Tuning the Marimba Bar and Resonator

Jeff La Favre

jlafavre@jcu.edu

Before we discuss the marimba in particular, it is helpful to review some aspects of the science of sound. In physics, sound can be described as a wave. We cannot see the vibrating movement of air molecules, so we do not have a visual sense of sound. However, wave phenomena in water are familiar to everyone. If a stone is dropped into a pool of water, waves propagate away from the stone in a circular pattern. A casual observer might reason that the wave is caused by horizontal movement of water away from the stone, but this is not true. Leonardo da Vinci described it eloquently: "it often happens that the wave flees the place of its creation, while the water does not; like the waves made in a field of grain by the wind, where we see the waves running across the field while the grain remains in place." In the case of the water wave, the visual effect is caused by the cyclical rise and fall of the water level and can be depicted as a sine wave graphically.

The graph is useful in defining some of the dimensions of the wave. The x axis represents the direction of wave propagation and the y axis represents the amplitude (height) of the wave. The normal water level is represented by a value of y = 0. The top of the wave is +ym above the normal water level and the bottom of the wave is -ym below the normal water level. The wavelength () is represented by the distance at which the wave pattern begins to repeat itself. Key positions on the wave are designated with the terms node (place where the wave crosses the x axis) and antinode (the position of maximum distance above or below the x axis).

A dimension of sound waves that is of particular interest in music is the frequency. If we graph the wave with time as the x axis, then it is possible to determine the frequency (which is related to the musical note). The period T is defined as the time interval in which the wave motion begins to repeat itself.

The frequency is calculated by the following formula:

f = 1/T.

The standard unit for frequency is Hertz or Hz (cycles per second). The notation that I have used on these web pages sets middle C as C4. Then the next C above middle C is C5, etc. The A above middle C (A4) is used as the standard note for tuning purposes and in the United States a tuning standard of A4 = 440 Hz is usually used (however, my marimba and many commercial marimbas are tuned to A4 = 442 Hz to yield a brighter sound that is supposed to sound better with the orchestra).

The Marimba Bar

The marimba bar vibrates in complex patterns that produce a sound uniquely characteristic to the instrument. An understanding of the sound quality (timbre) is gained by study of the vibrations. Each type of vibration is called a mode of vibration. In a scientific study of a C3 (C below middle C) marimba bar, Bork et al. (1999) identified 25 modes of vibration in the range of 0 to 8,000 Hz.

The tuning of a marimba bar may be elementary (only one mode tuned) or complex (several modes tuned). In the early 20th century, commercial marimbas were tuned only in the fundamental mode (the named note of the bar). During the 1920's a higher level of tuning was initiated by tuning the fundamental mode and the first overtone (the second transverse mode). This improvement in tuning yielded a more desirable sound from the bar because the first and second transverse modes were tuned to a harmonic interval. Later, master marimba builders started to include a tuning of the third transverse mode, a procedure sometimes called triple tuning. The finest modern marimbas are triple tuned and may have additional modes tuned or partially tuned to create an even more harmonious sound from the bars.

The tuning level adopted in fabricating a marimba bar will depend on the goals of the builder. For a simple instrument, all that is needed is a tuning of the fundamental mode. On the other end of the spectrum, a concert marimba will at least be triple tuned and further tuning refinements will yield the best possible sound. An understanding of several modes of vibration is essential for the builder who aspires to create a high quality instrument. Careful study of the 12 modes discussed below is recommended for those who are interested in tuning bars to a high level.
However, you may wish to restrict your reading only to the first three transverse modes and then skip down to the Tuning the Marimba Bar section

This section explores some of the more important modes of vibration in the marimba bar. The first 12 modes of the C2 bar for the La Favre marimba are presented below. Several methods were used to discover and measure the frequencies of vibration and to determine the mode responsible for each frequency. The marimba bar was struck at various locations on the top surface as well as the bar edge. The audio resulting from mallet blows was recorded digitally and the frequencies determined by Fast Fourier Transformation (FFT). Prominent frequencies found by FFT were confirmed by strobe tuner. Modes of vibration were identified by a "salt" method as follows: 1) bar was supported by foam blocks at the nodes of the fundamental mode, 2) salt was sprinkled on the top surface of the bar, 3) bar was subjected to sound at the frequencies determined by the FFT analyses. The sound source was a speaker connected to a tone generator. When exposed to sound at a frequency that matches a mode of the bar, the bar will begin to vibrate in that mode. The bar vibration causes the salt to accumulate at the places on the bar where vibration is minimal (the nodes). The pattern of salt on the bar after exposure to a specific frequency of sound is diagnostic for the mode. Check this page for more information on the salt method.

 

First Transverse Mode

The first mode (mode 1) of the bar is the mode with the lowest vibration rate (cycles of vibration per second or Hz). Mode 1 is a transverse out-of-plane type of vibration or simply, transverse. Since it is also the lowest vibration rate for a transverse mode, it is named the first transverse mode. Transverse modes of vibration are the type where the bar vibrates up and down along its length. As an aid in visualizing the vibration, look at the illustrations and photo below. The first illustration is a diagram of the bar, looking at the bar edge along its length. The bar is represented by a single line, with the curved lines representing the extreme positions in the vibration cycle. The photo shows the location of the nodes on the top surface of the C2 bar where salt accumulated after striking the bar repeatedly in the center with a mallet. The second illustration is animated with the same viewpoint as the first illustration. Movement of the bar is exaggerated in both illustrations. The bar moves up and down to the greatest degree in the center of its length. This location on the bar is called the antinode. There are two positions along the length of the bar where there is no up and down motion and these are called the nodes. The first transverse mode gives rise to the fundamental of the bar. The first transverse mode of vibration is always tuned in a marimba bar, regardless of its location on the keyboard. The La Favre marimba was tuned to a standard of A4 = 442 Hz and in this standard, the C2 bar is tuned to 65.70 Hz.

First Transverse Mode (Fundamental)

 

(illustration above derived from Bork, 1995 for a C3 bar - nodes do not line up with bar photo below because the illustration was drawn for a different bar - the exact location of nodes varies with bar dimensions and other factors)

C2 bar La Favre marimba - salt sprinkled on bar and struck with mallet in the center of the bar - salt accumulates at the nodes

Striking the bar in the center results in maximum excitation of mode 1 vibration because this is the location of maximum deflection. Conversely, striking the bar at one of the nodes results in very little excitation of mode 1 because there is little movement of the bar in this zone for mode 1. Therefore, the bar can be supported at the nodes with little damping effect on mode 1 vibration (glockenspiel, xylophone, vibe and marimba bars are supported at the nodes of mode 1).

 

First Torsional Mode

The next mode (mode 2) of vibration in the C2 bar was found at 102.2 Hz and identified as the first torsional mode of vibration. Yoo et al. (2003) studied the transverse and torsional modes of vibration in two commercial marimbas. They compared the higher modes of vibration to the fundamental by calculating a ratio (higher mode Hz / fundamental mode Hz). For the La Favre C2 bar, the ratio for the first torsional mode is 1.56. In other words, this mode vibrates at a rate 1.56 times greater than the fundamental. For a Malletech C2 bar, Yoo et al. (2003) measured the first torsional mode with a ratio of approximately 1.9 (judging from their graph and information in the text).

A torsional mode of vibration has a twisting type of motion. If you were to hold both ends of the bar and twist it along its length, the bar would move in a pattern of the first torsional mode. In the first diagram below the torsional motion is depicted with colored arrows. The red arrows depict the movement of the bar during half the vibration cycle and the blue arrows depict the movement during the other half. As the left front corner of the bar moves down (red arrow), the right front corner moves up. At the same time the left rear corner is moving up while the right rear corner is moving down. This mode has two nodes, represented by dashed lines. One node runs down the center of the bar length and the other perpendicular to the bar length at the center. There is no movement at the node lines during vibration. The second illustration below is an animation of the bar vibration, again exaggerated to help you visualize the movement.

The first torsional mode of vibration is usually not tuned. The first torsional mode is not excited into vibration to any great degree unless the bar is struck near a corner, where the antinodes are located. Here is what Bork et al. (1999) had to say about this mode of vibration in their C3 bar "This mode radiates weakly, since adjacent regions of opposite phase cancel the sound radiation at long wavelengths (here 1.06 m)." Therefore, this mode may not cause much of a problem if left untuned, because the sound waves radiating from adjacent corners of the bar tend to cancel each other.

