The art and science of instrument making involves many factors, each of which, when taken as a whole, will ultimately define an instrument's sound and playability. In this article, Franz Berwein explores the physics of sound as it applies to the zither. Originally published in German, this translation has been kindly provided by Jane Curtis.
This article appeared in 1991 in Saitenspiel, the publication of the German Zither Society DZB. It is translated and reprinted with their permission and that of the author. Although it is written with particular reference to the zither, its discussion of hearing, physics of the string, energy transfer, wood, vibration patterns, the transition of modes to sound, the vibrating air space, temperature and humidity, varnish, acoustics - all make it of interest with regard to other stringed instruments as well. Parentheses are as used by the author; brackets indicate translator notes.
Zither builders require instinct and long experience in order to build fine-sounding instruments. The science of acoustics can also contribute, for the known sound patterns of wood panels to be used for vibrating zither surfaces can reveal whether a given piece of wood will produce an instrument of superior tone.
The clear transparent tone of the zither results from the complex interplay of the strings with wood, air, and the human ear. Most important are the two resonance surfaces (especially the lower). That the instrument produces such a rich spectrum of partials is due primarily to its numerous modes or resonances. It is well understood how resonance occurs in the piano, violin, and guitar and one might perhaps think that it could not be so very different with the zither. All of these instruments have a resonator with its lower surface made of wood; all of them transfer the vibration energy of the strings to the resonating parts by means of bridges. The zither, however, is unique.
The Physiology of Hearing
The human hearing range lies between 16 and 20,000 Hz (Hertz) and declines sharply after about the age of forty. Tone differentiation is the most successful between 1000 and 3000 Hz (0.3% or one fortieth of a whole tone). This means, for example, that 1000 Hz can just be distinguished from 1003 Hz; we speak of a frequency differentiation threshhold of 0.003. Incidentally, the middle ear transmits only frequencies up to about 2000 Hz. However, since sound also causes the whole skull to vibrate (see tuning fork test), with the vibrations being transmitted directly to the inner ear, soundwaves with frequencies above 2000 Hz can also be heard. In general about 1500 pitch differences can be recognized, depending on the individual person.
On the other hand, the intensity threshhold, at 0.1, is much less refined. A sound is only perceived as louder or softer when its intensity is changed by more than ten percent (i.e. the sound pressure by more than five percent). The ear differentiates about 325 volume gradations. Thirty thousand nerve fibers transmit about 340,000 values over the aural nerve to the brain, by means of electric impulses called action potentials (up to 900 Hz per fiber). The sum of all the impulse frequencies determines the loudness.
The Physics of Strings
Strings are poor sound emitters, because they are poor in energy; they require appropriate resonators to pick up the vibration and amplify it. The form of the resonator has a decisive influence on the timbre of the instrument. Thus, for example, larger, more powerful zithers with longer dimensions (e.g. bass or alto zithers) sound darker and softer than descant or prime zithers. [The descant zither has a tonal range from C to about g'' on the fretboard (the higher tones are much less resonant and are less often used) and a range from about AA (depending on how many contrabasses are present) to f' in the accompaniment, bass, and contrabass strings.]
The zither changes the mechanical energy of the vibrating strings into sound waves, which propagate in the air and cause ear drums and skull bones to vibrate correspondingly.
When a fretboard string is struck with the ring, or an accompaniment string pulled with the side of the fingertip, it is stretched to a point like the string of a bow, until it slides over the tip of the ring or the finger. The string then begins to vibrate, and, as Hermann von Helmholtz (1821-1894) proved with bowed strings, the point remains. In the course of one vibration it runs through the entire amplitude curve of the string, which is the only part of the process seen by the lazy eye.
The point runs along the string to the bridge, which lies on the resonator. Over the hitchpins part of the energy is transferred to the bridge and from there to the thin flexible resonance surface. The rest of the energy causes the point to be reflected toward the opposite bridge [saddle]: the string swings out in the opposite direction. The left bridge [called the saddle on the zither, the equivalent of the nut on necked instruments] is attached to the solid tuning block (in which the tuning pins are placed), so that hardly any energy is transferred; but the point is reflected again.
