Identifying and Tuning Lateral and Torsional Modes in Marimba bars

Part Two - Identifying and Measuring the Modes

Jeff La Favre
(jlafavre@jcu.edu)

Description of some methods used in the study

Part One - Establishing Tuning Standards (36 images, 1.17 MB total)

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

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

Below I have added the tuning standards that were developed in Part One.


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.


 

Now that we have a set of tuning standards, let us examine the case of the A2 bar for the La Favre marimba. This will serve as an example of a bar that does not meet the tuning standards.

Below is the audio spectrum of the A2 bar, struck at the center.

 

FFT spectrum of A2 bar struck at the center, sample time 0 - 186 milliseconds (no resonator in place).

The spectrum above is dominated by two frequencies, 1090 Hz and 1121 Hz. One of the frequencies is due to the third transverse mode and the other to the first lateral mode. The assignment of each frequency to a mode is difficult because the modes are close together in the spectrum. The frequency at 112 Hz is the fundamental and at 445 Hz, the second transverse mode. The amplitude (strength) of the 1090 Hz frequency is 1399 and the amplitude of the 1121 Hz frequency is 1982. Since both frequencies have strong amplitudes and are closely spaced, they represent a tuning problem in the bar. Let us use this example to explore the roughness of complex tones with closely spaced frequencies.

In addition to FFT spectrum analysis, an inspection of the wave trace of the audio recording may be helpful if the closely spaced frequencies are the only strong modes in the spectrum. This happens to be the case for the A2 bar recorded without the resonator in place. The wave trace is provided below.

The wave trace above reveals the classic profile of two closely spaced frequencies, which produce a pattern of beats. The beats can be calculated from the wave trace, but it is easier to derive the beat frequency if you already know the frequencies of the two strong modes. The beating rate is calculated as the difference of the two frequencies, which is 1121 Hz - 1090 Hz = 31 beats per second. That rate translates to a beat period of 0.032 second, which is in close agreement with the beat period seen in the above wave trace.

Turning our attention back to the FFT spectrum, the mode at 1090 Hz has a relative amplitude of 1399/1982 = 0.71 or, we can say that the 1090 Hz mode is approximately 70% of the strength of the 1121 Hz mode. Now listen to the A2 bar recording using the link below (the resonator was removed from the instrument in order to keep the fundamental low, which is helpful for analyzing the third transverse mode)..

Listen to the A2 bar (no resonator in place)

You may need to play the above recording repeatedly. Listen carefully to the first part of the sound. Does the sound of the third transverse mode sound rough?

It is helpful if we just concentrate on the first part of the sound. Synthetically produced tones of one second duration are useful to emphasize the sound of the tone combination.

Now listen to the synthetic tones.

Listen to 1121 Hz pure tone

Listen to 1121 Hz 100% plus 1090 Hz 70%

Listen to 1121 Hz pure tone followed by 1121 Hz and 1090 Hz combination tone

 

The roughness of the complex tone should be very obvious by listening to the tones above. Since the A2 marimba bar does not produce a sustained tone like the ones above, we need to listen to synthetic sounds closer to the A2 profile. In the case of the two modes under consideration in the A2 bar, the vibrations damp out to 10% of maximum amplitude around 0.15 second and are almost gone by 0.3 second after the mallet blow. Therefore, we can create a synthetic sound that matches more closely that of the A2 bar using this information.

Listen to a 1121 Hz tone damping to 10% at 150 milliseconds

Listen to the same 1121 Hz tone combined with a 1099 Hz tone of the same profile

Listen to a 1121 Hz tone followed by the combination tone

The roughness of the complex tone is not as obvious in the synthetic sounds above as those with constant volume for 1 second. However, with careful listening, you should be able to recognize the difference between the pure tone and the complex tone.

Now listen to the first 120 milliseconds of the recording for the A2 bar

Listen to first part of A2 bar recording

Now listen to a synthetic combination tone of 1121 Hz and 1090 Hz

Listen to 1121 Hz plus 1090 Hz

The sounds are not the same because the A2 bar has additional frequencies. If we examine the spectrum carefully, there are some minor peaks at 112, 315, 412, 445 and 716 Hz. So we will add those frequencies to see if we can come closer to the real A2 bar.

