (e.g., 264 and 396), a difference tone is 132 (an octave below).
Key Terms:
Difference Tone: f1 - f2. Summation Tone: f1 + f2.
20) Original: The theory of beats explains the law that the smaller the two numbers are, which express the ratio of their vibrations...
Rewritten: Simpler frequency ratios (e.g., 2:1, 3:2) sound smoother because beats are fewer or faster (inaudible above ~32/second). Dissonance peaks at ~33 beats/second, varying with pitch—e.g., 528 vs. 495 (33 beats) is harsher than 99 vs. 66 (33 beats) due to a tighter interval.
Key Idea: Dissonance ties to beat rate and pitch.
21) Original: The sound of the piano, violin, etc., is only in a small measure due to the actual vibration of the strings themselves...
Rewritten: Piano and violin sounds come mostly from their bodies (soundboard, air cavity) amplifying string vibrations. A string alone on a rigid wall is quiet—too little energy moves the air. A soundboard’s large surface boosts the sound.
Key Idea: Amplification makes strings audible.
22) Original: The vibrations of straight rods may be either longitudinal or transversal...
Rewritten: Rods vibrate lengthwise (longitudinal, rare in music) or sideways (transverse, e.g., tuning forks). A ring (curved rod) tapped at points vibrates with a base tone (2n) and harmonics like 3n√6, 4n√13, etc.
Key Pattern: Ring harmonics follow a unique formula.
23) Original: The nature of the vibrations of a bell may be partly inferred from those of a ring...
Rewritten: A bell acts like stacked rings, vibrating in four segments for its base tone. Its full vibration pattern is complex and not fully predictable from a ring alone.
Key Idea: Bells build on ring-like vibrations.
24) Original: The vibration of plates is not, musically speaking, a subject of much interest...
Rewritten: Plates (e.g., gongs) and membranes vibrate musically but are less studied. Chladni showed plate nodes by bowing an edge, sprinkling sand—sand settles on still lines, forming patterns.
Key Term: Nodes: Non-vibrating areas on a plate.
See Also
Acoustic Impedance
Acoustic Levitation
Acoustic
Acoustics
American Acoustical Society
Analytic Study of Harmonic Intervals
Bioacoustics
Compression Wave
First Principles of Physics
Handbook of Acoustics
Harmonics
Keelys Forty Laws
Laws
Laws of Vibration
Oscillation
Overtone Series
Principles
Quantum Acoustics
Sound
Table 14.03 - Ranges of Forces Vibration Forms Types and Governing Laws
Vibration
17.20 - Acoustic Levitation
8.25 - Triune Acoustic Forces