By A.H. BENADE
In the second half of chapter 16 we examined the interplay between the physics of vibrating strings and the tuning behavior of various kinds of keyboard musical instruments. In the present chapter we will turn our attention to the way in which the strings of a piano, harpsichord, or clavichord communicate their carefully tuned vibrations to the soundboard and thence to our ears. Thus we will be focusing our attention on the nature of the sounds produced by these instruments. In broad outline, the chapter will begin by examining the requirements that must be met by a string belonging to a note in the middle of the piano keyboard. This is followed by an account of what is accomplished by the use of more than one string per note and a description of the changes and compromises necessary for satisfactory production of lower and higher tones. Having dealt with sounds produced on the piano, we will be ready in chapter 18 to adapt our understanding of these basic principles to the clavichord and the harpsichord.
17. 1. The Soundboard As Seen by the Strings; The Concept of Wave Impedance
Anyone looking into a grand piano will notice that each note in its scale has one or more strings stretched across a soundboard in the manner shown in figure 17.1. The so-called vibrating length of the string extends from a rigid capo dasrtro bar (or from a fixed agraffe) which is found at the keyboard end, to the bridge, which is elastically supported by a broad, thin soundboard. A hammer is arranged to excite this length of string by striking it at a suitably chosen point near the fixed end. Not shown in the figure is a felt damper that normally rests on the string near the hammer to keep it from vibrating when it is not in use.
In chapters 7 and 8 we learned some of the basic principles guiding the excitation of string modes by hammers of various sorts acting at various distances from the fixed end. Now it is time to look at the way in which these characteristic vibrations of the string are communicated to the soundboard and thence to the concert hall. The string and the soundboard meet by way of the bridge, so we need to know what sort of a termination the string "sees" as it looks at the place where it runs over the bridge. The bridge actually functions acoustically as a part of the soundboard, which is attached to its lower side. The soundboard is a two dimensional wave-carrying medium of the sort we first met in chapter 9. Carefully profiled ribs run across the grain on the underside of the soundboard to make its stiffness approximately the same across the grain as it is lengthwise. We have already acquired at least a general idea of the way in which such a uniform two dimensional object will respond to an excitation applied at some point on its surface (see figs. 10.13 and 10.14). In section 16.5 we also learned a little about how the driven motion of the soundboard can react back on the string to alter its natural frequencies.
There is more to what the string sees at the bridge end than just bridge and soundboard. There are some 240 other strings running over the top surface of the bridge, and these also form a kind of two
dimensional wave-carrying medium that is "visible" to our vibrating strings, albeit in a more limited way because waves are not able to run easily in a crosswise direction from string to string. (There is only the bridge to connect them to each other, with no ribs to equalize what we might in this case call the cross-grain and along-the-grain properties of the system.) The fact that most of the strings are damped by pieces of felt need not concern us at present, any more than does the question whether the edges of the Soundboard function as hinged or as clamped boundaries.
The playing string, the bridge- plus- soundboard, and the sheet of silent but down bearing strings can each be thought of as a wave-carrying medium. Acoustical theory tells us that any wave-carrying medium can be characterized fully by two specifications: the velocity with which waves are propagated along the medium, and the wave impedance. We will briefly review the first and more familiar of these before considering the idea of wave impedance.
When any sort of acoustic disturbance is made at one point in a wave-carrying medium, it takes a little while for the disturbance to make its appearance at another point a little distance away. The rate at which the disturbance travels from its source to the point of observation is what is known as the speed of sound or wave velocity (for example, we learned in sec. 11.8 that the speed of sound in air is about 345 meters/sec). The wave velocity always depends on the springiness or elasticity with which one small part of the medium acts on its neighbours during a disturbance; the wave velocity depends also on the inertia of the material (i.e., the amount of mass belonging to each of these small parts).
These are related to each other by the following formula:
wave velocity = Ö springiness / inertia
Question 2 in the final section of this chapter will help you understand why this formula for the wave velocity (the speed of sound) looks so remarkably like the formula found in section 6.1 for the natural frequency of oscillation of a spring-and-mass system.
The second concept we need for an understanding of how a struck string communicates with the soundboard is the idea of wale impedance. When a disturbance is set up in some medium and travels to the boundary between it and some other medium (as when disturbances travel along a slender wire to a thicker wire or a soundboard), a certain fraction of the disturbance is transmitted into the new medium and the remainder is reflected back into the original medium. The amplitudes of the reflected and the transmitted waves, and also the amounts of energy carried by them, all depend on the ratio of the wave impedances of the two media.' If these impedances are very different, there is almost complete reflection, with only a small share of the total energy being sent on. On the other hand, if the two media have wave impedances that are approximately equal, then there is very little reflection and the disturbance is almost completely transmitted across the junction. It turns out that wave impedance depends on the same two properties of the medium as does wave velocity, although they are arranged differently, thus:
wave impedance = = Ö springiness x inertia
Digression on Terminology-Wave Impedance vs. Characteristic Impedance.
In order to keep things clear for the lest technically oriented readers of this book, I hate chosen to use the slightly oldfashioned name wave impedance rather than the more current terns characteristic impedance. Everywhere else in this book, the perfectly customary adjective characteristic has been resend for use as a way to advertise certain attributes of some mode of vibration belonging to a particular finite system of springs and masses. Thus, for a given system, its entire behavior can be understood in terms of its modes of vibration, each of these having its own characteristic frequency of vibration, its own characteristic vibrational shape, and its own characteristic (internally caused) damping. There characteristic properties are determined jointly by the nature of the vibrating medium and by the way in which its boundaries are constrained. It is only under very special circumstances that one finds a characteristic vibration taking place in an infinitely extended system, and then it exists only in a restricted region of it. The watt impedance, on the other hand, is a way of specifying (along with the wave velocity) one of the attributes of the wavecarrying medium itself, without reference to its boundaries. As a matter of fact, one of the easy ways to measure a wale impedance is to experiment on a very extended piece of material and to conclude the measurements before any echoes can be returned by its boundaries (see sec. 17.4 below for an example of this). I might remark that those of us who use today's more conventional terminology in our daily work are in the habit of identifying the special attributes of bounded systems by use of the German prefix eigen in place of the English word characteristic that we employ in this book.
Let us illustrate the ideas of wave velocity and of wave impedance by considering the case of waves on a flexible string made of some material whose density is d and whose radius is r (i.e., crosssectional area = šr2). The string is long, and it is kept under a tension T.
wave velocity = Ö T/ šr2d = (1/r) Ö T/šd
wave impedance = Ö (šr2d) T = r Ö šTd
Here the tension T serves to supply the springiness, and the product šr2d will be recognized as the mass per unit length, which is a measure of the relevant inertia property of the string. Notice that we can trade tension for density or radius while keeping the impedance the same, but it is not possible at the same time to preserve the speed unchanged.
An analogous but somewhat simplified formula for the wave impedance of a soundboard at its driving point is:
= t Ö Yw dw x a , (a= numerical constant)
Here t is the thickness of the board, dw is the density, and Yw is the modulus of elasticity for the wood . We will assume that the ribs and bridge have been so designed that they properly take care of the difference in stiffness in the two directions relative to the grain, and chat the thickness t is also properly averaged to take these extra pieces into account. I should remark that the wave impedance of the board taken by itself is very considerably larger than that of a string. Because it will have little utility for us here, we will say nothing about the wave velocity in the soundboard beyond remarking that it is proportional to and that it has the unusual feature of having a different value at different frequencies.
We must also consider, besides the playing string and the soundboard, the aggregate influence of the damped and inactive strings. Their influence is best though[ of in two parts. The simplest but least important part is the wave impedance of the collection considered as a peculiar two-dimensional sheet; this turns out to depend on the strings' spacing along the bridge, and it has a magnitude only three or four times the impedance of a single string. The second and rather larger influence comes from the way the downward pull of the slanting strings between bridge and hitch pins alters the elasticity of the otherwise slightly arched soundboard, subtly modifying the soundboard wave impedance formula given above.
The string layout between the bridge and the hitch pin is illustrated in the top part of figure 17.2. This silent portion of the string (which is provided with a damping strip of felt) has a length Q and a "downbearing" P that is carefully proportioned to vary along the scale of any properly made instrument. Makers of the finest instruments find that the downbearing must be meticulously adjusted string by string on each individual piano, as a part of its final regulation. Errors in the trend of relationship among P, Q, and the string tension can cause as much trouble to the overall sound of the piano as can errors in the stiffness and curve of the bridge, or in the thickness of the soundboard. If the ratio P/Q is locally too small, the instrument acts somewhat the way it would with a thin spot in the soundboard. Notice that the downbearing is not simply a matter of getting adequate contact between the bridge and the strings; the string tension acting together with the offset on the bridge where the string runs zigzag past two steel pins is already quite sufficient for this contact, as is suggested by the lower part of figure 17.2.