Listen to synthesized A2 bar

Now we are closer to the real A2 bar sound. The sounds are not exactly the same because the synthetic one lacks some of the additional frequencies in the real A2 bar. Nevertheless, the point of this exercise is to identify the beating element in the real A2 bar recording. These audio files should allow you to do that.

Listen to first part of A2 bar followed by synthetic A2 bar

Listen to A2 then synthetic A2 then 1121 Hz plus 1090 Hz combination tone

 

The timbre of the A2 bar could be improved by retuning. The retuning effort is described on the following page.

Now that we have examined a problematic lateral mode in one bar, let us continue to evaluate additional bars of the La Favre marimba. A professional marimba tuner may be able to evaluate the bars by just listening to the sound they make when hit at different locations. I don't have that skill and need to rely more on an instrument-centered method, which is what I present here.

The frequencies of the various modes of concern can be determined by a strobe tuner. In order to do this, you need to know where to strike the bar and where to hold it for each mode. You may consult my page on this topic for more details. Hunting for frequencies of vibration with a strobe tuner can be difficult and it is easy to miss a mode if you are not careful. In this study I relied on spectral analyses to find the frequencies of vibration. To do this, the bar is struck at various locations and the resulting audio is recorded digitally. The recorded audio files are subjected to Fast Fourier Transformation (FFT) to find the frequencies of vibration. The value of this approach is that the analysis also provides a quantitative measure of the strength of each mode of vibration (evaluating the strength of vibration is possible to a limited extent with the strobe tuner, but FFT is a much better way to do this because it provides a quantitative value). If you have a computer, with an investment of about $100, you can obtain the FFT software necessary to do analyses. Whether you use the strobe tuner or FFT to measure the frequencies for modes, you still need another method for identifying the mode. The manner in which you hold the bar and where you strike it can be helpful in identifying the mode, but some uncertainty may remain when trying to assign a specific frequency of vibration to a specific mode. In order to develop a tuning strategy, it is very helpful to know which mode gives rise to which frequency. I used a "salt" method to identify the modes. My page on methods contains more information on the salt method.

A graph of vibration rates is helpful for evaluating the large data set. The appropriate scale is obtained by converting the vibration rates to ratios based on the fundamental.

 

In the graph above, we can identify a number of potential problems. For most of the bars between C2 and C4, the ratios for the first lateral mode and the third transverse mode are very close. On the left side of the graph, the second torsional mode is also close to the third transverse mode, while on the right side it approaches the second transverse mode. The third torsional mode approaches the third transverse mode on the right side of the graph. The first torsional mode may also appear to be close to the fundamental, but this is not a problem, as I discuss on a page dedicated to the first torsional mode.

Now that we have identified some potential problems, we must investigate further in order to establish which bars, if any, have problems (we have already discussed the problematic A2 bar). Primarily, we must establish whether or not the potentially problematic bars vibrate with a significant lateral or torsional mode under performance conditions. In order for a problem to exist, the lateral or torsional mode must vibrate with enough intensity to interfere with a tuned transverse mode. When these conditions are found, a tuning of the offending mode should be considered.

In order to determine if the lateral or torsional mode of concern does or does not cause a problem, audio recordings of the bars struck at different positions were analyzed by FFT. The wave traces of the recordings were also examined with the purpose of finding any beating patterns in the traces.

 

Finding the problem bars

This portion of the study is not complete at this time. Nevertheless, some helpful data are presented below.

In the lower register (C2 to C4) the second torsional mode does not cause any problems with the third transverse mode. The C2, D2 and E2 bars have second torsional modes and third torsional modes that are separated by less than 1.1 times the critical bandwidth. However, the peak amplitudes of the second torsional modes are below 20% of the respective third transverse mode amplitudes. Therefore, no retuning of the second torsional mode is needed. The second torsional mode of the F2 bar does not meet the tuning standards and a retuning is indicated in the table. However, the third transverse mode for the F2 bar is relatively low, indicating a possible problem with the mallet blow. Further testing is needed before condemning the second torsional mode in the F2 bar. When we reach the G2 bar, the interval between the second torsional and third transverse modes is greater than 1.1 times the critical bandwidth. The interval continues to grow in the A2 and B2 bars. Therefore, past the F2 bar, there is no need to retune the second torsional mode on the basis of critical bandwidth (paired with the third transverse mode).