Thus the point of the string runs forth and back many times between tuning block and bridge until the string comes to rest (unless it has been damped), having imparted most of its energy to the resonance surface, a small portion to the air. How rapidly a point propagates depends on various parameters.
Now the actual vibration pattern of the string is substantially more complicated. Namely, it vibrates not only with its primary frequency but simultaneously with partial vibrations at higher frequencies (also called overtones). The partials are called harmonics if their frequencies are whole-number multiples of the fundamental frequency.
This only applies in principle, however. Ideal strings would vibrate in harmonic intervals, but in actual practice the frequency proportion is not completely pure. Because of the stiffness of strings, the upper frequencies are slightly too high. The typical zither sound is one of the effects to which this gives rise. Without this special tone quality, we would probably perceive it as flat and boring.
Transfer of String Energy
The energies of all the partials meet on the bridge, which reacts in various different ways. The vibration of many harmonics is transferred well to the lower surface, that of others poorly or not at all. The speed of transfer is also determined by the bridge. The effect exerted by the bridge depends on how well matched the sound impedances of string and resonance surface are (impedance designates the forces that limit the diffusion of sound in a medium). If they were correlated exactly, and if the bridge did not limit or reflect any vibration, all of the string's energy would be transferred immediately to the lower surface. In this case, only a short loud stroke would sound. In the opposite case, if impedance were poorly balanced, the vibrating string would only be able to deliver its energy so slowly that very little of a tone could be heard at all.
Some Facts About Wood
The lower surface of a zither, usually of solid long-grained spruce (Picea abies), is about 4 to 6 mm thick and takes various forms. Each piece of wood is unique in its own way, and even two boards cut from the same tree, even from the same part of the trunk, are different. Therefore just building an exact copy of a fine-sounding instrument is not enough to produce an instrument with the same sound characteristics. If you wish to build a good zither, you must not only keep to the exact measurements but also consider the vibration characteristics of the wood. The sound properties are determined not only by contour, thickness, density and grain, but also by the characteristics of the [wooden] bridge and of the various [wooden] ribs or braces glued to the underside of the upper surface.
How zither builders select the appropriate tree, and how choosy they are in the process, is a story in itself. The chosen trunks are cut lengthwise into four and then cut to approximately the desired length. The block-like wooden pieces - the rind - must then be stored outdoors in woodpiles with protective roofing. Tradition requires a storage time of at least five or ten years for spruce. Some zither builders maintain that the wood should mature for at least fifty years. This opinion is supported by the conjecture of some wood experts that during the aging process the crystalline components inside the cell structure of the wood increase in relation to the amorphous components. This would explain the advantages of seasoning, as amorphous material readily absorbs and releases moisture, while crystalline material does not. This may also be the reason why many old instruments are better able to withstand changes in atmospheric humidity.
What physical properties distinguish the wood panels approved by a zither builder for the upper and lower surfaces, and what influence do they have on the tone of the individual instruments produced? Materials engineers look primarily at five quantities:
Elasticity is distinguished by means of the elasticity modulus for the two directions along and across the fibers. They show the material's resistance to bending and pulling forces, and they are defined as the proportion between the local force exerted per surface unit and the relative change in length that results from it. Shearing forces produce similar deformations, as can easily be demonstrated with a thick book by pushing one cover sideways against the other. There is also a measure for resistance to shearing forces - the shear modulus. Zither builders establish damping by tapping on the zither and noting how long it takes for the tone to die away. Good zither panels should show a maximum vibration time after the tapping. Other quantities beside vibration time can also be used to measure damping. If the zither panel is made to vibrate and the exciting frequency is changed consistently, the result is a resonance curve whose amplitude gives information on damping. The so-called Q(uality) factor can also be taken into account - the greater the Q factor, the smaller the damping. Finally, in order to determine sound velocity, it is necessary to divide the elasticity modulus by the density of the material and take the square root of the quotient. Spruce to be used for the upper surface of a zither should have as high a sound velocity as possible.