The wave impedance ratio between the struck string and the soundboard must be chosen to meet two conflicting requirements. First of all, there must be sufficient transmission of vibratory energy from the string to the soundboard that our ears are ultimately provided with a sound of satisfactory loudness. If the soundboard were a plate of steel 4 cm thick instead of a wooden board about 1 cm thick, its wave impedance would be increased several hundredfold and we would hear almost nothing from the soundboard, nor would the string produce much sound directly in the air. If on the other hand the disturbance excited on the string by the hammer were communicated to the soundboard at too rapid a rate, these vibrations would die down so quickly that we would hear little more than a tuned thud, a louder version of what is produced by hitting a note while a wadded handkerchief is firmly pressed against the vibrating part of the string next to the bridge. We also want the soundboard impedance to be high enough that its resonances will not play an unacceptably large role in the tuning of individual string modes, a phenomenon chat we met in section 16.5.
Digression on the Vibrational Modes of Segments of a Larger System.
In chapter 6 and in the latter half of chapter 10 the idea was developed that any finite-sized system of springs and mattes would hair its own Particular set of characteristic vibrational modes, these modes being attributes of the system as a whole. This implies that it makes no sense to consider apart from the whole a particular matt or even a subset of mattes at a separate vibrational system. In this chapter I have apparently violated this principle of the unified behavior of a complete system by discussing the string modes and the soundboard modes at though these were in fact separable. Let us tee why this very convenient separation of ideas proves to be acceptably accurate at a piece of physics. If a certain complete system (e.g., string and soundboard) consists of two parts or regions with drastically different wave impedances, the communication of vibrations from one of these parts to the other it small enough that the two behave very much at though they were fully isolated. When this condition it met, then, it is possible to pick out of the complete set of characteristic erodes a subset in which the overall vibrational shapes ordain that the predominant share of the vibration takes place in the high-impedance region, while the remaining modes involve chiefly the rest of the system, which it constructed of low-impedance material. Once the approximate vibrational shapes associated with each region alone are well understood, it is then easy enough to correct for the mutual influence of the two regions. It is in precisely this spirit that we corrected the string erode frequencies for the effect of soundboard resonances in section 16.5; the soundboard has a wave impedance to much higher than that of the strings that we are justified in thinking of them at quasi-separate entities. Notice, however, that the wave impedances of the bridge and the soundboard are similar enough that we would not be justified in dealing with them separately-the two act together with the ribs at a single wooden vibrating system (see sec. 9.5 for another example of a two- part system that cannot be dealt with piecemeal).
17.2. The Proportions of a Mid-Scale Piano String and the Necessity for Multiple Stringing
In section 16.5 we learned that the stiffness of real strings gives rise to a slight inharmonicity in the ratios between their characteristic frequencies and that this inharmonicity was less in long, taut, thin strings than in short, slack, thick ones. We have seen how a small amount of string-type inharmonicity serves a useful purpose-it can help disguise the necessary errors of keyboard temperament or can even convert some of these errors into musical virtues. Moreover, numerous experiments have shown that a certain amount of inharmonicity is necessary if the listener is to be satisfied that what he hears is an impulsively excited string sound. Nevertheless the history of keyboard instrument development from the earliest times reveals an intense though not always conscious interest in reducing the inharmonicity. Because of this, we will begin our discussion of the proportioning of midscale strings by postulating that their tension is to be made as large as is reasonably possible, short of breaking the string. On the basis of this choice, the vibrating length L of the string turns out to be a fixed length that is independent of the string's thickness. This is the reason that the length of the C4 string is close to 62.5 cm for all stee strung pianos. The minimum inharmonicity associated with a string tightened nearly to breaking tension depends in a simplified way on its radius r and length L as follows (compare with the formula for J given in sec. 16.5):
Jmin = r2 / L2 x a, (a numerical constant)
This suggests that we should use the thinnest possible string, since L has already been fixed by the frequency requirements laid down for the string. However, if we make the string too chin we are speared on the other horn of our dilemma. The transmission of vibration from our string to the soundboard is proportional to the wave impedance ratio, and so depends on the wire radius and soundboard thickness (as influenced by the ribs) in accordance with the expression:
string wave impedance / soundboard wave impedance= ( r2 / t2 ) x a , (a numerical constant)
This relationship holds only if the string tension is always maintained fairly close to breaking. The equation indicates that making the wire thin will mean that it will be able to drive the soundboard to only a small fraction of its own amplitude, so that only very weak sounds will be radiated into the room.
Some numerical values for the soundboard and strings of a real piano should be of interest at this point. A good piano has a soundboard made of beautifully finished spruce that has a density dw close to 0.4 grams/cm3. The soundboard is often tapered and is generally thinner at the bass side, but in the main its thickness is a little less than 10 mm. The radius of the C4 string is close to 0.5 mm, and its density d is close to 7.8 grams/cm3. A single such string on a piano having a soundboard of this description sustains its tones very acceptably and shows tuning behavior almost identical with that described in chapter 16. However, the loudness of the sound of the single string is inadequate and the tone lacks a certain liveliness that we have become used to in pianos having three strings instead of one for most of the keyboard notes. An obvious way of simultaneously meeting the least- inharmonicity requirements (which call for thin strings) and the loudness requirements is to use several strings, each of which will have acceptable inharmonicity and each of which can join with the others in driving the soundboard to a greater vibrational amplitude. The physics of the multistring piano note turns out to have surprising aspects that lead to two important features of the tone of a piano; a description of these matters is the subject of the next section.
17.3. The Effect of Multiple Stringing on the Sound of the Piano
We will introduce ourselves to some of the consequences of multiple stringing on a piano with the help of experiments you can easily try. Repeatedly strike the C4 key of a piano while alternately pressing and releasing a finger (or pencil eraser) against two of the three strings, so that part of the time only one string is free to vibrate and the rest of the rime all three strings are sounding. With any reasonably well-tuned piano, the perceived loudness at your ears (expressed in sones) should be roughly 40 percent higher when three strings are active than when only one is producing a sound (see curve B of fig. 13.5), which is a quite significant change. The next experiment consists in verifying in a crude and informal way that the total audibility time of the decaying tone is roughly the same whether three strings are active or only one. So far everything appears to be in accordance with our expectations. We also notice that the tone is a little thinner and perhaps less interesting when only one string is allowed to sound than it is when all three are set into vibration. To be sure, if the piano is badly out of tune the three strings will beat against one another to give the jangling sound conventionally associated with a barroom piano, while on a freshly tuned instrument there is only a hint of beats among the lower partials and a pleasantly shimmering suggestion of beating among the higher ones.
In 1959 Roger Kirk of the Baldwin Piano Company reported the preferences of a large group of people for the tuning relationship among the three strings of each so-called unison of a piano.' He found that :
the most preferred tuning conditions . . . are 1 and 2 cents maximum deviation among the strings of each note in the scale. Musically trained subjects prefer less deviation . . . than do untrained subjects. Close agreement was found between the subjects' tuning preferences and the way artist tuners actually tune piano strings.
He also found that a piano tuned so that the group of strings for each note of the scale covered a spread of 8 cents was acceptable to many listeners, and that the overall spread between the lowest and highest frequency strings was of more importance than the tuning of the intermediate string. The beat frequencies between the first five components (partials) of two C., strings tuned 2 cents and 8 cents apart are: )s 335 Hz beat that one uses in setting the equal-temperament fifth to G., (see sec. 16.6, part D). Note that partial 2 of the G., strings will have a similar bearing rate to obscure further the departure from just tuning. With the 8-cent interstring spread, on the other hand, the fifths become pretty diffuse.
Let us turn now to the interval of a major third in equal temperament. Using a 2-cent detuning, the fifth component group of Ca has within it a 1.5-Hz maximum beating frequency, as does the fourth component group of the note E., if its strings similarly have a 2-cent detuning spread. Taking these together we see the possibility of beat frequencies as high as 1.5 + 1.5 = 3 Hz among the components upon which the interval is chiefly based. In section 16.7, we learned that the beating rate for a piano tuner's third in equal temperament is about 8 Hz, a little more than twice the smearing produced by the detuned unison. If the spread among members of a three-string "unison" were increased to 8 cents, the beating would become rapid enough to
Component: 1 2 3 4 5
2-cent difference: 0.30 0.61 0.91 1.20 1.50 Hz
8-cent difference: 1.21 2.42 3.53 4.80 6.10 Hz
Notice first of all that with the 2-cent detuning the beating rate for the first pair of partials is quite slow, as are those for the second and third pair of partials. As a result the tone sounds reasonably smooth when played by itself. The 8-cent spread gives a rather brighter sound, but it is not yet the sort of jangle one gets with a spread of 15 to 20 cents.
