This essay discusses acoustics relevant to the practical building of marimbas, particularly in the unexplored low register. It was part of my work in 1994 to build two bass marimbas which between them have a range A⁰ to G³. This is one octave below the range of a standard bass marimba. (If there is such a thing.)

In recent years the quickly developing marimba seems to be developing further into the lower register. The upper limits of bar percussion are already well explored by the xylophone and glockenspiel.

"The story of the marimba is an ancient and colourful one. Few people realize that the marimba is one of the oldest instruments in the world - that it preceded the violin by about 2000 years." (*1.)

Musicologically classified as a xylophone, the marimba is similar in many aspects to that instrument. Only known to the Western World for a century, the two instruments share a common ancestry. Since the late 1920s the further refined marimba has become the more important instrument.

The principal of the instrument is that of a tuned timber bar, amplified by a resonating chamber. As the marimba develops further, especially into the lower register, the importance of the resonator becomes more pronounced.

Perhaps the earliest record we have of the marimba is the illustrations from ancient Madagascar of a woman sitting on the ground with two to three slabs of stone placed across her outstretched legs. Evidence of this type of instrument, (often with a shallow pit in the ground acting as a resonating chamber) has been found in many places around the world. (*2.)

Although the early development of the marimba has been confined to Central America, its origin, preceding its recorded history, has many possibilities. It is now widely excepted that the marimba originated in Africa and is closely related to the African xylophones.

By the beginning of the 20th century, the marimba had begun its exposure to the rest of the world, through North America. It was already a sophisticated instrument, and remains greatly unchanged in Central America today. Western culture has further refined the marimba in the last ninety years.

The modern concert marimba of today reflects the European idea - the sharps placed slightly to the right of the naturals. The Central American Marimba Doble however, remains the way it was originally built, the sharps directly behind the naturals.

Perhaps the most important contribution to the rise of the xylophone was made by George Hamilton Green and his brother Jo. Born in Omaha, Nebraska in 1893, George was proclaimed "the greatest xylophonist in the world" by the time he was 21.

The extensive composition, teaching and performing of the brothers raised the profile of the xylophone to a high level, providing a market for keyboard percussion instruments. (*3.)

The Deagan company made its first chromatic xylophones in 1903, and a few years later began manufacturing marimbas as well. At about this time, other companies also began to manufacture keyboard percussion instruments. Gradually marimbas gained space in the catalogs and by the '30s and '40s had clearly overtaken the xylophone. (*4.)

The biggest contribution to the marimba world in this period was by Clair Omar Musser who joined the Deagan company in the early '30s. During the following decade Musser personally built over 200 marimbas, and was responsible for most of the innovations in commercial instruments. Through his teaching and conducting, he introduced the marimba to much of Europe and the United states. (*5.)

The full size concert marimba of today is 4 octaves and a sixth - E² to C⁷ with some of the repertoire requiring a five octave instrument C¹ to C⁶, as used by virtuoso Keiko Abe. The bass marimba built by Clair Musser in 1950 covered this bottom octave, and some bass marimbas today extend as low as A¹.

1. James Dutton.	'Marimba.' in Percussion Anthony. 4th ed. (Northfield, Illinois: The Instrumentalist Company, 1998) 18.
2. David Vela.	Information on the Marimba. Ed. + Trans. Vida Chenoweth. (Auckland, New Zealand: Institute Press, 1958) 55.
3. Barry Bridwell/Scott Lyons	'Focus on performance: Marimba technique - a salute to George Hamilton Green, Xylophone genius.' Percussive Notes. (25#5, 1987) 54-56.
4. Linda Pimentel.	'The Aristocracy Of Manufactured Marimbas.' Percussive Notes. (21#1, 1982) 61-64.
5. David Eyler.	'Clair Omar Musser and his contributions to the marimba.' Percussive Notes. (28#2, 1990) 62-63.

When building a marimba, there are certain aspects to be considered which will effect the design.

"Bars, membranes and plates are three classes of vibrators whose modes of vibration are not related harmonically. Thus the overtones they sound will not be harmonics of the fundamental tone. The inharmonic overtones of these complex vibrators give percussion instruments their distinctive timbres, quite unlike those of the string and wind families."^1.