Second Torsional and Third Transverse Modes

FFT Spectra of bars struck at center-edge

Bar
2nd Torsional Mode
3rd Transverse Mode

Critical Bandwidth
x 1.1

(Hz)

retune?*
Frequency
(Hz)
Peak Amplitude
Frequency
(Hz)
Peak Amplitude
C2
595
155
666
1012
121
no
D2
642
244
748
1667
131
no
E2
720
159
840
1095
144
no
F2
747
99
888
398
150
yes
G2
795
187
1000
1314
162
no
A2
880
60
1121
617
178
no
B2
939
146
1261
232
193
no
*The decision to retune is based on the critical bandwidth and the relative peak amplitude of the torsional mode compared to the transverse mode.

The first lateral mode does cause a number of problems in the range of C2 to C4, as is evident in the table below. All bars in the range are listed in the table if their spectra had a peak for the first lateral mode when struck at the center-edge or the bar center. Bars D2, E2, F3, A3, B3 and C4 did not have a peak in their spectra for the first lateral mode. However, that does not mean that the first lateral mode is never active in these bars during a performance. Further testing is needed to confirm that these bars never vibrate to any significant degree during a performance of the instrument.

 

First Lateral and Third Transverse Modes

FFT Spectra

Bar
Bar struck at center-edge
Bar struck at center
Critical Bandwidth x 1.1
(Hz)
Retune?*
1st Lateral
3rd Transverse
1st Lateral
3rd Transverse
Frequency
(Hz)
Peak Amplitude
Frequency
(Hz)
Peak Amplitude
Frequency
(Hz)
Peak Amplitude
Frequency
(Hz)
Peak Amplitude
By CBW
By Amplitude***
C2
790 [786]
140
666
1012
NP
NP
666
4285
136
yes
no
F2
904 [907]
159
888
398
908
490
888
2881
162
yes
no****
G2
980 [991]
695
1000
1314
MP?
MP?
999
3115
176
yes
yes
A2
1091 [ND]
826
1121
617
1090
1400
1121
1982
194
yes
yes
B2
1213 [1219]
134
1261
232
1213
131
1261
876
214
yes
yes
C3
NP
NP
1335
378
1265 [1279]
279
1335
1038
223
yes
yes
D3
1410 [1410]
69
1510
171
1408
149
1499
234
248
yes
yes
E3
1610 [1604]
(1645)**
59
(85)**
1693
237
1645**
232
1693
555
277
yes
yes
G3
1960 [1965]
104
2016
487
NP
NP
2019
198
328
yes
yes

*The decision to retune is based on the critical bandwidth and the relative peak amplitude of the first lateral mode compared to the third transverse mode. A retune is not needed if either the critical bandwidth or the peak amplitude indicate no need for retuning.

**This frequency does not match the frequency found for the first lateral mode by striking the side of the bar. The assignment of this frequency to a mode is therefore in question, but the first lateral mode appears to be the best fit. There are two frequencies (1610 and 1645) present in the center-edge struck spectrum that might be assigned to the first lateral mode.

***Amplitude of first lateral mode must be more than 20% of third transverse mode to retune by peak amplitude

****Amplitude data for center struck bar do not indicate a need to retune

NP = no peak in the spectrum

ND = not determined

MP? = possible merged peak, the first lateral and third torsional frequencies may be too close to separate into two peaks

frequencies in brackets [ ] are the first lateral mode as determined by a strike on the side of the bar at the bar end.

Wave traces for the C2 bar

 

Wave trace of C2 bar struck at center-edge with mallet, no resonator in place. A careful inspection of this wave trace does not reveal any obvious beating pattern. However, there is some cyclical variation in wave amplitude that may be due to the weak second torsional mode or first lateral mode.

FFT spectra of C2 bar - sample time 0 - 186 ms, bar struck in center with no resonator in place. The third transverse mode is 666 Hz and the fourth transverse mode is 1289 Hz.

 

Wave trace of C2 bar struck at center with mallet, no resonator in place. There is no obvious beating pattern in this wave trace. There is an undulating pattern in the waves of the third transverse mode but this is due primarily to the presence of the fourth transverse mode with a frequency of 1289 Hz. Compare the above trace with the one below, which is a synthetic mix of 666 Hz and 1289 Hz.