When we use a 2-cent detuning between strings, the 0.91-Hz beat frequency belonging to its set of 3rd components is just able to cover up the 0.89 Hz beat that one uses in setting the equal-temperament fifth to G4 (see sec 16.6, part D). Note that partial 2 of the G4 strings will have a similar bearing rata to obscure further the departure from just tuning. With the 8-cent inter string spread, on the other hand, the fifths be come pretty diffuse.
Let us turn now to the interval of a major third in equal temperament. Using a 2-cent detuning, the fifth component group of C4 has within it a 1.5-Hz maximum bearing frequency, as does the fourth component group of the note E4 if its strings similarly have a 2-cent detuning spread. Taking these together we se, the possibility of beat frequencies as high as 1.5 + 1.5 = 3 Hz among the components upon which the interval is chiefly based. In section 16.7, we learned that the beating rate for a piano tuner's third in equal temperament is about 8 Hz, little more than twice the smearing produced by the detuned unison. If the spread among members of a three-string "unison" were increased to 8 cents, the bearing would become rapid enough to drown the temperament error completely. Clearly there is a trade-off of musical virtues between the two kinds of unison spread as one compares various musical intervals. In any event we have provided ourselves with another reason stringed keyboard instruments are so well-adapted to musical performance, despite the problems with fixed pitch that at first seemed insurmountable.
As a practical matter it proves to be exceedingly difficult to tune a set of unison strings to a true zero-beat condition (one even meets cases where it is literally impossible to do so). The question arises then whether or not people's preference for a slight detuning of the unisons is simply a favorable response to the most familiar type of sound, or whether something more fundamental is involved. Kirk finds that piano tuners and musicians are unanimous in their verdict that too-close tuning gives a tone that not only sounds dead but dies away too rapidly. Laboratory measurement confirms the auditory impression we gained in our initial experiments that slightly detuned (normal) strings die away in about the same total length of time as a single one of these strings when the other ones are prevented from vibrating. However, when three strings are tuned exactly together they will actually die away much more rapidly. The presence of other precisely intune strings encourages each string to transfer its vibration more rapidly to the soundboard and thence to the room! Let us first make use of our knowledge of wave impedance to verify its consistency with these observations and then go on to an example of the same kind of physics displayed in an everyday experience far removed from acoustics.
In section 17.1 we learned that the wave impedance of a string is equal to the square root of the product of tension T and mass per unit length (šr2d). How do we find the corresponding impedance for a triplet of identical strings acting together? The top part of figure 17.3 indicates the appearance of our three strings as they are normally seen in a piano. The middle part of the diagram shows them moved so close together that they are on the verge of touching. If they were identical- tuned strings, they would stay precisely in step with one another, and there would be no frictional or other force acting between them to change things in case they did touch. In other words, the three closely spaced strings will behave exactly like their more separate cousins. In particular, the aggregate impedances are the same in both cases. The bottom part of figure 17.3 shows the last step in our imaginary set of transformations: here the strings are fused together into a ribbon like whole, with no change of total mass or tension. An extension of our former reasoning shows that this new sort of string also retains the acoustical properties of its ancestor at the top-as long as we confine ourselves to vibrations of the normal type (up and down, as shown in the diagram).
Having done a little thinking about three strings acting precisely together, we are now ready to calculate. Clearly, the total tension acting on our composite string is three times the tension acting on each of the original strings, so we must write 3T under the square root sign where formerly there was a T. Similarly, any short length of the composite has precisely three times the mass of a corresponding length of ordinary wire, so we must also write 3(šr2d) in place of šr2d in the formula. Putting all this together, we get:
(wave impedance of a tricord )= 3 x (šr2d) x 3T=
= 3 x ( wave impedance of a single wire )
This shows us that three strings acting precisely together produce a threefold increase in the wave impedance, and thus a threefold increase in the amplitude of the bridge motion, which ultimately leads to a threefold reduction in the decay time of the vibration. You might find it worthwhile to deduce this last assertion on the basis of the principles outlined in section 6.1.
The expected difference in sound between a struck single string and a perfectly tuned triplet of strings is not hard to figure out on the basis of what we have just learned. First of all, the tone of the precisely tuned triple strings will die away much more quickly, which matches actual experience. Second, we would expect on the basis of curve A in figure 13.5 that the perceived loudness of the fundamentals of the tone (as expressed in sones) would be very nearly doubled because of the threefold increase in source (soundboard) amplitude. The cone would not actually appear this much louder, however, because a short or decaying sound always sounds less loud that a steady one. In the three string case the increased rapidity of the decay partially offsets the perceived effect of the larger amplitude.
We seem by now to have left the slightly detuned strings of a real piano in a sort of unexplained limbo between the single string and a perfectly tuned triplet. The true behavior of detuned triplets will be easy to understand once we have looked ac the everyday example I promised a few paragraphs ago. Suppose you have undertaken to push your friend's small car along a fairly level road. If the rolling friction of the car is large, you may find is barely possible to keep the vehicle rolling, and yet you will be able to move the car quire a distance under these conditions without much strain and without becoming winded. Suppose on the other hand that you have acquired a helper in the pushing, so that the two of you together can get the speed up to a fast walk. Pushing at this faster pace will soon leave you winded and panting for breath, even if you are not pushing any harder as an individual than you were during the solo performance. The point is this: the energy you expend in pushing with a certain force over a given distance will be spent in a much shorter rime if your friend helps you make the trip more quickly. The rare at which you work is increased because of the cooperative presence of your friend.
The translation of this example to the case of vibrating strings is easy: one string pulling up and down on the soundboard and moving it corresponds to you pushing on the car alone. If two strings are less than precisely in tune with one another, the situation is like the case where your car-pushing friend sometimes pushes with you and sometimes pushes in opposition to you. In a semi-disorganized situation like this there is no absolute coherence to the undertaking and the aggregate accomplishment is simply equal to the sum of the separate contributions.
Daniel Martin and his research group at Baldwin Piano Company have shown that a very characteristic feature of the sound from a piano is a dual decay pattern. This is the second musically important result of the use of multiple strings. A blow from the hammer starts all three strings off exactly in step with one another, so that they radiate strongly to the outside world. Initially, then, each partial dies away quickly at about the rate expected for strings that are in precise unison. However, because of their slight detuning from one another, they soon get out of step, so that we might say that there are eventually three solo performances. The vibration of each string then decays on its own in isolation at the single-string race, and close cousins to ordinary beats are produced for us to hear.
When the strings of the C4 note on a good piano are tuned to a total spread of about 2 cents, the net sound pressure due to all the partials (see sec. 13.2) shows the presence of fast decay for about 1 second our of the coral time of 20 seconds (crudely speaking) that is required for the net pressure amplitude to be reduced to 1/ 1000th of its initial value.
I will close this section with a brief explanation of the pitch rise that is often perceived in a piano tone as it dies away. To begin with there is a clearly audible change in tone quality, explainable in part by the fact that the lower-frequency partials become unimportant and then inaudible more quickly than do the higher partials, simply because of the greater sensitivity of the ear at high frequencies (see fig. 13.3). Furthermore, the amplitude of the lowest partial generally falls away more quickly than the higher partials, chiefly because the slow beating rare between the strings for this component keeps their vibrations in step for a longer time, during which they suffer the accelerated decay characteristic of the cooperative effect. This gives us an additional, mechanical reason to expect a listener's attention to transfer itself to the higher partials of a decaying tone. Because of string inharmonicity, these higher partials heard by themselves imply a higher pitch than that which our ears assign when they base their "calculation" on the lower partials (see also sec. 16.7). However, the decay patterns of individual notes of a keyboard scale differ enough from note to note, even on a very fine instrument, that we should not expect the invariable presence of a pitch rise during the decay of every cone.
17.4. The Action of Piano Hammers
General principles were developed in chapter 8 to guide our understanding of vibration recipes produced when strings are struck ac various places by various kinds of hammers. In particular we found that the duration of contact in a hammer blow exerts a considerable influence on the number of characteristic modes that are excited. Modes having frequencies high enough that one or more of their oscillations could cake place during the contact time are, as a result, only weakly excited. On a piano the time of contact is only partly influenced by the softness of the hammer felt; the predominant influence arises from the way the string itself pushes back against the hammer. Our thinking about this influence can conveniently be divided into what we might call an elastic version and a wavelike version, the second version being used to refine our conclusions from the first.
If someone were to force a piano hammer slowly and progressively into the exaggerated position shown in the upper part of figure 17.4, the tension of the deflected string would act on the hammer, exerting a downward restoring force whose magnitude would grow as the hammer is displaced farther and farther upward. It should be apparent from the diagram that the greater slant of the shorter, left-hand segment of the scrim; means that this segment exerts the major portion of the restoring force. For example, on a piano string whose hammer strikes at a distance H equal to 1/9th of the string length L, the two forces are in the ratio (L - H) / H =8 / 1, meaning that in this particular case the restoring force of the short segment acting on the hammer will be eight times as great as that of the long segment.