The many modes of vibration in a marimba bar affect the sound it produces and give it its distinctive quality, however a strong fundamental is essential for a good tone.

The purpose of the resonator is to amplify the fundamental frequency produced by the bar. This becomes increasingly important as the frequency extends lower and the radiation power of the bar becomes weaker. The relatively pure tone of the vibrating air column is dominant in good low pitch marimbas.

The wide pitch range of marimbas necessitates consideration of the human ear's ability to function at different frequencies.

The loudness of a sound experienced by the ear is not simply related to the intensity of the sound in decibels. Subjective loudness is different for every person and varies with the frequency of the sound. At around 2500 Hz the ear is at its most sensitive and very little sound pressure is required for a comfortable level of volume to be perceived. As the frequency becomes higher or lower, and increasing amount of sound pressure is required to achieve the same subjective volume. Subjective loudness is best shown by what is known as the Loudness Level Curves.^2.

The difficulty in compensating for this affected is heightened by the fact that sound pressure is an exponential function of the energy that produces it. In other words, to double the sound pressure of a sound requires more than double the energy.

The pitch of a marimba bar thus has a great effect on how it is tuned. In the extreme upper register, the harmonics become so high as to make them inaudible. Because of this, these harmonics are rarely placed, while by contrast, the lower bars can have as many as four vibratory modes carefully tuned.

Unlike the xylophone, the bars and resonators of the modern marimba become wider at the lower end. This helps to compensate for the drop in aural sensitivity and to achieve a constant volume over its whole range. This is important for the marimba as it sounds at pitch rather than at the octave like the xylophone.

In addition to the problem of reduced aural sensitivity to lower pitch, the lower marimba bars actually tend to radiate their fundamental weakly relative to the upper partials. This is because the dimensions of the bar are smaller than that of the wavelength of the fundamental frequency in air.

From these factors alone, it is obvious that as the marimba develops into the lower register, the resonator becomes increasingly important.

At the closed end of a tube the air cannot move, which forces that part of the air column to be a nodal point in any sound wave that is vibrating in the tube. Similarly, an open end must be an antinode. This means that the wave sections that can fit into a tube closed at one end are multiples of 1/4 wavelength. Likewise, in an open tube, multiples of 1/2 wavelength can vibrate, as represented in Fig. 3.

From Fig. 3 we can determine two important things. Firstly that for a given frequency, the minimum length of open ended tube that will resonate is twice that of a closed tube. Secondly, that the open-ended tube is capable of vibrating at all of the harmonics of its fundamental frequency, whilst the closed tube can only produce the odd numbered harmonics.

A marimba generally uses closed resonators because of both these considerations. In the lower register, where the resonators are particularly important, the wavelength is such that the space and length of tube saved with a 1/4 wavelength resonator is considerable. A closed tube also means that the harmonics of the marimba bar can be tuned to suitable notes without being amplified along with the fundamental.

The maximum possible amplitude of the vibration in a resonator, is dependent on the tube's diameter. Although its possible to increase the loudness of a resonator by increasing the diameter of the mouth only, the effect is not as pronounced as that of making the whole tube wider. As the resonator is generally the maximum width that can fit beneath the bar already, the idea of horn loading its mouth is worthless in most cases. This practice would also increase the difficulty of calculating the physical resonator lengths.

"At the open end of an air column containing a standing wave, the air is moving in and out of the open end, and its motion actually extends a little way past the end. This makes the tube appear longer than it actually is by an amount called the end correction. For a cylindrical pipe of radius r, the end correction has been calculated to be 0.61r"^3.

This is the approximate clearance between the bar and resonator which should produce the best sound in most cases. Reducing this clearance slightly has the effect of increasing the volume of the note. This is because less of the bar's energy is lost outside the resonator. The placing of the bar inside the acoustic length of the vibrating air column however, is often enough to flatten the note quite significantly.

There are essentially three ways in which a bar may vibrate: transversely, longitudinally and torsionally.

Marimba bars are concerned mostly with the transverse modes of vibration.

