Identifying and Tuning Lateral and Torsional Modes in Marimba bars

Part One - Establishing Tuning Standards

Jeff La Favre
(jlafavre@jcu.edu)

Description of some methods used in the study

Part Two - Identifying and Measuring the Modes (38 images, 1.8 MB total)

Part Three - Tuning the Modes (13 images, 0.48 MB total)

Part Four - Lateral and Torsional Modes for Bars D4 through C7

 

The discussion I provide below assumes you are familiar with the transverse, torsional, and lateral modes of vibration in marimba bars. For information on these modes, please see my page on tuning.

The position of the mallet blow on the marimba bar is known to affect the relative excitation of various modes of vibration (see my web page on this subject). There are locations on the bar where a blow can excite a lateral or torsional mode. For example, a blow near the edge of the bar, at the center of the bar length, will excite the second torsional mode in bass bars. A strike at this position may be intended by the performer or not. In any case, the tuning of the bar should include adjustment of lateral and torsional modes when they degrade the sound quality of the bar. Eliminating a problem lateral or torsional mode is easier if it is tuned during the fabrication of the bar. Trying to fix the problem in a retuning effort can be more challenging, perhaps impossible. If the problematic mode cannot be tuned satisfactorily, one must decide whether the problem is serious enough to justify replacing the bar.

How do we decide that a lateral or torsional mode degrades the sound quality of a bar? I don't believe there is a simple answer to this question, but I am sure that there are differing opinions. The primary concern is judging whether a lateral or torsional mode produces a consonant or dissonant sound in combination with other modes active in vibration. In my use of the terms here, a consonant sound is desired and a dissonant sound is not. The controversy centers on the definition of what constitutes consonant and dissonant sounds. I am certainly not in a position to serve as an authority on this subject because I don't have any formal training in music. Nevertheless, I will try to provide some information here that can be useful as guidance in tuning the lateral and torsional modes.

When a lateral or torsional mode vibrates at a frequency relatively close to the frequency of a tuned transverse mode, a potential problem exists in the bar tuning. The consonance or dissonance of the complex tone is evaluated on the basis of the interval between the two tones. Two pure tones of the same frequency, when mixed, result in a very consonant complex tone. As the interval between the tones increases, the dissonance rises to a maximum point and then starts to fall until maximum consonance is reached again at a certain interval. The controversy comes into play when we try to define which intervals fall into the consonant category and which intervals fall into the dissonant category.

There are a number of theories on the cause of consonance and dissonance (see Huron, David - Literature Cited at bottom of this page). The Tonotopic Theory, a psychophysical theory, is based on the physiology of the human ear. While not free of controversy, a physiological approach provides relatively concrete definitions for consonance and dissonance. Other approaches may tie definitions of consonance and dissonance to elements influenced by culture and music, which can be ambiguous. For purposes of tuning the lateral and torsional modes, I will adopt the Tonotopic Theory. In doing so, I can develop tuning standards based on a clear definition of dissonance.

When the human ear processes a complex tone of two closely spaced frequencies, the two constituent pure tones cannot be distinguished separately. In order to be audible as two distinctly separate tones, each tone must be outside the critical bandwidth of a frequency midway between them. Tones separated by more than the critical bandwidth are considered to have maximum sensory consonance. Tones separated by 25% of the critical bandwidth are considered to have maximum sensory dissonance (Plomp and Levelt, 1965). Two zones exist where the complex tone gradually changes from consonant to dissonant (0 - 25% critical bandwidth) and then dissonant to consonant (25 - 100 % critical bandwidth).

The most demanding tuning standard based on critical bandwidth would require that the frequencies of the two modes be exactly the same or be separated by more than the critical bandwidth. The standard can be relaxed somewhat for tones that are very close (i.e., not insisting that they be exactly the same frequency). Frequencies with an interval of 6 Hz or less are not considered to be dissonant to any significant degree (Helmholtz, 1877). On the other end of the standard, perhaps we can specify that the interval be at least 110% of the critical bandwidth. These are the standards I will adopt (6 Hz maximum between the tones or a minimum of 110% of the critical bandwidth). Others are free to argue for other standards, which might be less demanding, while still maintaining acceptable sound from the bar. I don't claim to have any special expertise in setting the standard.

