In the top three octaves the salt method cannot be used to identify the mode type. Nevertheless, the frequencies of torsional and lateral modes can be determined by FFT analysis of bars struck on the side at the end of the bar. The FFT spectral data are converted to ratios of the fundamental and graphed below. Data from bars struck on the top surface at the antinodes of the second transverse and third transverse modes are also graphed below. Data from bars struck on the corner for the first torsional mode are also graphed below.
All data graphed above was obtained by striking the bars with an Encore 42YB mallet, which has a small, yarn-covered head and is relatively hard. The first torsional mode data are from bars struck at the corner with the microphone approximately 2 inches from the struck corner (recordings with the microphone 12 inches away resulted in no peaks for the first torsional mode due to phase-canceling from adjacent bar corners). The second torsional and unidentified modes data are from bars struck on the side at the end of the bar. The second and third transverse modes data are from bars struck at the antinodes for the respective mode on the top surface of the bar (center of bar width).
As discussed on other pages, the lateral and torsional modes can represent a potential problem when they have ratios close to a transverse mode. From the graph above, we can see that the second torsional mode intersects the second transverse mode at the F4 bar. Therefore, we need further investigation of the bars in this region to determine if the second torsional mode represents a problem.
The unidentified modes (2 and 3) are the first lateral and third torsional modes. It is not possible to identify each with certainty due to the lack of a confirmation with the salt method and the fact that they have frequencies relatively close to each other in the spectrum. Nevertheless, we can see that the unidentified mode two intersects the second transverse mode at the A6 bar. Both of these modes are weak in the zone of the A6 bar and are unlikely to contribute audible components to the bar timbre.
The third transverse mode is weak between D4 and G5 and could not be found above G5. Therefore, it is unlikely to be of concern regarding tuning conflicts. The first transverse mode has a ratio of one for all bars and it is obvious that only the first torsional mode comes close in ratios. However, the first torsional mode is not a problem because it is only found when striking the bar corner with the microphone close to the corner. When the microphone is placed 12 inches from the bar, most bars did not have a detectable first torsional mode, and those that did had weak modes. These results suggest that the first torsional mode is subdued by the mixing of sound waves 180 degrees out of phase from adjacent bar corners, as suggested by Bork et al. (1999 ).
In the lowest octave of the La Favre marimba, using a medium mallet, the third transverse mode is the strongest mode for center struck bars and the second transverse mode is strongest for bars struck off center, at the antinode of the second transverse mode. In the next octave the third transverse mode weakens and the fundamental becomes the strongest mode for center struck bars. In the third octave, the third transverse mode becomes very weak and is a challenge to tune. It is now completely overpowered by the fundamental. At the E4 bar, even the second transverse mode is surpassed in strength by the fundamental for bars struck at the antinode of the second transverse mode. We are now in the region of the keyboard where the fundamental is the most significant contributor to bar timbre.
Due to the strength of the fundamental in the top three octaves, it is important to compare the lateral and torsional modes with the fundamental. This can be readily accomplished by comparing peak amplitudes in the FFT spectra. The table below presents the relative amplitudes of two modes compared to the fundamental (given as a percentage of fundamental peak amplitude).
Relative strength of two modes compared to the fundamental (data listed as a percent of peak amplitude of fundamental)
M114 = Vic Firth M114 - medium
mallet; 42YB = Encore 42YB - hard mallet
With a center strike on the bar, we should not expect strong excitation of the second torsional mode. The data in the table above confirm a low level of vibration. The strongest peak was found in the C5 bar (42YB mallet) with a peak of only 7% of the fundamental peak amplitude. In most bars, the second torsional mode was well below 7%, the majority without any detectable peak. Therefore, the second torsional mode does not represent a tuning problem for center-struck bars in the range D4 to C7.
The second torsional mode should be excited to a greater degree by a mallet blow to the center-edge of the bar. The data in the table show this to be generally true. However, the second torsional mode is still weak for bars in the range D4 to C7. With an exception of two bars (D4 and A5), all bars were 5% or less in amplitude compared to the fundamental. These results indicate that in general, the second torsional mode is not a concern in the range of D4 to C7. However, we will examine the two bars further.
The D4 bar has a second torsional mode amplitude that is 8% of the fundamental. Is this a tuning problem? Fist let us look at the nearby second transverse mode. The second torsional mode peak is 117% of the second transverse mode peak. Both of these modes fall within the critical bandwidth. Therefore, by the standards we have established, the second torsional mode should be tuned. However, due to the overpowering fundamental, it is unlikely that the second torsional mode represents a tuning problem. In the case of the A5 bar, with an amplitude of 9% compared to the fundamental, the frequency distance between the second transverse and second torsional mode exceeds 110% of the critical bandwidth. Therefore, the tuning of the second torsional mode meets our tuning standard.
We can't be certain of the mode identity for mode 2 (either first lateral or third torsional). However, the lack of mode identity does not prevent an evaluation of the mode to determine if it represents a problem in tuning. The mode 2 results for center-struck bars are similar to the second torsional mode. There are only two bars with amplitudes exceeding 5% (F4 and C5 for 42YB mallet). For center-edge struck bars, only the D4 bar exceeded a 6% amplitude level. Therefore, we will examine the D4 mode 2 further. The mode 2 frequency is close to the third transverse mode and has a peak amplitude that is 129% of the third transverse amplitude. In addition, the frequency distance between mode 2 and the third transverse mode is only 33% of the critical bandwidth. Therefore, of all the bars presented here, the mode 2 for the D4 bar represents the strongest case for retuning. Nevertheless, we must keep in mind that mode 2 amplitude is only 22% of the fundamental. It is not certain that this mode causes an audible tuning problem.
In general, it is safe to say that the torsional and lateral modes do not cause any tuning problems in the upper three octaves of the La Favre marimba. The reason is that these modes are too weak in this part of the keyboard, at least when struck with yarn covered mallets. With harder mallets, there may be some concern yet with the torsional and lateral modes, but then this is mostly the realm of the xylophone, not the marimba.
If we conclude the D4 to C7 tuning analysis at this point, there may be cause for celebration in that there appears to be no tuning challenges outside a tuning of the proper interval for the fundamental and second transverse mode. That would be a nice thought. But the wave traces for some bars in the D4 to C7 range have some strong beating patterns. An examination of the FFT spectra in the area of the fundamental reveals that some bars have one or more peaks in addition to the fundamental. These additional peaks are the source of the beating patterns. Use the link directly below to follow the investigation from this point.
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.
Part One - Establishing Tuning Standards (36 images, 1.17 MB total)
Part Two - Identifying and Measuring the Modes (38 images, 1.8 MB total)
Part Three - Tuning the Modes (13 images, 0.48 MB total)
Last update: 4/6/07
© 2007 Jeffrey La Favre