The perfect rectangular bar of uniform material and dimensions has the first three transverse modes along its length with ratios m1-1.00, m2-2.756, m3-5.404.^4. The closest one can get to this in practice is a glockenspiel bar, which also has a torsional mode lying between the second and third transverse modes.

It is in the lower range of the marimba that many partials fall within the audible frequency range; in particular the first three transverse modes. In earlier instruments only the fundamental was tuned. Over the last eighty years the tuning process has developed considerably, and it has been found that the best placement of the second and third modes is two octaves, and three octaves plus a major third above the fundamental respectively. These are both even multiples of the fundamental frequency, so are not amplified in a closed resonator.

Some of the most important work in this area has been done by Ingolf Bork and Jurgen Meyer^5., with their investigations into the partials of timber bars and their effect on pitch recognition.

They found that the intensities of the various partials depended mostly on the manner of exciting the bar and had no effect on the recognition of the fundamental, as long as the second partial was correctly tuned to the double octave within fifteen cents.

They also found that with a reasonably strong fundamental the placement of the other partials had no effect on the pitch recognition, unless very close to the fundamental.

The decay time of the partials from their points of maximum amplitude plays an important part in the perception of the note. Bork and Meyer found in a typical system, the decay times of the first and second overtones to be approximately 1/2 and 1/7 that of the fundamental respectively.
Fig. 5. Shows this.

The above Fig. 5 shows that there is a considerable period for which the fundamental is the only frequency present. However it also shows, that because of the relatively slow build up of sound in the resonator, the fundamental has not reached its maximum amplitude by the time the third partial has completely decayed. The conclusion reached therefore, was that as there is a period where only the overtones are present, their harmonic placement could effect pitch recognition in extreme cases.

In the extreme cases that my own bass marimbas present, the lower frequencies require a relatively long time to build up sound in the resonators. As the number of cycles per second becomes less, more time is required for the critical number of cycles. This means that the partials play a large role in pitch recognition.

Over a whole instrument however, the consistency of the tuning is important.^6. Over the average range of the marimba the tuning of the overtones is not just for easy recognition of pitch, but for evenness of tone and the avoidance of harmonic clash.

The first step in building a set of marimba bars is selecting the appropriate dimensions of the highest and lowest bars. Once these bar's have been cut, the other bars are graduated in between.

The next step is to find the nodes of all the partials (in the bars) to be tuned. Some of the fundamental nodal points will have to be compromised to a degree, but if the bars are graduated properly the nodes should also align closely between the extreme bars.

The fundamental's node lies approximately 1/5 of the bar's length from the ends. The exact points can be found by the gathering of small particles on the bar's surface under vibration. The best place to drill the bar's suspension holes is actually slightly inside the exact nodal point. This allowing the bar to vibrate for a longer period.

A node for the second partial of course lies in the center of the bar, which is also the antinode for the fundamental. The same logic dictates that the second partial's antinodes lie halfway between the nodes discussed so far.

Removal of material from the bar under an antinode will reduce the bending stiffness, thus lowering the pitch of the relevant partial.

The bending stiffness of timber in general is much less across than along the grain. In some cases this results in the first transverse node across the relatively short width of the bar, being actually low enough to be close to the second transverse mode along the grain. Usually this only occurs in the center of the common marimba range.

Although the excitation and radiation of this "Cross" mode is generally weak, it becomes noticeable when it comes within a tone or so of a stronger mode causing a harmonic clash. It is possible to tune this mode away from the problem pitch area by a process called wedging.

Wedging makes the Cross node sharper than it normally would be, by removing timber from under the outer edges of the bar's width before its center. Thus a cross section through the middle of the bar looks to be wedge shaped.

The initial idea that a marimba can go as low as other instruments, as long as its made a it louder to compensate for reduced aural sensitivity, is misplaced.

Most musical instruments, including those above, produce complex waveforms with a large number of strong partials that are harmonically related to the fundamental. The bottom A on a piano for example, has a fundamental frequency of 27.5Hz, which is completely inaudible to most people at the maximum sound pressure that even a good piano string is capable of producing. As a string is clamped at both ends however, it produces a full harmonic series of partials which comprise a large quotient of the total sound pressure. Even the first and second overtones at 55Hz and 82.5Hz fall into a greatly more sensitive region as can be confirmed by the loudness level curves.