If one strikes a piano key, the system of levers called the action accelerates the hammer to some final speed and then releases it, allowing it to continue freely upward until it strikes the string. When contact is made with the string, the hammer's upward motion persists, but the string exerts an increasingly large downward force on it as already described. If we temporarily set aside the force exerted by the longer portion of the string, is it apparent that the hammer and the shorter segment H of the string together constitute an elementary spring-and-mass system. The natural frequency fH of this system is determined by the string tension T, the length H, and the mass M of the hammer, as follows (see sec. 6.1):
fH= (1/2š) x T/MH
The lower part of figure 17.4 shows the motion of the hammer head (a) as it leaves its original rest position when the key is first pressed, and then as it continues to accelerate under the influence of the player's finger until the instant (b) when the action releases it. Following its release by the action, the hammer swings freely upward toward the string and meets it at (c), after which the string forces convert the motion into an up-and down movement of oscillatory type (c), (d), (e). If the hammer were somehow to glue itself now to the string, the oscillation would continue in the manner indicated by the dotted curve and the letters (f), (g), (h), and (i). In fact, the hammer comes loose from the string after about half a cycle of oscillation, at the instant marked (e), and then falls back down until (j) when it is caught and arrested by what is known as the check.
Clearly, if our calculation is correct, the all-important time of contact T, between hammer and string is about equal to one-half the time required for one oscillation of the hammer bouncing on its "spring," which is the string length H. When we modify the formula for ft, to take into account the three strings which act together on any given hammer, we get:
Tc= (1/2)( 1/fH) =š Ö MH/3T
Notice that, according to this formula, increasing either the hammer mass M or the striking distance H will lengthen the time of contact T, and thus reduce the number of higher partials excited in the tone, as explained in chapter 8. The "elastic" version of the hammer recoil analysis is now complete, and we must consider next how the wave behavior of disturbances on the long segment of the string alters the conclusions we have drawn thus far.
A hammer interacts with the longer segment (L - H) of a piano string in a way that can easily be understood if we begin by imagining the string to be extremely long, so that the hammer rebounds from it before an echo returns from the far end. For instance, it would take six seconds for an echo to return from the far end of a set of C4 piano strings one kilometer long (about 0.6 mile). During the time the hammer is touching the strings we have already noticed that it feels a springlike force exerted by the short string segment H. Wave physics tells us that as the hammer launches waves down the long segment of the strings, another force (in addition to the springlike force) acts to make the hammer feel exactly as though it were immersed in and plowing through an extremely viscous fluid. As a result, the half-oscillation discussed earlier is damped (in the manner described in sec. 6.1). In this case the lost oscillatory energy is transmitted out along the strings (for eventual return) instead of being frictionally dissipated. The viscous dissipation coefficient D defined for a springmass system in section 6.1 proves to be exactly the aggregate wave impedance of the long segments of the strings (see sec. 17.2)!
You may find it helpful to know that the combined wave impedance of three C., strings is roughly equal to the viscous coefficient D associated with two of your fingers moving broadside through a howl of molasses left outdoors in January. Despite the numerically large size of the viscous damping coefficient just described, calculation shows that the wave-type damping on a piano hammer produces only a few percent diminution in the amplitude of any one of its oscillations, so that the formula of our original, simple estimate of the hammer contact time T, does not yet need changing.
Having considered how the long string segment feels to the hammer before an echo has time to return, we are now ready to follow the progress of the half sinusoidal pulse impressed by the hammer blow on the long side of the strings as it travels to the far end and back. Figure 17.5 indicates that a completed upward blow from the piano hammer produces an upward pulse that travels to the bridge end and is then reflected back toward the hammer. Because the bridge has a very large wave impedance compared with char of the strings, this reflected pulse has very nearly the same amplitude as the original, but it is inverted. As long as the hammer is thrown clear in advance of the reflected wave (as is the case for notes below about CS on a piano), the pulse runs back and forth over the whole length of the strings, being reflected and re reflected ac the two ends. This particular motion is exactly the one we described in chapter 8, using language that derails the motion in terms of the vibration recipe belonging to a given hammer blow at a particular point on the three strings (see secs. 8.2 and 8.3 and also statement 6, sec. 11.9).
For the upper two octaves of the piano scale, the inverted pulse returns before the hammer has left the strings, and so adds its forces to those exerted by the short string segments H. As a result, the hammer is thrown off the strings earlier than otherwise, thus shortening the time of contact.
It is time now to go back and refine our view of what is happening on the short length H of the strings during the hammer blow. These do not really act exactly like the simple spring we assumed originally. The disturbance on this is actually a peculiar train of impulses rapidly echoing between the capo d'astro bar and the hammer itself (ac C4 these impulses make about four complete round trips during the time we calculated earlier for T,). When the net effect of these rapid echoes is properly worked out, we find we must change the half-sinusoidal hammer motion assumed earlier, which takes place between (c) and (e) in figure 17.4. The hammer motion is now seen to have a new but similar shape that looks as though is were made of roughly straight line segments, each lasting the time is takes the impulses to make one round trip between the fixed end and the hammer. The time of contact T, estimated earlier remains fairly accurate, however, as do our earlier conclusions about the effect on is produced by echoes coming back from the bridge.
The not-quite-sinusoidal (segmented) hammer motion can be thought of as a combination of the original sinusoid and an additional bouncing motion. This bouncing motion is of course caused by string vibrations set up in the short string length H during the time of hammer contact. These vibrations form a harmonic series whose frequencies are L/H times as high as the corresponding modes of the complete, full-length strings. During the course of the blow, then, the new high-frequency oscillations of the hammer and of the short part of the string are given to the complete strings in addition to the more familiar components of the vibration recipe. It is somewhat shocking to realize chat these extra components fill in the otherwise expected gaps in the recipe produced when the hammer strikes at nodal points at various modes. For example, in chapter 8 we learned that a simple, non segmented blow from a mathematically idealized hammer 1/4th of the way from one end of a string would eliminate modes 4, 8, 12, err., from the recipe. A real hammer blow restores these missing components. We have here the explanation of the century-old observation chat a piano-type hammer strikes in such a manner that no modes are ever missing from the recipe of a piano tone. 
17.5. Scaling the Strings of a Piano
The piano that has been studied so painstakingly thus far in the chapter would be of rather limited musical usefulness, for the simple reason that is can do little more than play the note C4! The extension of the basic design to the high and low limits of the scale is influenced by constraints of a mechanical sort and also by the fact that our hearing changes drastically as we go to these extremes. For example, the fundamental components belonging to the top octave (from 2100 to 4200 Hz) span the most sensitive range of our hearing, while the fundamentals of the lower notes (from 27.5 Hz) are only weakly heard under ordinary playing conditions.
The formula given at the beginning of section 16.5 for the vibration frequency of a flexible string suggests that for every octave one goes up in pitch, the string length might be halved (if the tension
and string size are kept fixed). We will see in a moment why it proves better on a piano to reduce the lengths by a factor close to 1/1.88 per octave, so chat if we start with our 62.5-cm C4 string, the Cs string (four octaves higher) has a length close to 62.5/(1.88); = 5.00 cm, instead of the 3.91-cm string length calculated on the basis of four successive halvings. In a similar vein, experience has shown the advisability of reducing the string diameter by a factor of about 0.946 per octave from the 1-mm diameter at C4, making the cop string a little under 0.79 mm in diameter. Strings proportioned thus have to be pulled to slightly lower tension ac the top of the scale than at C4.
As we have already learned in chapter 16, the inharmonicity of constant-tension strings proportioned in this way rises 2.76-fold for every octave we go up. For example, at the top note (Cs), mode 2 is more than 50 cents sharp compared with mode 1, instead of the 0.83-cent widening associated with C4. If many string partials for Cs were excited, the cone would be quite harsh ("metallic," i.e., reminiscent of the vibrations of steel bars), so the softness of the hammer felt must be carefully adjusted to give a suitable contact time during the blow in order to produce a tone of acceptable quality.
I will say little about the trend relating the upper strings' wave impedance to chat of the soundboard beyond remarking that at Ca the string impedance is only about 75 percent of the value at C4, which reduces the transfer of vibration from string to soundboard by the same factor. This reduction ac the upper end of the scale is desirable; not only does the maker raise the soundboard impedance by using progressively shorter inactive string lengths (between bridge and hitch pin) to increase their elastic contribution to the soundboard's own driving-point wave impedance, he also tends to increase it even further by thickening the soundboard at the treble end. Perhaps the increasing sensitivity of our ears for higher-frequency sounds calls for a reduction in the actual amount of vibration transmitted to them by the topmost strings. Another reason for reducing the string-to-soundboard coupling is that it leads, as we have already seen, to longer ringing of these strings. Recall that even so, the sound from these strings decays so rapidly that dampers are not normally provided for them.