CLICK HERE for more about tuning the first torsional mode (and the lack of a need to tune this mode)

First Torsional Mode

C2 bar La Favre marimba - salt sprinkled on bar and struck with mallet at corner while partially supporting the bar at the center to suppress the first transverse mode - salt accumulates along the node running the length of the bar.

Second Transverse Mode

The next mode (mode 3) of vibration in the C2 bar is the second transverse mode. The first, second and third transverse modes are the modes that are usually tuned in a marimba because they are key components contributing to the timbre of the bar. The second transverse mode vibrates in a pattern similar to the first transverse mode, but the second transverse mode has three nodes compared to two for the first mode. A node at the center of the bar is an important feature of the second transverse mode. If the bar is struck exactly in the center at this node, there will be very little excitation of the mode.

The second transverse mode of the C2 bar is tuned to 262.8 Hz. Recall that the first transverse mode of the C2 bar is tuned to 65.70 Hz. Dividing 262.8 Hz by 65.70 Hz, we obtain a value of 4.00. The calculation indicates that the tuned second transverse mode vibrates at a frequency 4 times greater than the first transverse mode, or two octaves above the fundamental.

Second Transverse Mode

 

(illustration above derived from Bork, 1995 for a C3 bar - nodes do not line up exactly with bar photo below because the illustration was drawn for a different bar - the exact location of nodes varies with bar dimensions and other factors)

C2 bar La Favre marimba exposed to sound from tone generator at 262.8 Hz - salt sprinkled on bar accumulates at the nodes - look carefully in the center of the bar where a small amount of salt has accumulated, marking the central node.

 

 

Second Torsional Mode

 

The next mode (mode 4) of vibration in the C2 bar was found at 595 Hz and identified as the second torsional mode of vibration. The ratio compared to the fundamental is 9.06. For a Malletech C2 bar, Yoo et al. (2003) measured the second torsional mode with a ratio of approximately 8.5 (judging from their graph).

Unlike the first torsional mode, this mode has antinodes at the center of the bar, located at each bar edge ( red and blue arrows located at the center of the bar in the first diagram below). This mode of vibration can be excited during normal playing if the mallet hits the bar more toward the edge in the center of the length. Here is what Bork et al. (1999) had to say about this mode in their C3 bar "This mode radiates better than the lower (1-1)-mode because its wavelength of sound is shorter. When the bar is hit vertically on the top, close to the edges, the (2,1) mode .... has almost equal amplitude as the (4.0) mode in the radiated spectrum." For clarification here, (1-1) = first torsional mode, (2,1) = second torsional mode, (4.0) = third transverse mode. Therefore, the second torsional mode may be more problematic in tuning due to its strong radiation of sound.

CLICK HERE for more information on the second torsional mode.

Second Torsional Mode

 

 

C2 bar La Favre marimba exposed to sound from tone generator at 595 Hz - salt sprinkled on bar clears from antinodes and accumulates in center

Third Transverse Mode

The next mode (mode 5) of vibration in the C2 bar is the third transverse mode. It was tuned to 662.2 Hz. This type of vibration is similar to the first and second transverse modes, but in this case there is four nodes. Importantly, an antinode occurs at the center of the bar. In contrast to the second transverse mode, this mode is excited when the bar is struck in the center, just as the first transverse mode. This mode is usually tuned as well, but only in the lower range of the keyboard. The third transverse mode is usually tuned to three octaves and a major third above the fundamental (i.e., a frequency of 10.08 times greater than the frequency of the first transverse mode).

Third Transverse Mode

(illustration above derived from Bork, 1995 for a C3 bar - nodes do not line up with bar photo below because the illustration was drawn for a different bar - the exact location of nodes varies with bar dimensions and other factors)

 

C2 bar La Favre marimba exposed to sound from tone generator at 662.2 Hz - salt sprinkled on bar accumulates at the nodes

 

First Lateral Mode

The next mode (mode 6) of vibration in the C2 bar was found at 786 Hz and identified it as the first lateral mode of vibration (also known as the first transverse in-plane mode). The ratio compared to the fundamental is 12.0. The lateral modes of vibration are transverse in type, but the movement is not up and down. The vibratory movement is left and right (from the player's perspective). In order to excite this mode of vibration to any great degree, it would seem that the bar must be struck on its edge, at the center of the length. However, the first lateral mode is known to contain a vertical element, at least in some bars. Therefore, a normal vertical strike on the bar can activate this mode, as was found for the La Favre C2 bar. Professional tuners are known to apply tuning methods such as "wedging" to this mode when it causes a conflict with a tuned transverse mode (Weiss, 2003). Bork et al. (1999) note that "A weak vertical component can be observed, probably due to the slightly asymmetrical scoop" in the C3 bar they studied. In other words, the bar they investigated did not have a perfectly symmetrical undercut arch, which they believe gives rise to some up and down movement in the bar for the lateral mode. Even when striking the bar with a normal vertical stroke then, the first lateral mode might be excited to some degree.

The first illustration below indicates the direction of vibration with red and blue arrows. The two nodes are marked with black circles. The nodes of the lateral modes are unique in that they run vertically through the bar (i.e. in the z direction). All other modes listed on this page have nodes that run horizontally (i.e., in the x and y directions).

CLICK HERE for more information on the first lateral mode.

First Lateral Mode

C2 bar La Favre marimba exposed to sound from tone generator at 786 Hz - salt sprinkled on bar is pushed off edges in the center

 

 

 

Third Torsional Mode

The next mode (mode 7) of vibration in the C2 bar was found at 1203 Hz and identified as the third torsional mode of vibration. The ratio compared to the fundamental is 18.3. For a Malletech C2 bar, Yoo et al. (2003) measured the third torsional mode with a ratio of approximately 16.5 (judging from their graph).

Third Torsional Mode

C2 bar La Favre marimba exposed to sound from tone generator at 1203 Hz - salt sprinkled on bar clears from antinodes and accumulates in center

Fourth Transverse Mode

Mode 8 was measured at 1287 Hz and found to be the fourth transverse mode. The ratio compared to the fundamental is 19.6. This mode is similar to the third transverse mode, except that it has five nodes instead of four. Also note that this mode has a node at the center of the bar, like the second transverse mode. Therefore, this mode will not be excited to any great degree when striking the bar in the center. I believe that some professional tuners also tune this mode in the lower bass bars and probably to a ratio of 20 times the fundamental frequency.

Fourth Transverse Mode

(illustration above derived from Bork, 1995 for a C3 bar - nodes do not line up with bar photo below because the illustration was drawn for a different bar - the exact location of nodes varies with bar dimensions and other factors)

C2 bar La Favre marimba exposed to sound from tone generator at 1287 Hz - salt sprinkled on bar accumulates at the nodes

Fourth Torsional Mode

The next mode (mode 9) of vibration in the C2 bar was found at 1585 Hz and was identified as the fourth torsional mode of vibration. The ratio compared to the fundamental is 24.1. For a Malletech C2 bar, Yoo et al. (2003) measured the fourth torsional mode with a ratio of approximately 25 (judging from their graph). The fourth torsional mode is unique compared to all other torsional modes presented here. It has two node lines running the length of the bar instead of one.

 

Fourth Torsional Mode

C2 bar La Favre marimba exposed to sound from tone generator at 1585 Hz - salt sprinkled on bar accumulates at the nodes

Second Lateral Mode

The next mode (mode 10) of vibration in the C2 bar was found at 1680 Hz and was assigned to the second lateral mode of vibration. The ratio compared to the fundamental is 25.6. Both the first and second lateral modes commonly include some vertical vibration in addition to the lateral element. That is the reason why they are activated to a certain degree when the mallet strikes specific locations on the top surface of the bar. Bork et al. (1999) found a significant vertical element of vibration in the second lateral mode of their C3 bar. I do not have the sophisticated instrumentation required to evaluate simultaneous vibrations in the horizontal and vertical planes. The salt method I employed is not clearly diagnostic for this mode in my hands. Therefore, the illustration I provide below includes the vertical element as detailed by Bork et al. (1999). The nodes of the lateral element are represented by black circles. The nodes of the vertical element are represented by dashed lines and are similar to the third torsional mode.