What physical processes are activated by tapping on the wood panels to be used for upper and lower surfaces? And can we, with the help of acoustic measurements, learn something about what tonal and playing qualities are to be expected from an instrument we intend to build with two panels whose acoustic properties are known? As early as 1830, the physician and physicist Felix Savart posed the question: "How high should the belly and back of a violin sound before the instrument is built?"
In order to answer such questions, the modes of unattached panels must be measured and compared with the qualities desired in the finished instrument. Savart used a method developed by his friend Ernst F. F. Chladni to make the vibration forms of the upper and lower surfaces visible. The panel is strewn with a fine powder and made to vibrate. At certain frequencies (the resonance frequencies or modes) a characteristic sound figure comes into being. Through the vibration of the wood, the powder is moved toward the non-vibrating zones (nodes) and gathers along the lines on which nodes lie. For each mode, the node lines produce a specific pattern - a Chladni sound figure. The frequencies at which a panel begins to resonate are determined by its solidity and mass.
Since then, vibration measurements have continued to be made on upper and lower surfaces before and after construction of the instrument, and we have learned to interpret the sound patterns. The violin builder and acoustic scientist Hermann F. Meinel in particular, during his research in the 1930s, discovered important relationships between the thickness of the panels, the vibration forms, the sound volume, and the timbre. He attempted to discover how the varnish, the properties of the wood, and the elasticity of the wood affect the sound of the violin. He was building here on previous research by Hermann Backhaus, and he attempted to discover whether an instrument can be improved in a particular frequency region by removing wood in some places. It turned out that this does not always lead to success, whereby Meinel encountered a basic problem of instrument building: Because several physical properties influence the totality [Zustand] of an instrument, the same minor change can have completely different effects on different instruments according to how it affects stiffness and the distribution of partials in each individual case. What leads to appreciable improvement in one instrument can worsen the quality of another.
If a zither is placed upon a loudspeaker, [powder strewn on the zither’s upper surface,] and a sinusoidal signal transmitted, so that the upper surface vibrates in sinus form, and if the vibration frequency is then slowly altered, the powder will arrange itself in characteristic sound patterns.
The sound pattern becomes much clearer when holographic photographs of vibrating upper surfaces are made by means of a laser. Through the vibrations, the modes take on typical interference patterns, which are similar to Chladni sound plates. With this technology vibrations with an amplitude of only a few micrometers can be clearly recognized.
It is also possible, with a tone generator on the hitchpin of each string, to create a sinus tone and thereby to measure the sound pressure. We thus obtain the corresponding resonance curves for the fundamentals of all tones. To be sure, this method gives only an approximation of the physical properties of individual instruments, but the typical characteristics of the zither are nonetheless already apparent. For one thing, the resonance surface has many - and fairly broad - resonance areas. With larger zithers (alto and bass) three to seven resonance peaks occur, all of them under 400 Hz. Descant zithers, on the other hand, never show more than three, which as a rule lie between 100 and 200 Hz.
The vibration modes of a resonating lower surface are represented here in accordance with a modal analysis. The various hatching patterns show how different areas vibrate.
In the frequencies between zero and 600 Hz alone, there were several dozen frequencies in which modes occur (i.e. at an average interval of about every 16.5 Hz).
Contrary to what one might think, the greatest vibrations occur mostly at some distance from the hitchpin. This shows how decisive it is that the impedances of strings and lower surface be well correlated. If the point at which the string crosses the bridge is set into strong motion, the energy of the vibrating string is transmitted too fast and an unpleasant sound results.
The fact that the vibrating zones on the lower surface take on a roughly elliptical form derives from the structure of spruce: In the direction of the grain, the vibrations propagate four times as fast as across the grain. The shape of the vibration patterns and therewith the timbre of the zither can be decisively influenced by special arrangement of the braces and bridges on the upper surface.