The true challenge to the piano maker's skill lies in the notes below C3. Even on a full concert grand with an overall measure of nine feet, the bottommost strings must be made less than half the lengths implied by the scaling rules used above Ca if the instrument is to have dimensions less than those of a battleship. For example, the bottom note (A) on a Baldwin concert grand I have examined has a string length close to 203 cm, instead of 486 cm. The same note on the six-foot-long model L Steinway grand in my living room has a somewhat shorter length-a little over 137 cm. On some small spinet pianos, the bottom string is a troublemaking 95 cm, only 20 percent of the "ideal" length.
How does one strive to meet the requirements for acceptable (if not good) tone, sufficient power, and adequate duration of sound in the lower strings? The need for acceptable tone implies not only a tolerably low value for the string inharmonicity factor J, but also a properly proportioned relationship among the hammer 's mass, breath, and softness, the string tension, and the point at which the hammer strikes the strings. To get sufficient power with an adequately long decay time one must in addition arrange to get a correct ratio between the wave impedances of the string and the bridge.
The formula given in section 16.5 for the natural frequencies of a flexible string suggests immediately that a proportional increase in string thickness will automatically offset a reduction in its length. For example, if we were to preserve our usual constant tension, the bottom string of our concert grand would have a diameter that is 486/203 =2.39 times the 1.22-mm diameter of the fulllength string called for by our basic midscale design. Such a 3-mm "string" would in fact be an impractically thick rod having nearly half the diameter of the tuning pins! The inharmonicity of this rod would he so large that it would emit a clanging sound when struck. The piano maker in practice avoids a great deal of the inharmonicity problem by using a slender steel string (to support the tension) which is wound with one or more layers of copper wire, so as to raise the mass per unit length without adding much stiffness. On a concert grand, carefully designed bass strings of this sort are held under a tension that is about 50 percent larger than the mid scale value. We find then (on the Baldwin concert grand, for example) that J calculated ' for the bottom string has a surprisingly low value, about equal to the inharmonicity coefficient belonging to C3 in the main part of the scale. On the smaller pianos, however, the problem remains serious, and what passes for good tone cannot be obtained from a bottom string shorter than about 130 cm. On the smallest spinets, J for the bottom string can he as much as ten times the concert grand's value.
Of particular concern to the piano maker is the problem of making a smooth transition from the full-length plain wire strings to the sequence of shortened wound strings that function for the lowest notes of the scale. Let us see how the problem is dealt with in the Steinway mentioned earlier. On this instrument the lowest triple set of plain wires is found at B2. The next note down the scale, B2♭, is provided with a pair of copper-wound wires having a length of about 91 cm. I have used the wire sizes and playing frequencies of these two sets of strings to calculate that the tension in the wound strings is 60 percent higher than in the plain ones, the latter being about 10 percent slacker than normal because they are already 10 cm shorter than the basic scaling rules would call for. The calculated string tension shows a rather large jump, so I checked the correctness of this calculated change in tension by comparing the pitches of sounds produced by plucking the non playing lengths of these strings between bridge and hitch pin. The copper windings of the B2♭ string do not extend into this region, and the core wire diameter is equal to the wire size for B2.
As one slowly plays down the chromatic scale in the vicinity of the break between nonwound and wound strings, one notices a slight but progressive deterioration of the tone below F3, where the strings first begin to fall short relative to properly scaled lengths of the sort used in the upper half of the instrument. The main alteration in tone is due to a growing inharmonicity associated with both a shortening of the string and the concomitant reduction of tension. The change of tone one hears in going between wound and nonwound strings is relatively small, the inharmonicity increase due to the greater stiffness of wound strings being offset by the increase in their tension. The calculated inharmonicity factors match within 5 percent across the break, which is less than the 9 percent change from note to note of the normal scaling!
The question of suitable gradation of string and soundboard wave impedance across the break is our next concern. Because the wave impedance depends on Ö [(inertia) x (elasticity)], it is apparent that in going from the "properly scaled" F3 down to the slightly slack strings at B2, we have a reduction of about 5 percent in the string impedance. Across the break, from the three slack wires at B2 to the two wound and very tight strings at B2♭, there is a 40 percent upward jump in wave impedance. Let us resort once again to observation in order to find out how the maker has dealt with these non uniformities.
On the Steinway in my home there is a trend of progressive increase in the decay times measured for sounds from single strings belonging to the notes running from F3 down to B2. The falling sequence of wave impedances for these strings, all mounted in a row on the same bridge, leads us to expect a 5 percent increase in decay time, roughly the observed amount. We must now compare the decay time for a single wound string belonging to B2♭ with a plain string belonging to B2; these two strings are found to vibrate for roughly the same lengths of time. On the other hand, the fact that the B2♭ strings have a 40-percent the string lead to piano tones that, for notes from the bottom of the scale tip to about C2, are made up of many (20 or 30) partials having significant strength in the room. That is, these partials average out to have pressure amplitudes that are more than 3 to 5 percent of the average amplitude of the three or four lowest-frequency partials.
2. Between G2 and C5 the sound pressure erciee typically contains partials of appreciable amplitude only up to about 3000 Hz. This means that at G2, 25 or 30 partials are tonally significant; at G3 the number is about halved; at C5 only 5 or 6 partials play much of a role in the sound.
3. For notes above C5 (fundamental frequency near 523 Hz) the number of significant partials decreases progressively, until at C8 (near 4200 Hz) the fundamental is accompanied by very little more than the second partial.
4. For notes below C2, the first few partials of a tone may have roughly equal amplitudes; however, because of the insensitivity of the ear below 60 Hz, the lowest-frequency partials of these tones contribute very little to the perceived loudness.
5. Along with the components having sharply defined frequencies (associated with the string vibrations), the piano produces a very considerable amount of fairly diffuse sound made up of closely spaced, even overlapping frequency components arising from (a) the thumping blow of the hammer as transmitted to the frame by the short part of the string and to the soundboard by the initial impulse carried to it by the longer string segment, (b) the short-lived oscillations of all the damped strings that are impulsively excited via the bridge motion at the beginning of the tone, (c) the analogous, higher frequency sounds associated with the "inactive" string lengths between bridge and soundboard, and (d) the excitation of fairly long lived oscillations in the topmost strings of the scale, which are not provided with dampers. Predominant among the sounds described in (c) is the contribution from the strings whose long part has been struck by the hammer. In many pianos the frequency of this contribution is harmonically related to that of the played note.
6. Due in part to the momentary compression and hardening of hammer felt in the course of a vigorous hammer blow, the vibration recipes associated with fortissimo playing show an augmentation of the strengths of the higher partials relative to the lower ones. This means that the number of significant partials in a given tone is increased when it is loudly played. The converse behavior is observed at the limits of pianissimo playing.
7. After a key is struck, the reverberative sound builds up in the soundboard, whence it radiates into the room. Each component of the sound comes out in its own way, but when our ears consider the sound in the aggregate, they are chiefly sensitive to the fact that there is an initial burst of sound associated with the combined effects of the struck strings and the various short-lived components listed in statement 5. At C4 this burst builds up in roughly 0.03 seconds. During the next few tenths of a second the short-lived components disappear, leaving the string sounds to decay in the manner described below. The time scale of these developments is longer when low notes are sounded and shorter for high notes. This is in part because as we go from one part of the scale to another, the predominantly excited components have different frequencies and therefore different "spreading times" across the soundboard.
8. The decay of any given partial in a piano tone has a complicated nature (see secs. 17.2 and 17.3). Taking the sound as heard overall, we find that 20 seconds is a good round number for the time required for the aggregate sound pressure of C4 to fall to 1/1000th of the original sound pressure. As we go up to about G7, the corresponding decay times become progressively shorter, decreasing by a factor of about 0.66 for each octave. Above G7 the decay times become very much shorter, falling to a value of less than 0.5 seconds at Cg. Going down the scale below C4, we find that the decay times become gradually longer, growing by a factor of about 1.2 for each octave.
9. When a key is released, the damper falls into place, but it is unable to kill off the string oscillations immediately. Each partial decays at its own race, because of the varying efficacy of damper action on each string mode. A characteristic part of the piano sound is the quickly damped tail at the end of each tone. The fundamental component disappears in a few tenths of a second, usually leaving the higher partials softly audible for several seconds.
10. Both top and bottom surfaces are active in coupling the soundboard vibrations to the air. When the cover is lowered, there is a distinct change in piano tone, because the top surface of the soundboard now communicates with the broad but not very high channel of air between it and the lid. Signals produced by parts of the soundboard far back in this channel take longer to come out into the room than do those which arise in regions near the open edge of the lid. The impulsive parts of the piano tone are influenced by the fact that in a narrow (almost two-dimensional) channel of air, a given frequency component is transmitted in several "propagation modes" each having its own (frequency - dependent) velocity. As a result of the altered coupling to the room, the build-up processes described in statement 7 are rearranged and spread over a longer period of time. Piano makers recognize the existence of a lid angle giving "best" tone in a concert hall. Curiously enough, the props installed on the piano to adjust the lid angle do not always provide this "best" angle.