The human ear and nervous system is used to hearing these harmonic relationships, and has the ability to assign a fundamental to a harmonic series even when the fundamental is week or missing. Benade^7. suggests that even if all but the sixth and seventh partials were electronically removed from the sound, the remaining frequencies at 165 Hz and 192.5Hz would be enough for us to easily hear a non-existent fundamental of 27.5Hz.

So as long as a string or wind instrument as the ability to produce a low note - no matter how weak the fundamental - it is usually enough for the listener to recognize it.

In the case of a low marimba however, the overtones are not harmonically related and tend to be avoided in favour of the pure resonator tone, so we come much closer to producing a pure single frequency. Some of the odd numbered harmonics which can (in theory) vibrate in the resonator, may actually do so. As they are not being excited however, they will be very weak. This means that for the same low A as the piano, the marimba must produce in the order of 65 decibels before a straining listener will even be aware of it.

In terms of building such a marimba note the implications are enormous. Not only does the resonator have to be over three meters long, but very wide, and with a bar over 1 metre long.

Quite apart from the problems of increased size and tuning requirements, there is a problem in tuning a low marimba bar. It is impossible to tune the fundamental by ear. Even when the antinode of the bar is next to the ear so the vibration can be heard, an accurate pitch cannot be assigned to the sound.

"Dr.Knudsen of the University of California at Los Angeles has investigated this problem, his findings shown graphically in" Fig. 7.

versus frequency.">
Fig. 7. Chart showing sensitivity of the ear as a function of the frequency.

Culver suggests that for a sine wave around 60Hz, the maximum pitch variation that the average ear can detect is almost a whole semitone.^8.

Although pitch recognition is greatly a matter of training, special equipment is necessary to tune a low bar and resonator with the required accuracy.

When the tuning of a low marimba bar and resonator match perfectly, the result is often subjectively unsatisfying. In this situation the resonator takes the energy away from the bar too fast, and the note does not last the duration we expect. Unfortunately an unaffordable loss in volume occurs if the resonator is detuned. This can be an appropriate time to utilize the flattening of the bar caused by closeness to the resonator mouth - detuning the note enough to give the desired length whilst actually increasing the sound pressure output. If the note is low enough its flattening is often appropriate anyway, being an extension in the lower direction of the stretch-tuning idea.

"Las Maderas Que Cantan" is the phrase used by the Central Americans to describe the timber and the instruments built with it, literally translating as "the woods that sing." Very few timbers actually do "sing", most of these being unsuitable for marimbas and xylophones for other reasons.

Some keyboards are made from American Grandillo which does not ring quite as long as Hormigo. One timber that works well for marimba is African Padoak, though it is not hard enough for xylophones and some of the smaller bars don't have the weight to sit properly on the frame.

Fairly recently (as far as the history of the marimba is concerned) a material called Kelon was developed by the Musser company for use in marching band instruments. The pultrusion process used to manufacture kelon is described by Art Ormaniec, Chief Engineer for Musser Industries as a "gathering of the glass fibres which are then pulled through a resin bath, pressed together, and finally drawn through a steel die and heat treated."^10.

Kelon is very durable, bright in sound quality, and produces plenty of volume. Not as suitable for a concert instrument, it lacks warmth and the punchy character that comes from the special elastic qualities of timber.

Once suitable timber is found it must be seasoned for a number of years, or dried in a kiln before an additional year or so in air. It must reach approximately 7% moisture content. "That is non as equilibrium moisture content (EMC) and is the point at which, given a specific temperature and humidity, the lumber will not absorb or desorb water." ^11. Even after this process the timber continues to change throughout the rest of its life, and new instruments generally have to be returned after a few years to compensate for the variation in water mass.

A resonator can be made of almost anything airtight. Probably the best material for custom resonators is PVC plumbing. Lightweight and durable, it comes in a range of sizes complete with fittings. Easy to work with and seal, fairly cheap, and most importantly, it has a smooth, hard surface providing low frictional damping, for the air vibrating through the open end.