Wave trace for a mixture of a 666 Hz tone with a 1289 Hz tone, amplitude ratio approximating the ratio for the C2 bar.

Wave Traces for the D2 bar

Tf

Wave trace of D2 bar struck at center-edge with mallet, no resonator in place. The very weak beating pattern of the second torsional and third transverse modes can be seen in this trace on the right side. With a beating rate of 106 per minute, one beat is approximately 0.009 second. This closely matches the cyclic pattern seen on the right side of the trace.

 

f

Wave trace of D2 bar struck at center with mallet, no resonator in place. There is no obvious beating pattern in this wave trace.

 

 

Wave Traces for the E2 bar

f

Wave trace of E2 bar struck at center-edge with mallet, no resonator in place. The beating pattern due to a weak second torsional mode can be seen in this trace between 0.065 second and 0.085 second. With a beating rate of 120 per second, one beat is approximately 0.008 second, which matches the beating pattern seen on the trace.

f

Wave trace of E2 bar struck at center with mallet, no resonator in place. There is some cyclical variation in wave amplitude in this trace, but no obvious beating pattern is present.

 

 

Wave Traces for the F2 bar

Wave trace of F2 bar struck at center-edge with mallet, no resonator in place. This trace has some variation in wave amplitude, but no obvious beating pattern. It is difficult to pick out a pattern when the wave amplitude is so low. The 143 beats per second caused by the weak second torsional mode results in one beat every 0.007 second, which is not readily apparent on the trace. A beating rate of 20 per second for the lateral mode is equivalent to one beat every 0.050 second, which is also not readily seen in this trace.

f

Wave trace of F2 bar struck at center with mallet, no resonator in place. This wave trace shows some mild variation in wave amplitude but there is no obvious repeating beating pattern. There are low amplitude points approximately at 0.025 second and 0.082 second, which yields a one beat pattern with a time span of 0.057 second, close to the 20 beat rate due to the first lateral mode.

 

Wave Traces for the G2 bar

f

Wave trace of G2 bar struck at center-edge with mallet, no resonator in place. Here is the first bar that has an obvious beating pattern The beating pattern in this trace can be seen with low amplitudes at approximately 0.030 second, 0.080 second, and 0.130 second. Thus, one beat equals approximately 0.050 second or 20 beats per second. In the FFT spectrum for this bar there are peaks at 1000 Hz and 980 Hz, which are the cause of the beating pattern seen here (1000 Hz - 980 Hz = 20 beats per second). Therefore, the beating pattern is due to the first lateral mode.

f

Wave trace of G2 bar struck at center with mallet, no resonator in place. There is no beating pattern present in this wave trace for the bar struck in the center, . This supports the finding of the FFT spectra for this bar, where the center-struck spectrum does not have peaks for the second torsional or first lateral modes. However, there is the possibility that the first lateral and third transverse modes were not resolved in the FFT spectrum. This could happen if they are very close in frequency. There is a slight reduction in wave amplitude at 0.095 second. Is this part of a beating pattern? Following the trace farther right (not shown), there is no hint of more beats.

 

Spectra and Wave Traces for the A2 bar

FFT spectrum of A2 bar struck at the center-edge (top) and center (bottom).

The spectrum for the center-edge struck bar has peaks at 880 Hz , 1091 Hz and 1121 Hz, and the center-struck spectrum has peaks at 1090 Hz and 1121 Hz. The results suggest a second torsional mode at 880 Hz which could not be confirmed with the salt test. Therefore, it was assumed that at this point in the keyboard, the second torsional mode becomes so weak that it cannot be excited into enough vibration to redistribute the salt on the bar. Nevertheless, the results of analysis of side-struck bars, presented in the graph farther up on this page, clearly establish 880 Hz as the second torsional mode.

 

 

f

Wave trace of A2 bar struck at center-edge with mallet, no resonator in place. A beating pattern is apparent in this wave trace. There are amplitudes near zero at approximately 0.065 second and 0.095 second (the exact points are too difficult to determine). This yields a beat every 0.03 second or 33 beats per second. The difference between the two frequencies obtained in the FFT spectrum (1121 Hz and 1091 Hz) calculate to a beating rate of 30 per second. Therefore, the beating rate seen on this trace is due to the 1121 Hz and 1091 Hz frequencies.