17.7. Examples, Experiments, and Questions
1. Find a freshly tuned piano and play the bottom note A0 along with its octave, A1. Get a fairly clear impression in your ears of the joint sound of these two carefully adjusted notes. Next listen to sounds produced by the two strings when you lightly touch the exact center of the A0 string with a finger tip so that its odd numbered modes are quickly damped out. You will be surprised at how many hitherto inaudible beats become easily apparent, producing a very rough sound. When the string is damped in its exact center, only those partials of A0 remain which directly compare themselves with the partials of A1 (see the digression on sounds with only even harmonics in sec. 5.5, and also experiment 2, sec. R.6). If a piano tuner were present, he could retune A0 so as to make the beats less noticeable in the altered circumstances, but then the normal A0-to-A1 octave tuning would become unacceptable. See if you can find some acoustical reasons why the presence of the odd-numbered partials in the normal tone of A0 leads the tuner to provide a setting different from the elementary minimum-beat conditions which we might naively expect. At the bottom of the keyboard, tuners generally check not only the octave, but also double octaves, as they proceed. They usually play their pairs of notes alternately as well as together. Why is this an admirable practice?
2. The number of round trips per second (shall we call it the echoing rate?) which a disturbance can make on a string of length L is v/2L, where v is the wave velocity for disturbances on this string. Replace the v in this echoing rate formula v/2L with the formula for the wave velocity of a flexible string given in section 17.1. Verify that the resulting formula for the echoing rate is exactly the same as the formula set forth at the beginning of section 16.5 for the mode-1 vibration frequency of a flexible string. To say there is a pulse echoing back and forth on a string is simply another way of saying that a repetitive process is going on, one which therefore can be described in terms of a set of harmonically related sinusoidal components. These components are the ones we have studied so much already from quite a different viewpoint. They are of course associated with the various vibrational modes of the string. In the real world of piano strings, the wire has stiffness, which means we cannot (strictly speaking) calculate a simple echoing rate. This is because a given impulse does not preserve its shape as it travels along the string, so that the disturbance on the string is not of a strictly repeating nature. As a result its recipe cannot be based on harmonically related frequencies. The reason for the changing form of the traveling impulse is that the various sinusoidal components making it up do not all travel at the same speed on a stiff string (see the remark about wave speed on a soundboard in sec. 17.1). It is the stiff string characteristic frequencies, therefore, which are the proper basis for constructing a vibration recipe.
3. A number of practical advantages are obtained from the practice of multiple stringing on pianos. Some very fine pianos are built using four instead of three strings on each note in the main part of the scale. You may find it interesting and worthwhile to speculate about the acoustical consequences of adding strings in this way. Assume first that no other changes are made in the instrument, and consider such matters as hammer rebound time, string impedances, tuning spread, etc., as they influence the overall tone. Why would it be very difficult to arrange for the use of multiple strings at the extreme bass?
4. Flat steel ribbons used instead of sets of round wires would appear to allow the piano maker convenience in adjusting the relations between inharmonicity, wave impedance, and the effect of string tension on hammer rebound. He would have ribbon thickness, width, and length at his disposal. The simple string frequency formula in this case becomes fn =(n/2L) Ö [T/ (rwd)] , where t is the ribbon thickness and w is its width. See if you can verify that the wave impedance becomes Ö Ttwd. The inharmonicity coefficient J uses the thickness t instead of wire radius r (the width w being irrelevant), and the numerical factor takes on a slightly larger magnitude. See if you can come up with a list of arguments for and against the use of tape like strings in a piano.
5. Many people find it possible to pick out fleeting parts of a musical sound if their ears have once been told what to listen for.
The various impulsive sounds coming at the beginning of a piano tone may be separated one from another by your ears if you will produce each one by itself before listening for it in a normally produced sound. For example, knock repeatedly at some point on the bridge by tapping with the eraser end of a vertically held pencil, and become familiar with the woody thump that results. Listen for this same thump when you play a vigorous note on some key whose strings pass over the bridge at your thumping point. Play only very short notes so that the sustained part of the tone does not distract you. Notice that the pitch of the thump varies with the position of the striking point along the keyboard. Now run your fingers lightly and quickly across a range of playing strings. Do this near the bridge and listen for the brief, harp like glissando that results. If you run upward in pitch along the top octave of damped strings it may be possible to imagine that the sound is reminiscent of someone whispering the word "whee !" After running your fingers back and forth across these strings and sounding them at random, you will probably be able to pick out their collective sounds when one of these triplets of strings is struck a short, sharp blow with its hammer in the normal way. The ringing of the damped strings is easily audible, and will be quite recognizable. Now see if you can hear these same sounds when the piano key is held down after each blow. Other aspects of the piano's initial tone can be tracked down with the help of similar ear-training experiments. You will also find it worthwhile to listen to a number of notes, each one being played by itself and allowed to die away. Damping one or two of the strings belonging to the individual note will produce tonal changes which will repay your close attention.
6. In 1949, Franklin Miller, Jr., of Kenyon College proposed that adding one or more lumps of material to a piano string might usefully reduce its inharmonicity and so improve the tone (see sec. 9.6, part 1). The reduction of frequency produced by loading a wire is one-half the shift associated with a similar loading on a plate (see sec. 9.4, assertion 3). You may wish to experiment by wrapping a few centimeters of wire solder tightly around a piano bass string near one end or the other. Why will wrapping it at a point 1/ 10th of the way along produce the desired result, a progressively greater alteration (lowering) of the first five modes? From your knowledge of hammer dynamics you may be able to decide whether it is better to add the load at the bridge end of the string or at the hammer end. Even though the steady state motion (made up of the modified string vibrations) can be made more nearly repetitive (i.e., more harmonic) by the use of an added mass, the initial echoing after the hammer blow will include among other things an early echo due to partial reflection from the added mass plus a later, modified pulse returning from the string termination. How will this affect the early impulsive part of the piano tone? Do you think that a normal bass string has its inharmonicity raised or lowered by the fact char the copper windings do not extend all the way to the two ends?
1. A. P. French, Vibrations and Watts (New York: Norton, 1971), pp. 259-64.
2. Eugen Skudrzyk, Simple and Complex Vibratory Systems (University Park: Pennsylvania State University Press, 1968), p. 253.
3. Roger E. Kirk, "Tuning Preferences for Piano Unison Groups," J. Acoust. Soc. Am. 31 (1959): 1644-48.
4. Daniel W. Martin, "Decay Rates of Piano Tones,"J. Acoust. Soc. Am. 19 (1947): 535-41.
5. Hermann Helmholtz, On the Sensations of Tone, mans. Alexander Ellis from 4th German ed. of 1877 with material added by translator (reprint ed., New York: Dover, 1954). See Appendix V, "On the Vibrational Forms of Pianoforte Strings," and Appendix XX, section N, part 2, "Harmonics and Partials of a Pianoforte String Struck at One eighth of Its Length." For references to more modern work see A. B. Wood, A Textbook of Sound. 3rd rev. ed. (1955; reprinted., London: Bell, 1960), pp. 100-101, and E. G. Richardson, ed., Technical Aspects of Sound, 3 vols. (Amsterdam: Elsevier,352 1953, 1:464-68.
The reader should be cautious in accepting the correctness of some of the work referred to by Wood and Richardson. To make matters worse, the summary of it given by Richardson is garbled at several points.
6. This calculation is based on equation 28c in the paper by Harvey Fletcher, "Normal Vibration Frequencies of a Stiff Piano String," J. Acoust. Soc. Am. 36 (196-I): 203-9. The effect of a winding taper at the string ends is incorrectly given by Fletcher as the explanation of certain inharmonicities that are in fact due to soundboard resonances.
7. Some of the information presented here is taken from the article by Daniel W. Martin (see note 4 in this chapter). Most of the remainder is to be found in Harvey Fletcher, E. Donnell Blackham, and Richard Stratton, "Quality of Piano Tones," /. Acoust. Soc. Am. 3- (1962): 749411. A less technical account of this same work is given by E. Donnell Blackham, "The Physics of the Piano," Scientific American. December 1965, pp. 88-99. The dependability of some of the conclusions drawn in these two papers is reduced by the fact that one of the pianos was badly out of tune at the time the experiments were done, thus deranging the excitation and decay processes of the strings, as well as the frequencies of their partials.
8. Franklin Miller, Jr. "A Proposed Loading of Piano Strings for Improved Tone," J. Acoust. Soc. Am. 21 (1949): 318-22.
The clavichord and the harpsichord developed earlier than the piano. We will devote only limited attention to the first of these, chiefly as a way to help us use our knowledge of piano-hammer dynamics to understand some of the tonal attributes of the harpsichord.