The vibration pattern of the lower surface becomes increasingly more complex as the frequencies rise. The vibration centers become smaller and appear in increasing numbers. The narrow strips of the grain in the lower surface between the two bridges [i.e. between the bridge on the right and the saddle on the left] behave like oval membranes. At these points there develop vibration centers, which unfold more or less symmetrically along the long axis. The almost
incomprehensible multiplicity and density of vibration patterns confirm a very high probability that each fundamental of a vibrating string excites a mode.
Very precise modes are responsible for the correct tuning and timbre of a zither (for example, modes 1, 2 and 5). The first mode corresponds to a vibration in which the upper surface is distorted in the direction of length. A zither builder can feel the first mode by holding the [panel designated for the] upper surface at both ends and twisting it. What this actually does is to determine the main stiffness of the first mode.
In order to test the second mode, some zither builders take one end of the panel in both hands, laying the thumbs on the top side parallel to the long axis, i.e. to the direction of the wood fibers, and the outstretched fingers on the under side pointing in the same direction. If the panel is carefully pressed and at the same time lightly bent, the stiffness across the direction of the fibers can be felt. Other zither builders can feel the second mode in their fingertips: Instead of placing
their fingers on the under side of the panel parallel to the fibers, they spread their hands and feel out the stiffness across and along the fibers as they press and bend. For the second mode - with good panels - a particular relationship between the two values is characteristic.
The fifth mode is determined by holding the panel firmly at both ends with the fingertips and bending it in the middle with the thumbs. This is also good for determining stiffness.
From Modes to Sound
A zither builder can cause different modes to sound by tapping on the panels. If the panel is held in the middle, and at a point that lies on a node line of the first mode, this mode can be excited by tapping on the upper or lower end with the fingertips. With the first mode, the two ends of the panel happen to correspond to vibration outcurves. With the second mode, the panel must be held at one of the four points at which the nodelines cut the edge. Here we must tap on the ends in the area of the vibration outcurves, i.e. in the vicinity of the longitudinal axis. The node lines of the fifth mode can be made to sound by grasping the panel at a point on the nearly oval node line of the fifth mode and tapping on the middle of the panel.
All of the modes contribute more or less strongly to the sound of an upper or lower surface, just how strongly depending naturally on the way in which the panel is held and tapped on. With a panel that has been fully processed, parts of different modes always sound, regardless of the tapping point. They are heard with particular clarity when the two fundamentals lie exactly an octave apart. It is often difficult to distinguish the frequency of the dominating sound. There are enormous differences from panel to panel, interpreted by zither builders in their own way. There are accordingly very different opinions on the sound of zither panels and what can be done with them.
How the characteristic vibration divisions change when unattached upper and lower panels slowly evolve into completed instruments can really not be followed with acoustical measuring methods either, let alone understood. In order to describe the vibration of an unattached panel mathematically, a minimum of nine parameters must be considered. To calculate all of them requires not only a great deal of technical knowledge but above all a great deal of time. With a completed zither this would naturally be even more complicated: top and bottom are glued to the frame, so that the edge of the two panels becomes a fixed nodeline. The effects of this on the various modes of the panel can scarcely be comprehended. All further additions inside or outside bring connected vibrations with additional resonances.
The Vibrating Air Space
Now on a zither, it is not only the strings and wood that vibrate. Between the two resonance surfaces air is enclosed, to which their vibrations are transmitted, and this air space too has favored modes. It could be described in simple terms as a trapezoid-shaped space bounded on the sides by fairly rigid walls and above and below by the more or less flexible resonance surfaces. There are however openings, differing according to how the instrument is built: Air can pass through the sound hole in the middle of the upper resonance surface and through openings near the tuning block.
The modes of this space can be predicted with the mathematical wave equation. There should be three types of modes: The first are vertical waves that occur at high frequencies, beginning at 945 Hz; more are added at harmonic intervals. The second type of modes expand from one side of the instrument and are reflected at the top of the tuning block, which again causes vertical waves. In this case the beginning frequencies lie fairly low; the higher frequencies appear at regular but not harmonic intervals. The third group has a node surface running across the middle of the hollow space, whereby every wave crest lies opposite a wave valley.