The salt pattern obtained on the bar after exposure to 1680 Hz is more difficult to understand than any other mode studied for the C2 bar. This is most likely due to the presence of significant vertical and lateral elements of vibration for this mode. Without a more detained evaluation of the interaction of vertical and lateral elements, obtained with sophisticated scientific instruments, I am not in a position to explain the salt pattern observed.

Second Lateral Mode

C2 bar La Favre marimba exposed to sound from tone generator at 1680 Hz - salt sprinkled on bar accumulates along the bar edges in the center.

Fifth Torsional Mode

The next mode (mode 11) of vibration in the C2 bar was found at 1957 Hz and identified as the fifth torsional mode of vibration. The ratio compared to the fundamental is 29.8. For a Malletech C2 bar, Yoo et al. (2003) measured the fifth torsional mode with a ratio of approximately 34 (judging from their graph). This mode is similar to the third torsional mode, except it has four node lines across the bar width instead of three.

Fifth Torsional Mode

C2 bar La Favre marimba exposed to sound from tone generator at 1957 Hz - salt sprinkled on bar clears from antinodes and accumulates at center node and at nodes across the bar width

Fifth Transverse Mode

Mode 12 was measured at 2097 Hz and found to be the fifth transverse mode. The ratio compared to the fundamental is 31.9. This mode is similar to the fourth transverse mode, except that it has six nodes instead of five. Also note that this mode has a antinode at the center of the bar, like the third transverse mode. Therefore, this mode will be excited when striking the bar in the center. As far as I know, this mode is not tuned in any commercially available marimbas.

Fifth Transverse Mode

C2 bar La Favre marimba exposed to sound from tone generator at 2097 Hz - salt sprinkled on bar accumulates at the nodes

 

 

 

The graphs below plot the relative rates of vibration (ratios) for modes of the La Favre marimba. An analysis of this type is useful for the marimba builder who wishes to refine the tuning of the bars. The untuned torsional and lateral modes can interfere with the harmonious sounds of the tuned transverse modes if the ratios have similar values. For example, in the La Favre marimba the first lateral mode has ratio values very close to the third transverse mode in the bottom two octaves. Therefore, the first lateral mode may degrade the tuning of the third transverse mode. Further investigation is needed in order to evaluate the level of degradation. If the lateral mode vibrates at an intensity much less than the third transverse mode, then there is no problem. The relative intensity for both modes must be measured for several locations that are reasonable spots that a player would hit during a performance. The graphs are a starting point for investigating problem torsional and lateral modes. My investigation in this area is ongoing. You can see some of my latest findings on my page covering torsional and lateral modes.

graph

 

 

Marimba bars may vary in the order of certain modes depending on a number of factors, particularly the shape of the undercut arch and general bar dimensions (length, width, thickness). The ratio values for the torsional modes found for the La Favre C2 bar are similar to those found by Yoo et al. (2003) for a Malletech C2 bar. In addition, the order of the first four transverse and first four torsional modes are the same for the La Favre C2 bar and the Malletech C2 bar (bottom two octaves). Yoo et al. (2003) did not study the lateral modes in their bars, which prevents a comparison for the lateral modes. The mode order for a C3 bar studied by Bork et al. (1999) was not the same as the La Favre bar. Bork et al. (1999) found the following order of modes: first transverse, first torsional, second transverse, first lateral, third transverse, second torsional, second lateral, fourth transverse, third torsional, fifth transverse, fourth torsional. Nevertheless, the modes discussed above are those that the tuner should be aware of if bars are to be tuned to a high standard. The tuner needs to be aware that the order of modes in their bar may be different than the order for the La Favre C2 bar.

It should be noted here that the bar vibrates in more than one mode at a time and in fact usually vibrates in several modes when struck. The modes active in vibration and their relative strength depend on where the bar is struck and the kind of mallet used. For bass bars, soft mallets that have a relatively high mass will excite the lower modes of vibration more than hard mallets of lower mass. Thus, soft heavy mallets are commonly used at the bass end of the instrument while hard mallets are used at the treble end. Furthermore, the lower modes of vibration will be active for longer periods of time after the bar is struck than the higher modes. This can be easily heard in a bass bar struck with a medium mallet. The higher modes (second and third transverse) are easily heard immediately after the bar is struck, but in a fraction of a second they die out and the fundamental alone is obvious. Bork (1995) states the following: "When the frequencies of the partials of a bar behave like 1:4:10, the second partial decays four times as quickly, the third partial ten times as quickly as the fundamental." In other words, in a bar tuned like a marimba bar (1:4:10), the fundamental rings four times as long as the second transverse mode and 10 times longer than the third transverse mode.

Tuning the Marimba Bar

I provide this information primarily for those who are interested in building their own marimba and those who want to understand how a bar is tuned. If you are thinking about retuning an existing marimba, please be advised that it is easy to permanently ruin the tuning of a bar, which then must be replaced. If you want to retune an out of tune marimba, I would suggest that you fabricate several practice bars first. If the possibility of ruining a bar is not an acceptable outcome, it would be best to send the bars to a professional tuner. In any case, you must be willing to spend a considerable amount of time learning the tuning process. It is not something that can be done quickly.

I have tried to provide enough information to enable you to tune a bar to a relatively high degree. The more modes you attempt to tune, the more difficult the tuning will be. If you want to build a marimba for serious music, then I would suggest you consider tuning at least the fundamental and second transverse mode. For more complete tuning, tune the third transverse mode and perhaps even the fourth transverse mode. To achieve the highest standard, you may need to partially tune some problem torsional or lateral modes. I did not tune any torsional, lateral or fourth transverse modes on my marimba at the time of bar fabrication. Currently (January, 2007), I am examining the torsional and lateral modes in my bars and have started some retuning work ( second torsional and first lateral modes   information on first torsional mode)

In order to achieve an accurate tuning, you probably need to use a strobe tuner (a real strobe, not an instrument that attempts to mimic a strobe). Unfortunately, strobe tuners are expensive. If you don't have access to a strobe tuner, you might try using a less expensive electronic tuner (the type with a needle and scale and/or LEDs that indicate "in tune") but you may find it difficult, if not impossible, to tune overtones with these tuners. Before the development of strobe tuners in 1942, professional tuners employed sets of tuned bars as standards, using their ears for tone comparisons. Even today professional tuners rely on aural technique for certain aspects of tuning. However, tone matching requires a keen ear. If you are gifted with sensitivity in tone perception, then you may be able to use your ear for tuning. Since I don't have this skill, I need to rely on a strobe tuner.

  The information in the previous section can be used to formulate a tuning strategy for the marimba bars. The transverse modes of the marimba bars are harmonically tuned (that is, the transverse mode overtones are whole multiples of the fundamental frequency). The first transverse mode of the marimba bar is tuned to the fundamental, the second transverse mode is tuned to the fourth harmonic (two octaves above the fundamental) and the third transverse mode is tuned to the tenth harmonic (three octaves and a major third above the fundamental). I triple tuned the bass register on my marimba, C2 to G#3 (first, second and third transverse modes tuned to fundamental, 4th harmonic and 10th harmonic respectively), which imparts a more consonant complex tone. Bars in the middle register (A3 to C5) were double tuned (first and second transverse modes tuned to fundamental and 4th harmonic respectively). Overtones of the higher register (C#5 to C7) reside above the range of the instrument and are sustained for very brief time periods or may even be inaudible. For these reasons, I did not tune the overtones in the higher register.

Tuning of overtones becomes more difficult in the higher registers due to the very brief sustain. The third transverse mode has the shortest sustain time and becomes difficult to tune above G#3. Higher up, the second transverse mode also becomes very difficult to tune. With my strobe tuner, I found it difficult to get a fix on the frequencies for overtones above the limits stated above. A more skilled tuner might push the tuning to higher points on the keyboard. Nevertheless, there comes a point in the keyboard where the bars no longer vibrate in a specific overtone, which limits the tuning of an overtone mode to bars below that point.