A search of the internet for the subject of critical bandwidth will yield various formulas and tables with values that differ to some degree. I will use the formula established by Greenwood (1961), which according to Huron (see Literature Cited at bottom of page), is currently considered to be accurate for calculating the critical bandwidth.

CBW = 22.9 (0.006046f + 1)

Where CBW is the critical bandwidth in Hz, f = frequency at center point of bandwidth

For example let us calculate the critical bandwidth at 600 Hz:

CBW = 22.9 [(0.006046)(600 Hz) + 1]
CBW = 22.9 ( 3.6276 +1)
CBW = 22.9 (4.6276 Hz)

CBW = 106 Hz

 



At this point it would be helpful to listen to tone combinations of various intervals and to inspect the wave traces of the combinations. If the interval between two pure tones is less than approximately 20 Hz , the complex tone will have a distinct periodic variation in volume, often called beating. The rate of beating is calculated simply as the difference between the frequencies. For example, if we mix pure tones of 500 Hz and 485 Hz, the beating rate will be 15 beats per second (500 - 485 = 15 beats). For a more detailed explanation of beats, read this page. Use the link below to listen to this combination of tones:

500 Hz plus 485 Hz


Compare the sound above to the pure 500 Hz tone:

500 Hz

Note that the one second tones are attenuated to zero amplitude at both ends, which eliminates a distracting sound at the start and end of the tone. The individual waves cannot be seen in these wave traces because they are highly condensed in the time scale. Rather, what you see is the wave envelope. The envelope for a pure tone at constant volume is just a rectangle. The envelope for the tone combinations resolves the individual beats, which can be counted. However, when there are many beats, as in the 500 Hz and 400 Hz combination, then the individual beats are difficult to count (at the time scale used here).


Now listen to a mixture of 500 Hz and 475 Hz (25 beats per second)

500 Hz plus 475 Hz

The beating rate of 25 per second can be described as a fluttering sound rather than a cyclical variation in volume. The human ear is limited in resolving beats at this rate. The beats are audible as a fluttering sound rather than individual beats. The critical bandwidth for this set of frequencies is 90 Hz. Therefore, the 25 Hz interval between the two tones is approximately one quarter of the critical bandwidth. Recall that this interval is considered to have maximum sensory dissonance.


Now listen to a mixture of 500 Hz plus 450 Hz (50 beats per second)

500 Hz plus 450 Hz

Here the fluttering is replaced with a sound that can be described as rough. The beating rate is 50 per second, which is too fast to be audible as a strong fluttering. I believe that most people would consider this sound less objectionable than the 500 Hz and 475 Hz mixture. In other words, this sound has an intermediate quality between maximum sensory dissonance and maximum sensory consonance. Plomp and Levelt (1965) would assign a position midway between dissonance and consonance for the 500 Hz plus 450 Hz combination (according to Figure 10 in their publication).


Now listen to a mixture of 500 Hz and 400 Hz: (100 beats per second)

500 Hz plus 400 Hz

I think you will agree that this sound is fairly consonant and you should be able to distinguish two tones. The critical bandwidth for this combination of tones is 85 Hz. Therefore, these tones are separated by an interval exceeding the critical bandwidth (118% of CBW). This is slightly more than the minimum separation that a torsional or lateral mode should have from a tuned transverse mode in order to produce a consonant sound when both modes are active at relatively high levels (alternately, the two modes could be tuned to very close frequencies for a consonant sound, but this may be more difficult to achieve).


Now listen to the complete series: first 500 Hz then 500 + 485 then 500 + 475 then 500 + 450 and finally 500 + 400

500 complete series


All of the above tones are one second long at constant volume. This sound profile allows you to easily judge the difference between tone combinations. However, it is obvious that these sounds do not match those typical of a marimba bar. In the marimba bar, the strength (volume) of a mode builds to a maximum shortly after the mallet blow and then declines. A convenient way of specifying a decline rate is by finding the point in the wave trace where the amplitude of the wave is 10 percent of the maximum value. We can specify this as the time to decline to 10 percent. In the case of the La Favre C2 bar, the third transverse mode at approximately 666 Hz declines to 10 percent around 300 milliseconds after the mallet blow.