The size requirements of low marimbas present practical difficulties in the areas of instrument range and transportation. These can be overcome to an extent by limiting the instrument to its critical components and shapes. "Bruce" - who is one of my own lower pitch marimbas, for example, is seven notes in range A⁰ - D#^1., and has no vertical frame, the bars being supported by the resonators. The resonators are too heavy to support, and heavy enough to "sit" by themselves. Each of the resonators is "L" shaped and extends along the floor for as much as two meters. As they come apart at the join, transportation is relatively easy. For performance however, the instrument is an awkward shape to place on stage.

For this reason it is important to have a clear idea of the likely uses and performance venues of a low marimba before the design is conceived.

for more information about "Bruce" and other bass marimbas I have built, click here!

One of the problems with a low marimba, is the size of the room needed to contain the sound. Using the example of A ⁰ with a frequency of 27.5Hz it becomes clear that a room in the order of twelve meters in at least one direction is needed to contain a single wavelength. I have found that the dimensions of most rectangular rooms are such that pronounced standing waves usually exist for one or more of the seven notes. This means that there is a dead spot in the room where a listener cannot hear the note. If the instrument is placed in such a spot, the note will not sound significantly in the room at all.

This is rarely a problem in a well designed auditorium, as it will be of sufficient size and have few parallel boundaries. Many smaller venues and rehearsal rooms can present difficulties though.

Perhaps one of the most important considerations is that of the music that can be played on such an instrument.

To produce an appropriate sound on a bar over a metre long, requires a very soft mallet head with enough mass to sufficiently excite the fundamental This extra weight along with the leverage of handles long enough to reach the notes, puts quite a strain on the small muscle groups generally employed in marimba technique. When the extra distance between notes is added to the equation, it becomes obvious that a low marimba cannot be played like the smaller instruments.

Facility is not likely to be a problem however, as there are very few occasions when faster passages on a low marimba work compositionally speaking. Not only does it take as much as 7/10ths of a second for the sound to build up in the resonator, but it can take as much time again before the sound is recognizable.

Culver ^12. suggests that although pitch recognition can take place in one twentieth of a second regardless of frequency, it takes two whole cycles before a sound can be recognized as such. It may take as many as twenty cycles before the complete characteristic of the sound can be deciphered.

For the low A, this is 0.73 seconds. Add this to the time taken for the bar to excite the air column, and its over a whole second from the time the bar is hit to the time the whole sound can be heard.

One method of overcome both the problems of standing waves, and low subjective loudness, is through amplification. This has its own range of problems however, and if not done carefully can actually lower the overall subjective sound.

Each of the stages of sound reproduction - microphone, amplification, equalization and speakers - all need to be able to deal with the required frequencies.

Firstly, a quality microphone with low bass roll off is needed to provide a sufficient signal. If the response in the required region is too little, the amplification will have to be so high that the white noise produced will raise the subjective threshold of hearing.

Once a quality input is achieved, the electronic processing must be kept to a minimum as each process tends to roll the signal of higher, and create its own quite noise. The power of the amplification generally needs to be relatively higher, as producing volume at low frequencies takes a large amount of energy. Allowing for double this amount will mean that the amplifier is not working in its peak region, and further reduce unwanted noise.

Any speaker cone 18 inches in diameter or larger is generally capable of reproducing a 27.5Hz signal. The performance of various cones in different enclosures is highly variable, and the manufacturer should be consulted for the exact specifications. Once a speaker has been found that will reproduce the desired frequencies with suitable efficiency, a power handling of double the amplifier's output is required to minimize distortion.

Once all of the amplification appliance have been achieved, the size of the wavelengths must be considered when placing the speakers. If the distance of the speakers to a listener is an odd multiple of a half wavelength different from the distance of the listener to the instrument, then phase cancellation will occur. The listener will hear very poor sound, or none at all.

As we are dealing with a range of frequencies over a keyboard, the best way to avoid this problem is to place the speakers as close as possible to the instrument.

When dealing with any aspect of a musical instrument, it is important not to be too reliant on acoustic theory. Although the principles of acoustics are absolute for any given situation, our knowledge of these principles can never be. Once theory has guided us to a finished instrument, the only test of success is that of the ear. If it subjectively satisfies, then it is not important why. If it does not, then the reasons should be studied until a solution is found.

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