A simple rectangular bar does not vibrate with harmonic overtones. In order to tune the bar to harmonic overtones, it is necessary to cut an arch from the bottom. The removal of wood changes two properties of the bar important in the tuning process: 1) flexural strength and 2) mass. Removing wood from the bar results in a reduction in flexural strength and a reduction in mass. However, flexural strength is proportional to the cube of the bar thickness and mass is only proportional to the thickness (Bork, 1995). A reduction in flexural strength results in a lower vibration rate while a reduction in mass results in an increase in vibration rate. These are opposite effects that occur when wood is removed from the bar. However, since the flexural strength is related to the cube of the bar thickness, this property is more influential than mass in areas of the bar where flexing occurs. For example, when wood is removed from the center of the bar, the vibration rate is reduced. Why? Because the flexural strength is reduced more than the mass. Since the bar flexes to a great degree in its central region, the reduction in flexural strength dominates over the loss of mass. In contrast, at the very ends of the bar there is no flexing in the transverse modes. Therefore, a drop in flexural strength at the bar ends does not effect a reduction in the rate of vibration. When wood is removed from the ends of the bar, the mass is the only important property for tuning. Since a reduction in mass results in an increase in the rate of vibration, removal of wood at the bar ends will raise the rate of vibration. A good understanding of the effects of wood removal at various locations on the bar is the key to tuning.

For a person who is learning the process, begin triple tuning by cutting a very conservative arch, which results in a bar several semitones above the target note. Then small amounts are removed from specific areas of the arch to achieve the harmonic intervals (the fourth and tenth harmonics, i.e. 4 and 10 times the fundamental frequency).

The fundamental is lowered more than the overtones when material is removed from the center of the arch (area 1 in illustration below). The second transverse mode is lowered most when material is removed somewhere between the center and end of the arch (area 2). The third transverse mode is lowered most when material is removed near the end of the arch (area 3). When all three frequencies are at the desired harmonic intervals, then material is removed evenly across the arch to approach the target note.

This tuning method should seem logical if you recall the modes of vibration for the bar. The first transverse mode of vibration (the fundamental) requires maximum flexing of the bar in the center (its antinode). By removing material in the center, the bar becomes more flexible (less stiff), which results in a slower rate of vibration. The antinodes for the second transverse mode of vibration (tuned to the fourth harmonic) are on both sides of the bar center, so removal of material there results in a more flexible bar in the locations where this mode requires maximum flexing. The third transverse mode of vibration (tuned to the tenth harmonic) has antinodes at the bar center and near each of the zones labeled 3 in the illustration above. Therefore, removing material from the areas labeled 3 results in more flexibility for two of the three antinodes of this mode.

For more information on the locations where wood should be removed to tune the transverse modes, CLICK HERE.

Cutting the arch

The amount of wood removed from the bottom of the bar varies, with more removed toward the bass end of the instrument (see photo of bars farther below). I selected the C4 bar as a starting point. An arch was penciled in along the bar edge, centered along the length, starting 1/4 of the length from each end and running along the center of the bar thickness in the middle of the bar length. For the first few bars I used a band saw to cut the rough arch. Unfortunately, I don't have a very good band saw, which made it difficult to cut a proper arch. So I developed a method using the table saw, which I will describe later. Once the rough arch is cut, the strobe tuner is used to determine the frequencies of the fundamental and other transverse modes. The modes tuned depends on the bar, for C4 only the first and second transverse modes were tuned - see below for measurement procedure. You must proceed carefully to tune the modes to the required harmonic interval(s). For double tuned bars, first tune the second transverse mode to two octaves above the fundamental, by removing more wood nearer the ends of the arch. When the proper two octave interval is obtained, then remove wood evenly across the arch to drop both fundamental and second transverse mode. For bass bars, the third transverse mode may have a frequency greater than ten times the fundamental. If so, then remove material near both ends of the arch (area 3), and be careful to remove equal amounts on both ends (my table saw method insures this). You may also need to extend the length of the arch as well. The key is to remove small amounts of wood and make frequency measurements after each removal. After the third transverse mode equals a frequency of ten times the fundamental (three octaves plus a major third), work on bringing the second transverse mode to an interval of two octaves above the fundamental. Then remove material evenly along the arch to drop the fundamental and other modes to the required frequencies. As you do this, you will probably need to make minor adjustments by removing more material from specific zones to maintain the proper harmonic spacing of the fundamental and other modes.

Continue the rough tuning until you are about 50 cents above the target note (then the holes are drilled for the cord and finish is applied to the bar and, then the final tuning is done). You must have patience. I spent up to 4 hours per bar doing the rough tuning on the first bass bars. As you do more tuning the process speeds up. In the end I could triple tune bass bars in 1.5 to 2 hours. I should note here that my work was done in the winter and my belt sander was in the garage while the tuner was in the house. Traveling back and forth consumed a lot of time. A professional can do the tuning in a matter of minutes and they usually have the tuner near the belt sander. Nevertheless, when you are learning, the process takes considerably longer. You might try using some scrap wood like pine to try tuning a bar or two before doing the actual bars for the instrument. This is advice I have read elsewhere, but I was too anxious to start the real tuning and skipped this step. I was lucky in that I only ruined one bar by tuning it too low (I had to reject a few other bars as well since they did not sound good after tuning). Again, the key is to work very slowly, especially on the first few bars you tune. After you get a feel for the tuning, you can become more bold and remove more material with each tuning cycle.

After I had rough tuned the C4 bar, I used it as a reference for the arch shape. I worked my way to the top of the instrument, using the tuned bar as reference for the next higher note. Then I worked my way down the bass region, starting with B3. I simply transferred the shape of the arch from a rough tuned bar to the adjacent uncut bar. But when I made the first cut, I did not cut all the way down to the line, I left about 1/16 to 1/8 inch extra wood. Remember, if you cut off too much, you will have to discard the bar and start over.

I used a table saw and belt sander (6 x 48 inches) to remove wood from the underside of the bar. After drawing the arch on the bar edge, I used the table saw to cut the rough arch. The process is slow, but insures a very accurate arch symmetrically (I know this method is not used in a production shop, where a band saw is usually used, but then they have better band saws than the one I own). Measure the length of the bar and divide by two. Set the table saw fence to this number. Now raise the saw blade to a height that is 1/16 to 1/8 inch less than the full depth needed to reach the arch line in the bar center. Position the bar against the miter gauge and butt the end of the bar against the table saw fence. Then run the bar through the blade to make a cut. Then retract the fence the thickness of the saw blade and make another cut (you will need to make two cuts for each fence setting, one with each end of the bar against the fence). Continue making cuts, and drop the height of the blade as you go to maintain the cuts 1/6 to 1/8 inch below the penciled arch line. Yes, this is a very slow way to cut an arch, but you can't beat it for symmetric accuracy and for uniform depth of cut across the width of the bar. After you cut the arch on the table saw, smooth the arch with the belt sander. Then take measurements with the strobe tuner. If a great deal of wood remains to be removed from the arch in specific areas, then use the table saw. Use the belt sander for removing smaller amounts. I alternated between using the table saw and sander as I worked the arch down. You don't want to use the belt sander to an excessive degree because this hand method can yield a non-symmetrical arch. You need to be very careful with the belt sander as it removes wood quickly. Sand just a small amount and then make measurements with the strobe tuner. After a while you will have a better feel for the amount of sanding required to drop the frequency a certain amount.

After you have drilled the holes and applied the finish to the bars, you are ready for the final tuning. Bars vibrate at different frequencies at different temperatures. Therefore, it is best to do the final tuning in a temperature controlled environment. If you don't have a temperature controlled environment, restrict your final tuning to parts of the day when the temperature is similar. In my case I did the final tuning at a time of year when my house was being heated. For final tuning the temperature was always between 71 and 75 degrees F and the very final tuning of each bar was done at 73 degrees F. Keep in mind that when you sand the arch, the bar will heat up. So you must wait for the bar to return to room temperature before you make a determination of the final tuning. You should do the final tuning at a temperature similar to that which you anticipate for a typical performance with the instrument.

Fixing the tuning of a bar that is below the desired note

If by accident you tune a bar slightly below the target frequency, you can raise the frequency to a limited extent by removing wood from the ends of the bar. Usually this is done by reducing the thickness of the bar between the nodes and the bar ends. Removing wood close to the bar end has the greatest effect. Therefore, you can try chamfering the end of the bar at the bottom to raise the tuning.