18.1. The Clavichord
On a clavichord each key forms a simple rocking lever whose far end carries a wedge-shaped metal tangent that rises up against the string. We might think of the tangent as being a narrow and hard piano hammer, but unlike a piano hammer, the tangent is not released by the action to fall away from the string. The sounding length of a clavichord string is the part between tangent and bridge, while the short portion of the string (marked H in the top of fig. 17.4) is provided with a strip of damping felt to keep it more or less silent. We can use the lower part of figure 17.4 to help ourselves visualize what happens when a clavichord key is pressed. The tangent (carrying its own mass plus those of the key and the player's finger) rises toward the string along the curve (a) (b) (c), exactly as does the hammer on a piano. Once contact is made, the string and tangent remain together and move tip and down in an oscillatory manner very similar to what is indicated by the letters (d), (e), . . . (i). The chief difference between what is sketched in the figure and the motion of a tangent is that the steady pressure of the player's finger causes the whole bouncing oscillation to take place above and below an average position somewhat above the original string position (c). Also, this oscillation is fairly heavily damped (chiefly by the player's finger). The formula in section 17.4 for the oscillating frequency f" of a hammer bouncing on its string applies to the present situation as well, and we can understand from it why the large mass M of tangent-plus-key-plus-finger joins with the clavichord's low string tension T and relatively long, felt damped string length H to give the tangent a very low bouncing frequency. The period of oscillation (1/ffl) of this bouncing is several tenths of a second instead of the few hundredths of a second that it is on a piano.
The tangent's oscillation is exceedingly slow in comparison to the string's musical vibrations. The initial kink imposed on the string at the instant of contact can therefore be thought of as echoing rapidly back and forth along a string whose "fixed" end at the tangent is gradually moving upward and downward. According to wave physics, the string modes that combine to give such a motion follow a recipe that is almost identical with the recipe of a string plucked very close to one end by a narrow plectrum. In other words, all the lower-numbered modes decrease with mode number very nearly as 1/n (instead of 1/n2) up to that mode whose wiggliness has a curvature matching the curve produced by the stiffness of the string at the plucking point (see sec. 7.3, sec. 7.4, part 6, and sec. 8.4).
The actual vibration recipe of a clavichord string is not quite like that described in the preceding paragraph. During the earliest instants after the tangent touches the string, the contact force is not very large, so that each of the first few echoes returning from the bridge actually makes the string jump off the tangent momentarily once per cycle of the lowest string mode. These jumps are rapidly damped out, however, because at each jump a considerable amount of wave energy escapes past the tangent to he eaten up by the felt damper on the string beyond. What we hear then is a very brief "tzip" of sound at the beginning of each tone. The components of this initial sound are of course in exact harmonic relationship to the echo repetition rate. The sustained portion of the clavichord sound quite resembles the tone of a harpsichord, though it is considerably softer.
18.2. The Harpsichord
The harpsichord has enjoyed a long period of popularity that extends to the present day. Its development began well before 1600 and peaked during the first half of the eighteenth century. Even in the early part of this period the art of harpsichord building and design was well developed and sophisticated, and the harpsichord gave way to the piano only when the latter instrument was improved to the point where it became competitive. Fine harpsichords made as long ago as 1618 by the Ruckers family show much of the subtlety of soundboard-ribs-and-bridge design and string scaling that we find in today's pianos. Wire of brass, iron, and steel was available in accurately graded sizes for use by early harpsichord builders. The predominant British and German system of the eighteenth century gave 9.4 percent reductions in going from one numbered size to the next, so that there were eight wire sizes for each doubling of diameter, quite enough to take care of the necessary changes for scaling harpsichord strings.'
Let us make a quick survey of the relation between the note C9 on a particular Ruckers harpsichord and the same note on a typical grand piano of today. On the old instrument, the string length is slightly greater-70 cm instead of 62.5-as comports with a lower overall tuning based on an A4 setting near 410 Hz. The steel strings for this note have a diameter of 0.32 mm instead of the piano's 1 mm, and the harpsichord's string tension is about 11 percent of that used on a piano. This reduction in string tension is almost entirely attributable to the use of thinner strings rather than to limitations of strength in the materials. Typically, for each octave one goes up from C4, the strings are 50 percent as long as the corresponding ones in the lower octave, in contrast to 53 percent on the piano. The diameter of a harpsichord string will be 73 percent of the measure of its mate an octave lower, while on a piano the higher note has string diameters that are 94 percent of those for the note an octave below it. As we go down from mid-scale, the bass strings of a harpsichord grow according to the rule described for the upper scale. The soundboard on this harpsichord has a thickness varying between 2.5 and 3 mm, in contrast to a thickness near 10 mm that is typical of a piano.'
The mounting of light, low-tension strings on a thinner soundboard gives a string-to-soundboard wave impedance ratio for the harpsichord that is higher by a factor of 1.3 than the ratio between a single string and the piano soundboard. This might lead us to expect the decay time on a harpsichord to be about 20/1.3 = 15.4 seconds (see statement 8, sec. 17.5; why is it correct here to use the single-string rather than the triple-string wave impedance for the piano?). I find by informal trial on a similarly proportioned modern harpsichord that the apparent persistence of the overall tone is in fact much less than this, the time being on the order of half a dozen seconds. However, the above prediction based on wave impedance ratios is oversimplified, since it does not take into account the damping of string vibrations by viscous friction in the surrounding air. This damping has only a small role to play in the behavior of a piano string, but it cannot be ignored on the harpsichord.
In 1856 the distinguished British physicist Sir George Stokes worked out the theory of such air damping of string vibrations by viscous friction, and showed among other things that the vibrations of small-diameter wires are more quickly damped than are chose of large wires (halving the diameter halves the decay time). He further showed that the high frequency modes die away more quickly than do the lower ones (doubling the frequency reduces the decay time by about 70 percent). If we confine our attention to the lowest-frequency component (mode 1) of sounds from the two instruments, it is possible to reconcile their decay times reasonably well by including the effects of air friction.' Numbered statement 4, below, implies the resolution of any remaining discrepancy in the perceived decay times.
Digression on Archimedes and Mersenne.
The inverse relationship between vibration frequency and both string length and diameter was recognized long ago by Archimedes (287-212 B.C.), at least in the tense that halving either dimension would raise the pitch by an octave. The fondness of Greek intellectuals and their successors in Europe for simple ratios at a means for expressing the perfection of nature obscured for many years the fact that the vibrations of bars and those of water in a cup do not follow such relationships, and (in particular) also obscured the square-root relationship between string tension and vibrating frequency. 1t it perhaps significant that the Frenchman Marin Mersenne (1588-1648), who is credited today with scientifically clarifying the nature of string vibrations, !iced at a time when craftsmen were already very expert in making use of musical strings. Mersenne was quite aware of the influence of stiffness on the effective lengths of strings. We should realize, however, that the class of ideas implied by our term wave impedance did not become well systematized until the latter half of the nineteenth century, following the laying of the Atlantic telegraph table. 
We are now in a position to describe the tonal nature of the harpsichord sound.
1. When a harpsichord key is depressed, the plectrum is in contact with, the string for a short time before the string slips off of it to vibrate freely. During this short contact time, small-amplitude but audible clavichord-like vibrations arc set up on the portion of string between plectrum and bridge, and also in the part between plectrum and fixed string end. In particular, the sound begins with a brief but complex buzz as the echoing impulses on both sides of the string cause it to tap against the plectrum. The sound recipe also contains harmonic components belonging to the characteristic vibrations of the short and long portions of the string acting independently. These are not generally in tune with the note eventually to be produced, the exact frequencies depending on the position of the plucking point along the string.
2. Once the jack has pulled the string aside and released it, ordinary plucked-string vibrations of the sore discussed in sections 7.3, 8.1, and 8.4 are set up on the whole string. Furthermore, the string shapes and velocities chat are present on the two sides of the jack before the string slips clear now become free to travel up and down the length of the entire string. The presence of these additional vibrations means chat. as in the case of the piano tone, the complete recipe has in it modes of vibration that have nodes at the plucking point.
3. When the player releases a key, the plectrum brushes past the string slightly before the damper comes down into action. During this interval of time an extra bit of sound arises from the momentary tapping (buzzing) of the string as the plectrum slips past it. Because this tapping takes place between the string and a relatively hard. narrow object, a great many of the string modes are excited to appreciable amplitude. This is particularly true because, in contrast to the effect of a single. metallic tangent blow, we have here repeated blows, all exactly in step with the natural vibrations of the string. The duration of the tapping excitation is somewhat longer than that of its clavichord like predecessor during initial plucking. The brief chirp that one generally hears at the end of a harpsichord tone is compounded out of the main tone plus the components added on the plectrum's return trip, these being permitted to decay over a period of 1/4 to 1/2 a second after the relatively narrow damper comes into action.