Use the link below to listen to a 500 Hz pure tone with a decline to 10 percent at 300 milliseconds.

500 Hz 300 ms

This sounds much more like a marimba bar, although it is a pure tone. Instead of a rectangular wave envelope for the pure tone, the envelope here is triangular. Now we can evaluate the tone mixtures with wave trace profiles that are similar to real marimba bars. Below you will find links to tone combinations with the 300 millisecond decline to 10 percent.


500 Hz plus 485 Hz - 300 ms 15 beats per second


500 Hz plus 475 Hz - 300 ms 25 beats per second (28% of Critical Bandwidth)


500 Hz plus 450 Hz - 300 ms 50 beats per second (56% of Critical Bandwidth)


500 Hz plus 400 Hz - 300 ms 100 beats per second (118% of Critical Bandwidth)

With a 300 millisecond decline time to 10 percent, we can still clearly hear the dissonances in the tone combinations. For higher frequencies, where the decline is more rapid, we will find that it is more difficult to identify the dissonant sounds. These are presented below.


First, listen to tone combinations of one second duration with constant volume:

1000 Hz


1000 Hz plus 985 Hz 15 beats per second


1000 Hz plus 962.5 Hz 37.5 beats per second (24% of Critical Bandwidth)


1000 Hz plus 925 Hz 75 beats per second (48% of Critical Bandwidth)


1000 Hz plus 850 Hz 150 beats per second (99% of Critical Bandwidth)


The La Favre marimba bars with third transverse modes near 1000 Hz have a decline time to 10 % amplitude of approximately 150 milliseconds.

Now listen to the tone combinations with a decline to 10 percent at 150 ms.

1000 Hz - 150 ms


1000 Hz plus 985 Hz - 150 ms 15 beats per second

It is difficult to hear the beating sound here because there are so few strong beats. Compare the wave trace above with the one below. The one below is for the 500 Hz and 485 Hz combination, which has a decline period twice that of the above trace. Also note that the trace below has a much stronger beat amplitude for the second and third full beats, compared to the above trace. This is the reason why it is easier to hear the beating pattern of the 500 Hz and 485 Hz combination compared to the 1000 Hz and 985 Hz combination.

Since the 15 beats per second rate is much less noticeable in the 1000 Hz zone, it is possible to relax the tuning standard for frequencies of 1000 Hz and higher to allow intervals of 15 Hz or less.


1000 Hz plus 962.5 Hz - 150 ms 37.5 beats per second (24% of Critical Bandwidth)

Unlike the 15 beats per second rate, the 37.5 beats per second rate is easily audible here. Therefore, a tone combination with this interval in the 1000 Hz zone is not acceptable.


1000 Hz plus 925 Hz - 150 ms 75 beats per second (48% of Critical Bandwidth)

This tone combination has a rough sound.


1000 Hz plus 850 Hz - 150 ms 150 beats per second (99% of Critical Bandwidth)

This tone combination has a sensory consonance.


Now, listen to tone combinations of one second duration with constant volume:

2000 Hz


2000 Hz plus 1985 Hz 15 beats per second


2000 Hz plus 1970 Hz 30 beats per second


2000 Hz plus 1930 Hz 70 beats per second (24% of Critical Bandwidth)


2000 Hz plus 1860 Hz 140 beats per second (48% of Critical Bandwidth)


2000 Hz plus 1720 Hz 280 beats per second (100% of Critical Bandwidth)


The La Favre marimba bars with third transverse modes near 2000 Hz have a decline time to 10 % amplitude of approximately 40 milliseconds.

Now listen to the tone combinations with a decline to 10 percent at 40 ms.

2000 Hz - 40 ms


2000 Hz plus 1985 Hz - 40 ms 15 beats per second

It is not possible to hear beats here because there is essentially only one half a beat audible.