 

Some Helpful Information from the Literature

Bork et al. (1999) used modeling to determine just how sensitive the bar is to removal of wood in the arch area. They calculated that a removal of only 0.5 mm from the thickness of the bar in the arch area would drop the frequency of the fundamental by about 8% (more than one semitone)! You need to think about this for a minute. A layer of wood 0.5 mm thick is not much at all, yet a removal of this small amount of wood drops the fundamental more than a semitone. This is why I state that you need to work very slowly when you are learning how to tune a bar. You need to appreciate how much the tuning changes with a small amount of wood removal.

 

Tuning Problem Lateral and Torsional Modes

Lateral and torsional modes can be tuned by cutting notches on the underside of the bar, at the bar edge in the center of the length (MacCallum, 1969). The arch in the center region of the bar can also be shaped so that the bar thickness is greater in the center than the edge (drawing below), sometimes called wedging or having a wedge cross section (Weiss, 2003). Properly placed grooves in the arch area can also help in tuning a problem lateral or torsional mode (Weiss, 2003). Keep in mind that the wedging technique will also affect the tuning of the transverse modes. If you are trying to wedge a bar in a retuning process, you may need to raise the tuning of the transverse modes first by removing wood from the ends of the bar (as described earlier on this page). Experienced marimba builders may know which bars in the keyboard need to be wedged in their design and incorporate the wedge where needed during the initial shaping of the arch. This is a much better approach than trying to fix the problem after the transverse modes are tuned.

  I have started to investigate lateral and torsional modes and how to tune them. Here is what I have so far.

CLICK HERE for more about tuning the first torsional mode (and the lack of a need to tune this mode)

Using the Strobe Tuner to Measure Frequencies of the Modes

The frequencies are measured with a strobe tuner. The different modes are emphasized depending on where the bar is grasped and where it is struck with the mallet. A mode is brought forth by holding the bar at a position that coincides with a node (a location where the bar does not vibrate in that specific mode). A mode is damped if the bar is held at a position that coincides with its antinode (the location of maximum vibration for the mode).

The Fundamental (First Transverse Mode)

To bring forth the fundamental, the bar is held at the node of the first transverse mode and struck at the center of the bar.

a

 

 

The Second Transverse Mode

To bring forth the second transverse mode, the bar is grasped at its center and struck as illustrated below. Trial and error with the mallet will allow you to locate the position where the second transverse mode will sound loudest when struck. Slightly adjust the position where you grasp the bar until the bar rings clearly when struck. The fourth transverse mode can be measured as well with this technique because it also has a node at the center of the bar.

a

 

The Third Transverse Mode

To bring forth the third transverse mode, the bar is grasped near one end and struck in its center. Note that the outer nodes for the third transverse mode are closer to the end of the bar than the nodes for the first transverse mode. Therefore, holding the bar near the end does not damp the third transverse mode very much but does damp the fundamental significantly. The second transverse mode is not excited when the bar is struck in the center because it has a node there. These are the reasons why the third transverse mode is excited with this method and the other two transverse modes are subdued . Some trial and error may be needed to find the proper place to grasp the bar.

a

In addition to a little trial and error in discovering the exact place to grasp a bar and strike it, you may need to try using different mallets to strongly excite the various modes.

 

Drilling the Holes for the Bar - Finding the Nodes of the Fundamental Mode

After you have rough tuned the bar to 50 cents above the target note, it is time to drill the holes for the cord. Place two blocks of foam under the approximate location of the nodes as illustrated below.

Sprinkle some salt on the bar in the area of the nodes as illustrated below and then tap very lightly with a mallet at the center of the bar.

Keep tapping until the salt accumulates over each node as illustrated below. Mark the node lines lightly with a pencil on the top surface of the bar. In the photo below you will note that the salt has not accumulated perpendicular to the bar length, but at an angle. In a set of bars, some will have nodes that are perpendicular to the bar length and some may be angled. This is due to the non-uniform nature of wood.

Mark the center of each node line with a cross tick (at the center of the bar width - see illustration below). Before you drill the holes, you should layout all of the bars with the proper spacing between each bar, as they will be on the instrument. The hole in the bar that is located near the center of the instrument width (inboard) is drilled perpendicular to the bar length. The hole near the end at the outside edge (outboard) of the keyboard must be drilled at an angle because the adjacent bars are not the same length. First line up all the node lines for the bars on the outboard side with a straight edge as illustrated below (i.e., move the bars inboard or outboard so that all outboard nodes are in a straight line). Then lay the straight edge over the node lines on the inboard side. Try to align the straight edge to minimize the distances from the true nodes to the straight edge. It may not be possible to pass a straight line through all of these nodes, in which case you must compromise a little. The object is to align each bar to minimize the distance between where the holes must be drilled and the true nodes. Move individual bars slightly toward the center or outside of the instrument to minimize the difference between the locations where the bar must be drilled and the true nodes. Then draw two lines across the bars with a straight edge as illustrated below. These lines mark the locations where the bars must be drilled. Measure the angle of the outboard line and use the angle measurement when drilling the holes on the outboard side.

Layout of bars for marking hole locations

Red lines represent the true nodes for each bar determined by the salt method described above. The top of the illustration represents the inboard side of the keyboard. The inboard line drawn to mark the locations for the inboard holes is a compromise that comes as close as possible to each true node while maintaining a straight line perpendicular to the bar lengths.

 

Look at this page for the jig I used to drill the holes.

 

Tuning Analysis of Two Commercial Marimbas

Yoo et al.(2003) examined the first four transverse modes of vibration and the first five torsional modes of vibration in a Malletech and a Yamaha five-octave instrument. For the Malletech instrument, the second transverse mode was tuned to the fourth harmonic for bars ranging from C2 to D5, after which the ratio (second / first transverse mode) slowly decreased to a minimum of about 2.5, and the third transverse mode was tuned to the tenth harmonic for bars ranging from C2 to C4, after which the ratio decreased to less than 5 in the highest bars. For the Yamaha instrument, the second transverse mode was tuned to the fourth harmonic for bars ranging from C2 to F5, after which the ratio slowly decreased to a minimum of about 2.6, and the third transverse mode was tuned to the tenth harmonic for bars ranging from C2 to C#4, after which the ratio decreased to less than 7 in the highest bars.

Yoo et al.(2003) found that the first torsional mode frequency ranged from about 1.9 times the fundamental frequency (lowest notes) to about 1.2 times the fundamental frequency (highest notes). Furthermore, the second torsional mode ranged between 9.4 times the fundamental frequency (F3 bar) to 3.9 times the fundamental frequency (C7 bar). The authors have this to say about the importance of torsional modes: "The large bars on these marimbas are quite wide, and thus the torsional bars (I believe they mean torsional modes here) can radiate an appreciable amount of sound. In normal playing, the bars are struck near their centers, where the torsional modes have nodes, and thus they will not be excited to any great extent. On the other hand, if the bars are struck away from the center, deliberately or not, the torsional modes could contribute to the timbre.

Bork and Meyer (1985) investigated the tuning of the third transverse mode. In the study, they compared tunings of the following six intervals above the third octave: major second, major second +50 cents, minor third, minor third +50 cents, major third, and major third +50 cents. Thirty individuals with training in music were used to judge a preference for the different tunings. In the composite listening results a preference was identified for a minor third + 50 cents (i.e. a ratio of 9.88 to the fundamental frequency). The second most preferred tuning was a major third +50 cents (ratio of 10.38), which yields a brighter timbre according to the authors. In summary, Bork and Meyer (1985) state that the tuning of the first and second transverse modes is critical for pitch perception and that the second transverse mode needs to be within plus or minus 15 cents of the double octave for proper pitch perception. Furthermore, "The third partial (third transverse mode) can influence the pitch slightly, but this partial mainly contributes to the timbre or tone quality." Apparently the manufacturers of the Malletech and Yamaha instruments prefer a brighter tuning for the third transverse mode. For the Malletech, the tuning was about a major third plus 17 cents and the Yamaha was about a major third plus 34 cents (frequency ratios of 10.18 and 10.28, respectively). But we should be cautious about drawing too much from the analysis of just one instrument from each builder. In fact, some builders offer voicing options for marimbas, which may include different tunings of the third transverse mode depending on the desire for a darker or brighter sound.