Wave trace of A2 bar struck at center with mallet, no resonator in place. Unlike the G2 bar, the A2 bar has a beating pattern when struck in the center. In fact, this beating pattern is the strongest of all bar traces presented on this page and represents a problem with the tuning of the first lateral mode. A measurement of the beating pattern on the trace also matches nicely with a beating rate of 31 per second by FFT spectrum analysis (1090 Hz and 1121 Hz).

Spectrum and Wave Traces for the B2 bar

FFT spectrum of B2 bar struck at the center-edge (left) and at center (right).

The spectra for the center-edge and center struck bars have peaks at 1213 Hz and 1261 Hz. The 1213 Hz frequency was assigned to the first lateral mode. The 1261 Hz frequency is the third transverse mode. The frequency of the lateral is within the critical bandwidth of the transverse mode, which is a potential problem. The wave trace for the center-edge struck bar might confirm the problem, but there is no hint at all of a beating pattern. The reason is that the spectrum contains other significant peaks that obscure the beating pattern (e.g., second transverse mode at 499 Hz and second torsional mode at 939 Hz). However, a careful inspection of the wave trace for the center struck bar does reveal a very weak beating pattern. The spectrum for the center-edge struck bar has a second transverse peak that is stronger than the third transverse mode. That is why we are unable to see a beating pattern for the 1213 Hz and 1261 Hz combination (see the illustrations below for a demonstration).

For the center-edge struck bar, the amplitude of the first lateral mode was measured at 134 while the the third transverse mode was 232. These amplitude results weigh in favor of classifying the bar with a problem first lateral mode.

Wave trace of B2 bar struck at center-edge with mallet, no resonator in place. With this bar we begin to see the waves of the fundamental, which make it difficult to check for a beating pattern of the third transverse mode. A beating rate of 48 per second is equivalent to one beat per 0.021 second. However, it is not possible to pick out a cycle of that period with visual inspection of the trace. Therefore, we cannot use a visual examination of the wave trace to support a decision to retune.

Mixed pure tones of 125 Hz, 499 Hz, 939 Hz, 1261 Hz and 1213 Hz with amplitude proportions similar to those of the B2 bar. Note that this trace of a synthesized mixture of tones like those for the B2 bar has a wave trace similar to the actual B2 bar, especially on the right side of the wave trace.

 

Mixed pure tones of 1261 Hz and 1213 Hz with amplitude proportions similar to those of the B2 bar. This is the wave trace pattern for the B2 bar if it vibrated only in the first lateral and third transverse modes.

Mixed pure tones of 939 Hz , 1261 Hz and 1213 Hz with amplitude proportions similar to those of the B2 bar. This is the wave trace pattern for the B2 bar if it vibrated only in the second torsional, first lateral and third transverse modes.

Mixed pure tones of 499 Hz, 939 Hz, 1261 Hz and 1213 Hz with amplitude proportions similar to those of the B2 bar. This is the wave trace pattern for the B2 bar if it vibrated only in the second transverse, second torsional, first lateral and third transverse modes. Here it is evident that the addition of the second transverse mode obscures the beating pattern of the mixture of 1261 Hz and 1213 Hz. In the case of the actual B2 bar there is the question of whether the strong presence of the second transverse mode masks, to any degree, the roughness due to the first lateral and third transverse modes. If so, then a retuning of the first lateral may be unwarranted. However, more testing must be done to determine if there is a spot along the edge of the bar where a strike would produce a spectrum with only a minor second transverse mode peak. In that case, we should expect to see the beating pattern in the wave trace and would need to conclude that a retuning of the first lateral mode is warranted.

 

Wave trace of B2 bar struck at center with mallet, no resonator in place, 0 to 0.125 second. In this trace the third transverse waves are stronger than for the center-edge struck bar. The FFT spectrum contains a strong peak for the third transverse mode and a weak peak for the first lateral mode. Therefore, the beating pattern in this trace is not obvious, but is seen more readily in the trace below. A careful inspection of the trace below in particular shows a periodicity in the strength of the third transverse waves which is consistent with the presence of the first lateral mode. However, the weak amplitude of the first lateral compared to the third transverse mode indicates that no retuning of the first lateral is warranted if we consider only this wave trace.

Wave trace of B2 bar struck at center with mallet, no resonator in place, 0.120 to 0.245 second.