2000 Hz plus 1970 Hz - 40 ms 30 beats per second

You have to listen very carefully to hear any rough sound here. Since the 30 beats per second rate is hardly noticeable in the 2000 Hz zone, it is possible to relax the tuning standard for frequencies of 2000 Hz and higher to allow intervals of 30 Hz or less.


2000 Hz plus 1930 Hz - 40 ms 70 beats per second (24% of Critical Bandwidth)

A rough sound is apparent here.


2000 Hz plus 1860 Hz - 40 ms 140 beats per second (48% of Critical Bandwidth)

A rough sound is apparent here.


2000 Hz plus 1720 Hz - 40 ms 280 beats per second (100% of Critical Bandwidth)

A consonant sound is apparent here.

This completes the analyses of tone mixtures with equal amplitudes. For a complete treatment of this subject, it is necessary to also analyze tone mixtures with different amplitudes.


At this point we have established which tone combinations result in dissonant sounds at 500 Hz, 1000 Hz and 2000 Hz, when the tones mixed have equal amplitudes. For the marimba, there will be many instances when the lateral or torsional mode is weaker than a neighboring transverse mode. In these cases we need to decide at which level a torsional or lateral mode ceases to be a problem regardless of the interval between the tones.

Let us listen to tone combinations of 500 Hz and 475 Hz in an effort to establish the threshold level where the 475 Hz tone causes a distinct dissonance.


500 Hz plus 475 Hz at 100% amplitude


500 Hz plus 475 Hz at 50% amplitude


500 Hz plus 475 Hz at 25% amplitude


500 Hz plus 475 Hz at 20% amplitude


500 Hz plus 475 Hz at 12.5% amplitude

After listening to the five samples above, it seems reasonable to set a standard at the 20% amplitude level. That is, a lateral or torsional mode with 20% or less amplitude compared to the neighboring transverse mode will be acceptable regardless of the frequency interval. Now we are ready to complete a tuning standards statement for the lateral and torsional modes.


Tuning Standards for Lateral and Torsional Modes

1. For tones less than 1000 Hz, the torsional or lateral mode must have a frequency spaced no more than 6 Hz from a neighboring transverse mode OR the torsional or lateral mode must have a frequency spaced at least 110% of the critical bandwidth from a neighboring transverse mode.

2. For tones less than 2000 Hz, the torsional or lateral mode must have a frequency spaced no more than 15 Hz from a neighboring transverse mode OR the torsional or lateral mode must have a frequency spaced at least 110% of the critical bandwidth from a neighboring transverse mode.

3. For tones of 2000 Hz or higher, the torsional or lateral mode must have a frequency spaced no more than 30 Hz from a neighboring transverse mode OR the torsional or lateral mode must have a frequency spaced at least 110% of the critical bandwidth from a neighboring transverse mode.

4. Any lateral or torsional mode that vibrates at an amplitude of 20% or less compared to the neighboring transverse mode is considered to be within the standard regardless of frequency interval. This part of the standard supersedes standards in items one through three.


 

Continue to Part Two - Identifying and Measuring the Modes (38 images, 1.8 MB total)

Part Three - Tuning the Modes (13 images, 0.48 MB total)

Part Four - Lateral and Torsional Modes for Bars D4 through C7

Description of some methods used in the study

 

 

RETURN TO MAIN TUNING PAGE

Literature Cited

Greenwood, D.D., 1961. Auditory masking and the critical band. Journal of the Acoustical Society of America 33: 484 - 501.

Helmholtz, H., 1877. On the sensations of tone, as a Physiological Basis for the Theory of Music. Dover Publications, New York. 576 p. (second English edition of 1954, rendered conformal to the fourth German edition of 1877)

Huron, David. Consonance and Dissonance - The Main Theories.
http://www.music-cog.ohio-state.edu/Music829B/main.theories.html

Plomp, R. and W. J. M. Levelt. 1965. Tonal Consonance and Critical Bandwidth. Journal of the American Acoustical Society of America 38: 548 - 560.

 

Last update 3/24/07

© 2007 Jeffrey La Favre