 

Wave forms of complex tones (containing more than one frequency)

As discussed earlier, the vibrating bar vibrates in more than one mode at a time. As an example, let us examine the C3 bar, which when tuned at a standard of A4 = 440 Hz, has a fundamental at 130.81 Hz, a fourth harmonic at 523.25 Hz and a tenth harmonic at 1318.5 Hz. First, listen to each of these frequencies in isolation (click the links below to hear each frequency - these tones were generated with computer software):

130.81 Hz

523.25 Hz

1318.5 Hz

Now listen to the combination of all three frequencies (simulated C3 marimba bar):

130.81+523.25+1318.5 Hz

Below you will find the wave forms of each of the three frequencies (these are copies of the wave forms from the computer software used to generate the tones). In generating these tones, I have attempted to mimic the type of wave forms that the marimba makes, but keep in mind that this is synthetically produced sound. The length of the wave form (the x axis) is a representation of the duration of the sound. The height of the wave form (amplitude) is an indication of the strength (energy) of the sound (not necessarily the loudness since the human ear has a variable sensitivity to different frequencies). The fundamental wave form has the longest duration (I made the duration equal to 0.7 second). The fourth harmonic has a duration of 0.35 second and the tenth harmonic is 0.1 second (these durations may not match exactly those found in a real marimba bar). The vertical line at the left of the wave form represents time 0, when the bar is struck. Notice that the fundamental builds more slowly in strength than the overtones. This is due to the lower frequency and the lag time required to build resonance in the tube. You should also note that the fundamental has the highest amplitude, followed by the fourth harmonic and the tenth harmonic has the lowest amplitude.

130.81 Hz, the Fundamental

523.25 Hz, the Fourth Harmonic

1318.5 Hz, the Tenth Harmonic

Below you will find wave forms for 130 Hz and 1300 Hz. Note that there is one wavelength for the 130 Hz wave form and ten wavelengths for the 1300 Hz wave form. These are the wave forms for each tone in isolation. If we mix the wave forms together to synthesize a complex tone of the fundamental (130 Hz) and one overtone at the tenth harmonic (1300 Hz), then we obtain a wave form like the second illustration below. The more frequencies we add together, the more complex the wave form becomes. In this case I have kept the example simple so that it is easier to make out the two frequencies in the combined wave form.

separate wave forms for 130 Hz and 1300 Hz

wave form for a tone with fundamental and the tenth harmonic

The wave form immediately above depicts a complex tone. If we were able to visualize the air molecule movement represented by the wave form, we would notice a pattern of strong pressure waves passing a point of reference with a frequency of 130 per second. But each of those strong pressure waves would not be smooth. Rather, they would have ripples of lower pressure variation with a frequency of 1300 per second.

In order for a complex tone to sound harmonious, the higher frequencies (overtones) must be whole number multiples of the fundamental frequency. Below you can listen to a complex tone with two harmonic overtones plus another complex tone that contains overtones that are not harmonic. Listen to both and decide which one sounds better. Hopefully you will agree that the first link sounds better. This is the reason why the marimba bars are tuned with harmonic overtones.

130.81+523.25+1318.5 fundamental with two harmonic overtones

130.81+587.33+1479.9 fundamental with two overtones that are not harmonic

 

Effect of Position of Mallet Blow on Bar Timbre

PLEASE BE ADVISED: the page linked below contains 40 graphic images for a total of 2.34 MB, which can take as long as 15 minutes to download over a phone modem.

Analyses of sound spectra from bars struck at different positions

 

Some details about tuning and musical scales

Modern western music employs an equal tempered scale, which is a compromise developed to facilitate music played in different keys. An older system, known as Just Intonation, employs a scale whereby each step up the scale is a whole number ratio as follows: C 1/1, D 9/8, E 5/4, F 4/3, G 3/2, A 5/3, B 15/8, C 2/1. Just Intonation results in a collection of notes that are very consonant due to the whole number ratios employed. However, if one attempts to shift keys while playing music on an instrument tuned in this manner, the results are not acceptable.

Some musical instruments, marimba included, have fixed tunings. In order to play in different keys with Just Intonation, these instruments must be retuned (or in the case of a marimba, a different set of bars must be used!). To solve this problem, the equal temperament scale was developed. In the chromatic scale of 12 semitones, each step is the 12th root of 2, which preserves the doubling of frequency with each octave. The 12th root of 2 is 1.059463. Therefore, you can calculate the frequencies of notes in the scale by starting at a given point and simply multiplying by 1.059463 to obtain the frequency of the next semitone. For example, suppose we are tuning to a standard of A4 = 440 Hz. Then to calculate the frequency of A#4, calculate as follows: 1.059463 x 440 Hz = 466.16 Hz. To calculate B4, multiply 1.059463 x 466.16 Hz = 493.88 Hz, etc. If you continue calculations in this way, you will find that A5 calculates to 880 Hz (i.e., the frequency doubles on the octave). Below you will find a table that compares the frequencies of an octave in Just Intonation and Equal Temperament.

Note Just Intonation (Hz) Equal Temperament (Hz)
C 264 261.63
D 297 293.66
E 330 329.63
F 352 349.23
G 396 391.99
A 440 440.00
B 495 493.88
C 528 523.25

The reason I have added this section is to clarify a detail on tuning the tenth harmonic of the bar. In the case of the fourth harmonic, the frequency should be exactly four times the fundamental frequency since the interval is two octaves. However, the tenth harmonic is not an interval of octaves, and with equal temperament tuning, the frequency of the tenth harmonic will not be exactly 10 times the fundamental (with Just Intonation, the tuning would be exactly 10 times the fundamental). To clarify further, a table of tunings in equal temperament for the bottom octave of the marimba is provided.

Fundamental Fourth Harmonic Tenth Harmonic
C2 (65.70 Hz) C4 (262.81 Hz) E5 (662.25 Hz)
C#2 C#4 F5
D2 D4 F#5
D#2 D#4 G5
E2 E4 G#5
F2 F4 A5
F#2 F#4 A#5
G2 G4 B5
G#2 G#4 C6
A2 (110.5 Hz) A4 (442 Hz) C#6 (1113.77 Hz)
A#2 A#4 D6
B2 B4 D#6

Unfortunately, we are not finished yet with the details. Apparently, human hearing is a bit quirky. We tend to hear high notes a bit flat. To compensate for this, an instrument with a wide compass can be tuned with the high notes slightly sharp. For my marimba, I employed the following adjustments (stretch tuning): starting with C#6, one cent was added to the tuning for each semitone (i.e., C#6 + 1 cent, D6 + 2 cents, etc.). A cent is 1/100th of the interval between semitones (i.e., there are 1200 cents to an octave).

Human Hearing at Different Frequencies and the Graduation of Bar Width

You will note that my bars are graduated in width, growing wider with lower notes. One of the challenges in designing an instrument which sounds in the bass region is to provide tones that can be easily heard. The human ear has a remarkable range of sensitivity to sound at different frequencies. Sensitivity in the bass region is much less than in the range around 1000 to 4000 Hz. In order to partially overcome this problem, the bass bars are designed with wider width so that they generate sound at a higher energy level.

The graph below depicts the threshold of human hearing at different frequencies. Note that the human ear is most sensitive to frequencies in the range of 1000 to 4000 Hz. The energy level of the sound is graphed in decibels (dB), which is logarithmic. An increase of 10 dB represents a 10 fold increase in sound energy and an increase of 20 dB represents a 100 fold increase in sound energy. Sound at a frequency of 62.5 Hz (lowest note on the marimba) must be about 30 dB greater than sound at the reference of 1000 Hz in order for the human ear to detect it. That is, sound at 62.5 Hz must have an energy level 1000 times greater than a sound at 1000 Hz at the threshold of human hearing!