Wave Traces for the C3 bar

Wave trace of C3 bar struck at center-edge with mallet, no resonator in place. There is no peak for the lateral mode in the center-edge struck bar and there should be no beating pattern present in the trace. There is no visually apparent beating pattern, but we have seen with the B2 bar that a beating pattern is not visually apparent in the wave trace. Therefore, wave trace analysis for bars in this region of the keyboard may not be as useful as with the bars in the lowest octave (at least as far as the third transverse mode is concerned).

In this trace the fundamental starts to overwhelm the third transverse mode at about 0.03 second. If there was a beating rate of 70 per second in this trace, there would be approximately two beats occurring during the first 0.03 second, when there is a relatively strong presence of the third transverse mode.

Wave trace of C3 bar struck at center with mallet, no resonator in place. The first lateral mode does beat against the third transverse mode in this trace and can be seen with careful inspection. With a rate of 70 beats per second, one beat spans 0.014 second. The fundamental for C3 is 131 Hz, one wave spans 0.00762 second, and two wavelengths have a time span of 0.015 second. This is very close to the span time of one beat at 0.014 second. Look on the trace at 0.050 second, where the wave line crosses the baseline. Here the waves of the third transverse mode have relatively low amplitude. Follow the fundamental wave pattern forward for one cycle to about 0.058 second. Here the waves of the third transverse mode have relatively high amplitude. Follow the fundamental wave forward for another cycle to about 0.065 second where once again the waves of the third transverse mode have relatively low amplitude. Therefore, there is a beating pattern visible, albeit not obvious, with a rate of approximately 70 beats per second.

 

Wave Traces for the D3 bar

The spectrum for the center-edge struck bar has peaks at 1410 Hz and 1510 Hz, and the center-struck spectrum has peaks at 1408 Hz and 1499 Hz. The 1408 Hz and 1410 Hz frequencies were assigned to the first lateral mode. The 1499 Hz and 1510 Hz frequencies are the third transverse mode. The beating rates for the two mode pairs are 100 and 91. The wave traces are not helpful in visualizing a beating pattern.

Wave trace of D3 bar struck at center-edge with mallet, no resonator in place.

Wave trace of D3 bar struck at center with mallet, no resonator in place, 0 to 0.125 second.

Wave trace of D3 bar struck at center with mallet, no resonator in place, 0.125 to 0.250 second.

Wave Traces for the E3 bar

Wave trace of E3 bar struck at center-edge with mallet, no resonator in place. The FFT spectra indicate that there are beating frequencies next to the third transverse mode. However, the amplitudes of these neighboring peaks are relatively low, and the fact that there is two peaks probably obscures any beating pattern that might arise. Even a careful examination of the wave trace above reveals no beating pattern. It is also important to note that the major activity of vibration for the third transverse mode occurs in the first 0.030 second. Therefore, even with a beating rate of 48, there is only time for one significant beat at most.

Wave trace of E3 bar struck at center with mallet, no resonator in place, 0 to 0.065 second. The third transverse mode is also strong only during the first 0.030 second in the center struck bar. This is the zone where a beating pattern has the most chance to be audible. But there is no visible pattern of beating during the first 30 milliseconds. Later in the vibration of the bar, as seen below, a weak beating pattern can be seen. For example, look at the peaks in the region of 77 milliseconds, where the peaks of the third transverse mode are relatively strong. In the next peak of the fundamental, centered at about 83 milliseconds, the peaks of the third transverse mode are relatively weak. The pattern of weak and strong peaks appear to match a time cycle of 48 beats.

Wave trace of E3 bar struck at center with mallet, no resonator in place, 0.060 to 0.125 second.

Wave Traces for the G3 bar

The spectrum for the center-edge struck bar has peaks at 1960 Hz and 2016 Hz, and the center-struck spectrum has a peak at 2019 Hz. The 1960 Hz frequency was assigned to the first lateral mode. The 2016 Hz and 2019 Hz frequencies are the third transverse mode. The beating rate for the center-edge struck bar is 56.

Wave trace of G3 bar struck at center-edge with mallet, no resonator in place.

There is no obvious beating pattern in the trace above.

Wave trace of G3 bar struck at center with mallet, no resonator in place.

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

Part One - Establishing Tuning Standards (36 images, 1.17 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

Last update: 3/24/07

© 2007 Jeffrey La Favre