At low frequencies, particularly in the lowest mode, the instrument behaves like a partly open system; the dominating vibration forms equal those in a pipe closed at both ends, which means that they make almost no direct contribution to the sound of the zither.
The air space however is not unimportant: It definitely influences the vibration mode of the lower surfaces, by exerting pressure and suction and by enabling air and wood vibrations to unite. Exactly what happens in this complex interchange has not yet been researched.
Acoustics and Zither Building
When zither builders begin to carefully remove wood from the inside, to give the zither the finishing touch, they keep picking up the panels and testing how they bend and how they sound when tapped upon. They feel the stiffness of the wood and listen for how the sound changes when a few tenths of a millimeter of wood are removed. The art of zither building consists of acquiring the right feel for wood and of learning how the two panels should sound before the instrument is assembled, in order for the instrument to posses good sound qualities. It takes many years of experience to develop the necessary sensitivity.
Everything that zither builders feel and hear with a successful panel can be understood and duplicated by keeping track of measurements made. This process gives rise to five guidelines:
1) Good instruments result when the amplitude of the fifth mode is relatively large and the derivative modes of the upper and lower surfaces vary less than a tone from each other. If the upper surface has a higher mode than the lower, the instrument normally acquires a bright radiant tone. Its sound is the opposite, dark and full, when the lower surface shows the higher frequency in the fifth mode.
2) If the frequencies of the second mode vary by less than 1.4% (about 5 Hz), the result is an easy-playing soft-sounding instrument.
3) If the fifth mode shows the same frequency on both surfaces, the two frequencies of the second mode must not differ from each other by more than 1.4%, or a difficult-playing instrument with a hard sound will result.
4) Outstanding instruments result if the frequencies of the second and fifth modes on both panels are about an octave apart and both panels show strong resonance at these frequencies.
5) The quality can be further improved if the upper surface is so tuned that the frequency of the first mode lies an octave below the frequency of the second mode. Then the frequencies of the first, second, and fifth modes stand in a harmonic relation to one another.
It is easy to list these criteria, but whether they can be fulfilled in actual practice is another matter. In order to achieve the ideal state for the three most important modes, it is necessary to bring together various factors that alter the sound of an instrument. Even when the panels are successfully thinned at certain places in such a way as to show the desired characteristics, they can still be ruined by the varnishing process. The sound of an instrument is of course also influenced by changes in relative humidity and temperature as well as by the physical properties of the wood chosen for the upper and lower surfaces.
To be able to tune the panels well, the influence of varnishing must also be taken into account. It increases mass and stiffens the topmost fibers of the wood, thereby causing stronger damping. The smaller the elasticity modulus of a piece of wood, the more strongly stiffness and damping will be altered by varnishing. The effect varies with different types of wood. Varnishing can damage the tone of an instrument, but this can be prevented by including it in the calculations for tuning the unattached panels. On the other hand varnishing can be a positive factor when pairs of panels are out of tune with each other. Varnish matures for more than two years before it acquires its final properties. This is doubtless one of the reasons why a freshly varnished zither requires several years to unfold its true sound qualities.
Temperature and Relative Humidity
Zither builders continually face the problem that some instruments sound dull and monotonous in hot damp summer weather and others take on a hard tone in heated living areas where the air is dry. An instrument "feels" best in the temperature and humidity conditions under which it was built - only then can it reach its finest tone. Wood is sensitive to humidity, because it absorbs and gives off moisture readily. Although this exchange is somewhat slowed by the varnish on the zither's exterior, it is not hindered by the unprotected interior, if we do not take loss of tone quality into account. Wood absorbs moisture very slowly (over time periods of several months) but can give it off within a few hours. A layer of varnish clearly protects the surfaces; they are also put out of tune by changes in humidity, but much less than surfaces without the protective layer. Different types of wood absorb differently, for example maple more strongly than spruce. A humidity fluctuation of 15% to 80% causes a deviation of about 20 Hz in the fifth mode. At a constant humidity of 50%, temperature changes between 15 and 25 degrees Celsius have hardly any effect - they change the frequencies by only about 1%. In order to tune zither surfaces exactly, temperature and humidity should be held as constant as possible, or at least humidity should be kept at a steady 50%.