Threshold of average human hearing at different frequencies

Actually the situation is not quite as bad as it might appear by looking just at the threshold of hearing (0 dB at 1000 Hz). As the sound level increases, there is less difference between frequencies in perceived volume, as the graph below indicates. For example, a tone at 1000 Hz and 40 dB would have volume approximately equal to a tone at 60 Hz at about 60 dB. In other words, the 60 Hz tone would need to have 100 times more energy than the 1000 Hz tone to be perceived at equal volume. In another example, a tone at 1000 Hz and 80 dB would have volume approximately equal to a tone at 60 Hz at about 90 dB. In other words, the 60 Hz tone would need to have 10 times more energy than the 1000 Hz tone to be perceived at equal volume. I am not sure what the sound levels of the marimba are in a typical performance, but I don't think the range of 40 - 80 dB is unreasonable (click here for chart of sound levels). Therefore, the C2 marimba bar must produce 10 to 100 times more energy than a bar near 1000 Hz to be perceived at equal volume. I think it is clear that the bass bars I have designed do not meet this level of energy production. Indeed, most bass instruments do not produce such sound levels and this is the reason that we commonly perceive them to have lower volume.

Equal loudness curves for average human hearing

The audio link below demonstrates the variable loudness of tones at the same decibel level. Each tone is 0.5 second in duration starting with C2 (65.41 Hz), followed by C3, C4, C5 and then C6 (1046.5 Hz).

C2 then C3 then C4 then C5 then C6

 

Marimba vs. Xylophone and History of Tuning

In the late 19th Century John Calhoun Deagan started manufacturing xylophones in the United States (Trommer, H., 1996). His enterprise flourished in Chicago as the J. C. Deagan company. J. C. Deagan Inc. manufactured a number of different musical instruments including musical bells, orchestral bells, xylophones, marimbas, vibes and chimes. Another prominent company manufacturing bar percussion instruments in the United States during the early 20th Century was Leedy Manufacturing Company. During the mid 1920's, both of these companies developed methods for tuning the second transverse mode of bars (Winterhoff, 1927 and Schluter, 1931). Prior to that time only the fundamental was tuned.

The practice of tuning the third transverse mode developed later but I have not been able to find a reference to the individual or individuals who developed tuning of this mode. However, it was established prior to 1969 because it is mentioned by MacCallum (1969) and Moore (1970). MacCallum states that:

"In the case of marimba bars modern tuning brings down the errant first two overtones to notes of the same lettered name of the fundamental note. Tuning the third overtone is often done, but is difficult and is a luxury."

In his appendix, MacCallum makes it clear that the first overtone (second transverse mode) is tuned two octaves above the fundamental and the second overtone (third transverse mode) is tuned three octaves above the fundamental. He does not indicate the tuning interval to be used for the third overtone (fourth transverse mode?). As far as I know, the modern standard used by marimba manufacturers for the third transverse mode is the tenth harmonic, not the eighth that MacCallum lists. However, Nakano and Ohmuro (1997) propose a tuning like that suggested by MacCallum in their patent (the patent is assigned to Yamaha Corporation). Moore (1970) states the following:

"For particularly accurate tuning, the third partial as well as the second partial may be tuned to a desirable harmonic relationship with the fundamental. Tuning of the third partial requires extreme care and skill on the part of the tuner and is done only upon special request from the customer. Tuning of the third partial is done only in the low register of the instruments, on bars with fundamentals of C4 downward. For marimba and vibe bars the desirable ratios of the second and third partials to the fundamental are 4:1 and 10:1. For the xylophone bars the desirable ratios of the second and third partials to the fundamental are 3:1 and 6:1."

I find it curious that two books published only a year apart (MacCallum, 1969 and Moore, 1970) have such differing views on tuning of the third transverse mode. The difference is probably due to the two sources the authors rely on for authority. MacCallum cites a Mr. Del Roper of Monrovia, California as his source for tuning details. Moore's source for tuning authority appears to be the Musser Division of Ludwig Industries. Clearly Musser would be an authority on tuning of that period since they were a major manufacturer of marimbas. MacCallum states that Del Roper "...is an expert marimba builder and a virtuoso on the instrument." However, I don't know if Mr. Roper was associated with a large manufacturer of marimbas. I suspect that Mr. Roper may have built custom marimbas but not on a large commercial scale. In that case, Moore (1970) should serve as the authority for a typical commercial marimba of the period.

The marimba and xylophone are both fabricated with wood bars, yet the instruments are quite different in character. The differences are due in part to the tuning of the harmonics. The transverse modes of the marimba are tuned as even-numbered harmonics while the xylophone second transverse mode is tuned as an odd-numbered harmonic (odd-numbered harmonic is the standard in the USA while in Europe the even-numbered harmonic is preferred for xylophones). Furthermore, xylophone bars tend to be somewhat thicker than marimba bars of the same note and are played with harder mallets. In addition, many xylophones have a compass of C4 to C8, while marimbas commonly have a range that starts below C4 and usually a top end at C7. The character of each instrument is further developed via the resonators. A xylophone can be characterized as an instrument with a bright timbre while the marimba has a dark timbre, particularly in the bass region.

Each bar of the marimba is positioned just above a resonator, which amplifies the sound of the fundamental. The resonator tubes are closed at the bottom end. A closed-tube resonator amplifies only the odd-numbered harmonics. Therefore, the 4th and 10th harmonics of the marimba are not amplified. This arrangement results in an instrument with a mellow or dark character. In contrast, the xylophone has a bright character partly because the second transverse mode is also amplified by the resonator (if tuned by the USA standard). This occurs because the second transverse mode of the xylophone is tuned to an odd-numbered harmonic. The term quint tuning is often used for this kind of tuning for the xylophone, where the second transverse mode is tuned to one octave and a fifth above the fundamental (ratio of 3:1). Quint tuning was developed in the United States by Henry Schluter, who was the master tuner for the J.C. Deagan company (Schluter, 1931). The illustration below shows the shape for a xylophone bar according to Schluter's patent.

In the illustration above, you can see that the bar is relatively thick in the center and there are two arches in the undercut rather than one. This shape is one method of achieving the 3:1 ratio for the second transverse mode and the fundamental. The 3:1 ratio can also be achieved with one undercut arch of a specific profile (Orduña-Bustamante, 1991). Thinking about the illustration above, and using the information provided on this page about marimba bars, one can understand the rationale behind the shape shown above. The deeper arch cuts on each side of the bar center are near the antinodes for the second transverse mode. Therefore, the second transverse mode is dropped relatively more in frequency compared to the fundamental, while the fundamental is maintained relatively high due to the great thickness of the bar at its center point.

 

The Marimba Resonator

Here we continue the discussion of the science of sound. The water wave is a transverse type of wave because the displacement of the water is perpendicular to the direction of travel. The sound wave is a longitudinal type of wave because the displacement of air is parallel to the direction of travel. The air molecules vibrate back and forth in the direction the wave is traveling, creating zones of compressed and expanded air, as depicted in the illustration below.

The animation below depicts sound waves, with the black dots representing air molecules:

Sound waves can be graphically depicted by two methods: 1) variation in air pressure or 2) variation in air displacement. Both methods are compared in the illustration below:

Note above that the waves in the two graphs do not line up (i.e. the peaks and valleys of one are not at the same place as the peaks and valleys of the other). A node of displacement (where wave crosses the x axis) is the same location as an antinode of pressure. The zone between an area of compressed air and a zone of expanded air (the pressure antinodes) is the zone of maximum air displacement (a displacement antinode). The zone of maximum displacement is the zone where air molecules are moving to the maximum degree. The distinction between displacement and pressure waves is particularly important when illustrating resonance phenomena for the marimba.

The marimba resonator is a tube, open at the end next to the vibrating bar and closed at the other end. A sound wave travels down the tube and is reflected back at the closed end. As a result, sound waves are traveling in two directly opposite directions in the tube. If the tube has the proper length (approximately 1/4 wavelength or odd multiples of 1/4 wavelength), a standing wave will develop. A standing wave is formed by the combination of the sound waves traveling in opposite directions. The combined wave form results in a wave that appears stationary, thus the name standing. The illustration below can be examined to aid in understanding the nature of a standing wave.

The yellow wave represents the wave traveling into the tube and the red wave represents the wave traveling out of the tube. When the waves combine, the standing wave depicted in green results. At time 0 the incoming and outgoing waves are in phase. At time 1/4 T (time for 1/4 of wave to travel past a point), the incoming and outgoing waves are completely out of phase and when added together, the wave pattern is a flat line. As we proceed through one cycle, we see a pattern where the waves move up and down, but in place. Interestingly, there is no movement at all at the nodes. The animation below depicts the standing wave pattern over time.