4. Besides the expected brief ringing after the damper touches the string, there is one more aspect of the damped sound of a harpsichord string that helps to establish the musical personality of the instrument. Since the damper is firm and narrow, the segment of string between the fixed end and the damper vibrates briefly at its own natural frequencies. In general the pitch of this short sound is not in any musical relation to the main tone. One finds, however, that certain strings of a harpsichord scale have their dampers located close to a node for one of their higher partials (typically the 5th, 6th, or 7th). For these strings, then, our ears arc provided with a more lingering, harmonically related reminder of the main tone, which may last for a second or two.
5. The tone color and (to a slight extent) the loudness are both altered when a key is struck more or less hard. That is in part due to changes in the amount and duration of the clavichordlike fraction of the tone. The remaining contribution comes from changes in the relative amounts of soundboard and damped-string sound that are produced in comparison with the relatively fixed amplitudes of the main string sounds (see sec. 17.6, statement 5).
6. The sound pressure recipe for a harpsichord note contains a much larger number of important partials than does the tone of a piano. The effect of frictional damping by the air on the slender strings of a harpsichord causes the high-frequency components of its tone to die away very much more quickly than do the lower partials, so that the perceived duration of the tone as a whole is very short. During the decay, the tone color changes because the vibration recipe rapidly loses its higher partials, exactly the reverse of the way in which piano tones are heard to survive longest via their higher partials.
7. Due to the slenderness of harpsichord strings, the inharmonicity of the partials of a harpsichord tone is generally very much less (e.g., l/14th as large at C4) than that found on a piano. The musical effect of the greater harmonicity is not particularly apparent, however, because partials 2, 4, 6, . . . of the harpsichord tone have, very crudely speaking, the same frequency shifts at C4 due to inharmonicity as do partials 1, 2, 3, . . . of the piano tone. The harpsichord's larger string impedance relative to that of its bridge also increases the random inharmonicity due to soundboard resonances. The harpsichord tone thus collects by means of its large number of important partials an aggretate inharmonicity that does not differ much from the inharmonicity associated with the fewer partials in a piano tone. Tuning discrepancies are perhaps a little harder to detect in the more diffuse but shorter-lived sound of a harpsichord.
18.3. Examples, Experiments, and Questions
1. On large harpsichords one finds stops that give the player a choice of varying tone colors. One of these stops arranges for the strings to be plucked a considerable distance from their fixed ends. Another arrangement presses a small block of felt against each string very near to the fixed end, so as to damp its vibrations lightly. See how many musical implications you can draw from the acoustical changes produced by these stops. Consider in particular that the influence of the added felt block at the end of the string increases progressively as we go to the higher modes.
2. Most harpsichords have at least a pair of strings for each note of the main scale, and the player has the option of plucking one or both of these. For both mechanical and tonal reasons, the distance between the fixed string end and the plucking point is different fits the two strings. If both strings of a pair could be plucked exactly together, the acoustical consequences would be very similar to those associated with multiple stringings on a piano. In practice the strings are not released precisely together, which at the very least eliminates the rapid initial decay. Think about the auditory consequences of having; strings excited a few hundredths to one-tenth of a second apart. Consider next what goes on if only one string of a closely tuned pair is excited directly by the player, the ordinary damper being lifted for both. Each vibrational mode of the plucked string their drives its originally silent counterpart into transient motion to the sort described in chapter 10. As. the driven oscillations of the second or "sympathetic" string build Lip, they in turn start driving the soundboard, and so produce a certain share of the audible sound. See if you can figure out why the vibrations of the plucked string may die out very rapidly at first, and yet leave its with an actual swelling of audible sound. What sort of tonal effect would you expect from the fact that the sympathetic string, receives an initial impulsive excitation when the first kink arrives at the bridge after the plectrum slips free of the other string?
3. Harpsichords are usually provided with a set of so-called "four-foot" strings in addition to the normal "eight-foot" ones that provide the basic scale. The four-foot strings are tuned to sound an octave above their nominal note names. Thus the C4 key of a harpsichord keyboard can pluck strings tuned to 261.6 Hz and also one tuned to 523.2 Hz. On the Rockers harpsichord described earlier, the string lengths of the four-foot strings are approximately half those of their eight-foot brothers. The wire sizes are not quite the same, however: at C6 the wires are alike, at C4 the higher-pitched string is about 10 percent thinner than the lower, while at C2 (the bottom note), the four-foot string is about 20 percent smaller. The thinner, high-pitched set of strings thus runs at a reduced tension, so that at C6 the wave impedance is 90 percent of the eight-foot value; at C4 and at C2 the figures are close to 80 and 70 percent. See what you can predict about the loudness and sustaining power of the four-foot strings, taking account of the fact that we hear better at high frequencies and also considering the fact that the short strings run over their own slender bridge, after which they are anchored with large downbearing directly to the soundboard in the region between the two bridges. The four-foot strings are generally plucked a little closer to their centers than is customary for the full-size strings.
4. Some concerts of baroque music are played at today's pitch, based on A-440, while at other times the choice favors a reference frequency near 415 Hz, which is a semitone lower. Harpsichords are sometimes built so that either pitch can be selected by mechanical transposition, the keyboard being slid sideways to operate on different plectra and strings. On other instruments it becomes necessary to retune the strings themselves to shift from high pitch to low pitch or vice versa.
Consider what happens to a satisfactory high-pitch instrument when the scring tension is slackened about 12 percent to bring it to the lower tuning. What will happen to the decay time of the tones and to their loudnesses? (Be careful here-there are several aspects to the physics and also to the perception process.) How will the stiffness and soundboard resonance contributions to the inharmonicity be changed? What musical consequences will these have? The initial thump from the soundboard and frame will have an audibly different relationship to the main sound. A serious builder of harpsichords might find similar cogications useful in suggesting ways to guide his proportioning of string gauges and lengths, soundboard thickness, bridge and rib dimensions, downbearing angles, etc., when he adapts a successful design intended for one tuning to the construction of an instrument tuned to the other pitch.
5. Unlike the piano, the bottom surface of a harpsichord case is closed by a large board traversed by several stiffening ribs mounted on its inner surface from one side plank of the case to the other. Soundboard vibrations communicated to the somewhat compartmentalized air cavity within the case are radiated into the room via the long, narrow opening left at the keyboard end of the soundboard. The overall balance of sound from different parts of a harpsichord scale (both loudness and tone color) can be influenced by the acoustical relationship of the air cavity modes to those of the soundboard (see sec. 17.6, part 10, concerning the lid on a piano). The effect is particularly noticeable at the bass end of the scale. Refer back to section 9.5 and see how much of the discussion there of the interaction of kettledrum cavity and drumhead can be adapted to the present situation. Why could the cutting of a hole in the case bottom, or of an elaborately carved "rose" in the soundboard, be expected to produce tonal changes?
6. One occasionally meets notes on a harpsichord that "beat with themselves" even when only a single string is permitted to vibrate. The simplest way in which this phenomenon can come about is the following. At the bridge, the stiffness of the anchorage appears considerably greater to a string that vibrates from side to side in a horizontal plane (parallel to the soundboard) than it does to a string vibrating more normally in a vertical plane. Because of this, any given mode of oscillation will have a slightly higher frequency when excited in a horizontal plane than in a vertical (see sec. 16.5). Furthermore, we find that the plane of oscillation for such a string will slowly rotate, at a rate equal to the frequency difference between the two versions of the mode. As a result, an initial vertically oriented oscillation produced by normal plucking will slowly rotate a quarter revolution into a horizontal oscillation (which cannot drive the bridge) before continuing another quarter revolution, at which time it will again become a vertical oscillation, etc. Verify char the sound waxes and wanes, as a result of this rotation, at twice the frequency we normally would associate with ordinary beats between the modes. Why is this whole phenomenon most likely to manifest itself for string modes whose frequencies lie close to resonances of the soundboard? Can you figure out why even a slight kink put into a wire during installation can give rise to a similar kind of bearing sound?
1. Kenneth Bakeman, "Stringing Techniques of Harpsichord Builders," Galpin Soc. J. 27 (April 1974): 95-112.
2. Friedrich Ernst, "Four Ruckers Harpsichords in Berlin," trans. David Jones, Galpin Soc. J. 20 (March 1967): 63-76, and J. H. Van der Meet, "An Example of Harpsichord Restoration," Galpin Soc. J. 17 (February 1964): 5-16.
3. A. B. Wood, A Textbook of Sound. 3rd rev. ed. (1955; reprinted., London: Bell, 1960), pp. 109-110. See also Irving B. Crandall, Theory of Vibrating Systems and Sound (New York: Van Nostrand, 1926), pp. 12-33.
4. A fascinating account of the historical development of these ideas is to be found in Sigalia Dostrovsky, "Early Vibration Theory: Physics and Music in the Seventeenth Century," Archive for History of Exact Science, in press.