The Experience of Hearing — Physiological Manipulation
All in all, we can now understand relatively well how the zither sound comes into being and how it can be shaped by the instrument builder. When the player strikes the string, it begins to vibrate in a complex way. The vibrations become audible only after transmittal to the lower surface. In this process the bridge acts as a selective filter. The lower surface then exerts the dominant influence on the selected frequencies, which are affected to a less important degree by the other components and the air in the hollow space.
The decisive factor for clarity of timbre appears to be the multiplicity of modes. In particular, because of the high vibration density of the lower surface [see Figure 1], a large portion of the partials spectrum of the struck string becomes active and can generate modes - be they in the lower surface or the air or both at the same time.
Finally, our hearing also plays a role. One of the things that distinguish a good zither is the strong resonance of the low tones. However, the fundamental of a bass or contra string does not have a very great amplitude, so that the tone is actually not particularly loud. [If the amplitude were greater, however,] the string would swing so far out that it would strike the next string or even the top surface. [That we do not perceive the low strings as less loud] is due to an illusion by our hearing system, to the so-called residual effect: The low strings namely generate a series of markedly higher partials, and from them our hearing constructs a correspondingly strong fundamental. The high strings, on the other hand, seem almost to vibrate only in the fundamental, because we are unable to hear the frequencies of their higher and highest partials.
With the contra G (50 Hz), for example, the fundamental vibration of the string in the frequency spectrum that strikes our ear is barely perceptible. But the brain organizes the further harmonics (of 100, 150, 200, 250, ... Hz) into a group belonging to the contra G, to which it can easily contribute a separate frequency of 50 Hz.
We all experience the residual effect in telephoning. We recognize low voice ranges correctly although the loudspeaker in the telephone is much too small to be able to vibrate strongly enough for the respective frequencies. Another property of our hearing has an effect that is taken into account in constructing a zither. In general we hear high tones better than low. To compensate, the low ranges must release more energy when music is played. Good zithers are so built that subjectively they sound equally loud at every pitch.
Science can not replace the art of zither builders - far from it. It is their secret how they can create instruments whose tone bewitches and inspires us. With their long years of experience and their instinct, often inherited over generations, zither builders have mastered the complex interrelationships, probably without knowing what is happening according to the laws of physics. It is astonishing.
Loudness is the subjectively perceived strength of a sound, given in sones. A 1,000-Hz tone with a sound pressure of 40 dB has a loudness of 1 sone. A sound perceived as twice this loud has a loudness of 2 sones, and so on. Volume is an objective measurement for the strength of sound perception. It is a linear function of the decimal logarithm of the sound's intensity, given in dB. Volume so defined gives no direct information on the subjectively perceived strength of a sound event. That is, the propagation velocity c and the distance to be traversed (the length l of the string) determine the vibration frequency (pitch). For example, when the point can move forth and back 440 times per second, the sound is a', familiar to all musicians as the tuning (or chamber) pitch of 440 Hz. [See next section]
For critical review and valuable suggestions, I thank Prof. Dipl.-Phys. Eberhard Meinel, Westsächsische Hochschule Zwickau, University of Applied Sciences, Study Course Construction of Musical Instruments Markneukirchen.
Index of References:
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|HUTCHINS, C.M.:||Spektrum der Wissenschaft Verlagsgesellschaft Heidelberg 12/1981, Seite 113-122|
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|KUCHLING, H,:||Physik, 13. Auflage VEB Fachbuchverlag, Leipzig 1976, Seite 140|
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