 

The animation below depicts a 1/4 standing wave in the resonator tube. The black dots represent air molecules. Notice that over the period of the cycle, the number of air molecules changes at the closed end but does not at the open end. The closed end is a pressure antinode while the open end is a pressure node.

Resonance (an amplification of sound) occurs because part of the sound energy is reflected back into the tube as it exits. In the vicinity of the open end of the tube, the impedance drops. The change in impedance is the cause of wave reflection back into the tube. The reflected wave undergoes a phase change of 180 degrees, which is required for maintenance of the standing wave condition. The sound energy that is reflected back into the tube adds to the energy entering the tube from the vibrating bar to increase the volume of the sound. Hermann Helmholtz described the reflection for a tube with two open ends in his book On the Sensations of Tone : "On our exciting a wave of condensation at one end, it runs forward to the other end, is there reflected as a wave of rarefaction, runs back to the first end, is here again reflected with another alteration of phase, as a wave of condensation, and then repeats the same path in the same way a second time."

The closed end of the resonator tube represents a pressure antinode and the open end a pressure node. If you think about it, this should be logical. The closed end is furthest removed from the outside atmosphere, and is the place where the pressure fluctuates the most. The open end is very close to the outside atmosphere, and thus can't support wide fluctuations in air pressure. Conversely, the closed end of the tube represents a node of air displacement while the open end represents an antinode of air displacement. This is also logical: the air is allowed to move more freely at the open end than at the closed end.

For someone unfamiliar with the subject, confusion may arise when a wave form is drawn inside the resonator tube. The conventional diagram places a node at the closed end. It is assumed that the reader understands that the wave form depicted is a displacement wave form, not a pressure wave form. A pressure wave form has the antinode at the closed end, where the maximum fluctuation in air pressure occurs. The illustration below shows the standing wave form for a 1/4 wave with the displacement node at the closed end.

The tones produced by musical instruments are actually a mixture of sound frequencies, usually in a harmonic series. The mixture gives rise to the timbre (tone quality) of the instrument. The lowest frequency sound wave of the tone is called the fundamental or first harmonic. The overtones (sound waves with higher frequencies) are called harmonics if they are whole number multiples of the fundamental. For example, suppose a tone is analyzed and found to contain the following frequencies: 440 Hz, 880 Hz, 1320 Hz and 1760 Hz. The 440 Hz frequency is the fundamental and establishes the tone as the note A4 (the A above middle C). The 880 Hz overtone is the second harmonic (i.e., twice the fundamental frequency). And 1320 Hz is the third harmonic and 1760 Hz the fourth harmonic.

An interesting property of a closed resonator tube tuned to the fundamental (a quarter wave tube) is that only odd-numbered harmonics resonate. The reason for this is that a displacement node must be located at the closed end and an antinode must be located at the open end. These conditions can't be met by even-numbered harmonics. For example, lets take a 1/2 wave, which is twice the frequency of the 1/4 wave illustrated in the tube above. The half wave could be drawn with a displacement antinode at the open end but then the node is located in the middle of the tube, not at the closed end.

So let us try to place the node at the closed end.

But now there is also a node at the open end, which won't work. Therefore, even-numbered harmonics cannot resonate in a closed tube.

What is the next wave pattern that will resonate in a closed tube? We can fit a 3/4 wave pattern in the tube with the node and antinode in the correct position.

Recall that the 1/4 wave form represents the resonance condition for the fundamental. Then the 3/4 wave form represents the resonance condition for the third harmonic (3 x 1/4 = 3/4), which is the first available resonance for an overtone.

 

So, how do you calculate the tube length for the resonator? The formula is given below. This formula is actually the formula to calculate the length of 1/4 of a sound wave at a given frequency. Since the velocity of sound varies with air temperature, the wavelength at a given frequency will vary with the air temperature. Many marimbas have stoppers in the tube that are not adjustable (at least not easily). Resonators of this design can only resonate properly at one air temperature since the wavelength varies with temperature. This becomes particularly critical for bass resonators, where the changes in wavelength are large. That is the reason why some high-end marimbas have adjustable resonator stops on the bass end. Some also have adjustable stops on all resonators, which allows for fine tuning.

L = v/f/4

Where:

L = length of column (from opening to closed end) in feet

v = Speed of sound (1129 ft per second @ 70° F, 1132 ft per second @ 72° F, 1146 ft per second @ 85° F.)

f = frequency of desired pitch in Hertz

4 = Quarter-wavelength factor

There is also an end correction factor that should be applied to the calculation. As it turns out, the wave is not reflected back into the tube at its opening, but slightly past the opening. There may be other factors that add to the end correction as well, such as the presence of the bar. Therefore, it is necessary to subtract an end correction from the length of the tube. The theoretical end correction for a resonator closed at one end is 0.61 r, where r is the inside radius of the tube. But keep in mind that this theoretical end correction is for a resonator without an obstruction near its open end. In the case of the marimba (and other similar instruments) the builder will likely find that the real end correction is greater than 0.61 r. Also see the section below for more about end correction. Keep in mind that the calculated tube length is from the surface of the stopper, so you will need to add extra length to accommodate the thickness of the stopper. And if you want to fabricate an adjustable stopper, so that the resonator can be tuned to the bar at different air temperatures, then you must accommodate the highest air temperature you want to be able to tune.

Actual air column lengths required for resonance of the La Favre marimba

Upon completion of the marimba, I set out on the task of tuning the resonators. The tuning was done by ear, listening for maximum resonance as the bar was struck with a mallet, adjusting the resonator stopper as needed. After the stoppers were all set, the distance from each stopper to the open end of the tube was measured (except for the mitered resonators, where accurate measurement is not possible). I found that the column lengths measured were short of the lengths calculated with an end correction of 0.61 r. The end correction required for my marimba was in the range of 0.78 to 0.85 r.

 

Cited References

Bork, I., 1995. Practical Tuning of Xylophone Bars and Resonators. Applied Acoustics 46: 103-127.

Bork, I. and J. Meyer, 1985. On the Tonal Evaluation of Xylophones. Percussive Notes 23: 48-57,

Bork, I, A. Chaigne, L.-C. Trebuchet, M. Kosfelder and D Pillot, 1999. Comparison between Modal Analysis and Finite Element Modelling of a Marimba Bar. Acust. Acta Acust. 85: 258-266.

MacCallum, F. K., 1969. The Book of the Marimba. Carlton Press, New York, New York, 112 pages.

Moore, J. L., 1970. Acoustics of Bar Percussion Instruments. Ph.D. thesis, Ohio State University. Permus Publications, Columbus, Ohio, 1978, 164 pages. (available from http://wwwlib.umi.com/dissertations/)

Nakano, M. and Ohmuro, H., 1997. Percussion Instrument with Tone Bars for Exactly Generating Tones on a Scale. United States Patent 5,686,679.

Orduña-Bustamante, F., 1991. Nonuniform Beams with Harmonically Related Overtones for use in Percussion Instruments. J. Acoust. Soc. Am. 90: 2935-2041.

Schluter, H. J., 1931. Vibrant Bar for Musical Instruments. United States Patent 1,838,502.

Trommer, H., 1996. John Calhoun Deagan. Percussive Notes February 1996: 84 - 85.

Weiss, L. V., 2003. Bill Youhass 30 years with Fall Creek Marimbas and still going strong. Percussive Notes August 2003: 38 - 40.

Winterhoff, H.E., 1927. Musical Bar. United States Patent 1,632,751.

Yoo, J , T. D. Rossing and B. Larkin, 2003. Vibrational Modes of Five-Octave Concert Marimbas (Proceedings of the Stockholm Music Acoustics Conference, August 6-9, 2003 [SMAC 03], Stockholm, Sweden).

 

Articles in Percussive Notes are available at Percussive Arts Society web site (members only). United States Patents are available online at Google.

 

©2007 Jeffrey La Favre

 

 

 

(last update: 4/6/07)

You may contact me at jlafavre@jcu.edu regarding these web pages.

 

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