This study examines if wine glasses can generate multiphonic sounds as a result of a hit with a metal stick on the bowl of the wine glass (hit-excitation). It demonstrates that at least two Eigenfrequenzen (= natural frequencies) of the excited wine glasses with frequencies in Hz of "F1" and "F2" (with F2<F1) will generate complex tones with frequencies in Hz of "F2+F1" and "F2-F1" – which is a typical characteristic of multiphonic sounds known from woodwind instruments. Based on the analysis of the decay kinetics of the Eigenfrequenzen and the complex tones after a hit-excitation of the wine glass, the hypothesis has been developed that the oscillations F1 and F2 build a power-profile which can be mathematically described as a multiplication of the amplitude-values of both oscillations and that this power-profile is the driving force to generate oscillations of some areas of the walls of the bowl of the wine glasses with the frequencies "F2-F1" and "F2+F1".

## Table of Contents

Summary

Introduction

Material & Methods

Results

a) Analysis of sounds of wine glasses

b) Analysis of multiphonic sounds of tenor saxophones and clarinets

Discussion

References:

## Summary

In this study, it could be confirmed that wine glasses can generate multiphonic sounds as a result of a hit with a metal stick on the bowl of the wine glass (hit-excitation). It could be demonstrated that at least two Eigenfrequenzen (= natural frequencies) of the excited wine glasses with frequencies in Hz of “F1” and “F2” (with F2<F1) will generate complex tones with frequencies in Hz of “F2+F1” and “F2-F1” – which is a typical characteristic of multiphonic sounds known from woodwind instruments. Based on the analysis of the decay kinetics of the Eigenfrequenzen and the complex tones after a hit-excitation of the wine glass, the hypothesis has been developed that the oscillations F1 and F2 build a power-profile which can be mathematically described as a multiplication of the amplitude-values of both oscillations and that this power-profile is the driving force to generate oscillations of some areas of the walls of the bowl of the wine glasses with the frequencies “F2-F1” and “F2+F1”. The fact that the phases of the oscillations of the complex tones match the phases of the oscillations forming the quasi periodic oscillation of the power-profile is an argument supporting the hypothesis. For multiphonic sounds generated with saxophones and clarinets, the same observation could be made, although one of the two complex tones shows a slight phase-shift vs. the respective oscillation of the power-profile. So, the developed hypothesis on the origin of complex tones as part of multiphonic sounds generated with wine glasses seems to be valid for woodwind instruments as well. Further investigations may be needed to confirm the hypothesis and to understand the slight differences observed in woodwind instruments concerning the phase of the oscillation of one complex tone and its respective oscillation of the power-profile.

## Introduction

Multiphonic sounds became an integral part of modern music, but also of musical science (Ref. 1-4). These sounds fascinate musicians, composers, audience and researchers, who want to understand and explain the physics behind this phenomenon. It has been described previously that “real multiphonic sounds” generated with woodwind instruments have a certain structure of the acoustic signals (Ref. 5, 6) containing “complex tones” and therefore differ significantly from “multiphonic sounding Harmonics” generated with string instruments like a piano or a cello which miss those complex tones (Ref. 7). In case a multiphonic sound of a woodwind instrument contains two basic signals (F1, F2) of different frequencies (Hz), the frequencies of the detectable complex tones follow the equation: n*F1 + m*F2 (n, m are integer numbers >0). A typical multiphonic sound generated with a woodwind instrument consists of the following three tone-groups: 1) basic frequency F1 (Hz) and related harmonics with frequencies of: n*F1 (n is an integer number >1); 2) basic frequency F2 (Hz) and related harmonics with frequencies of: m*F2 (m is an integer number >1) and 3) Complex tones which frequencies following the equation: n*F1 + m*F2 (Ref. 6, 7). Although strong evidence has been provided that a pair of complex tones with given frequencies of “n*F1 + m*F2” and “n*F2 - m*F1” (with n*F2 > m*F1) are real derivatives of the tones with frequencies n*F1 and m*F2 (Ref. 7), the concrete mechanism of this phenomenon has not been described yet. Early data describe that non-periodic oscillations of multiphonic sounds in woodwinds contain chaotic dynamics (Ref. 8) whereas Rehm et al. proposed a certain type of “Coupling of the two standing waves within the instrument” as the basic mechanism (Ref.7). Linke et al. described an ”Impulse pattern formation within the instrument” as the basic principle to generate a multiphonic sound in woodwind instruments (Ref.9). Vergez et al. presented some interesting findings on quasi periodic sounds generated with simple flute-instruments (Ref. 10) and because of the simplicity of these instruments, this might help to clarify how multiphonic sounds are generated in wind-instruments. The major problem in investigating the origin of the complex tones as part of the multiphonic sound generated with woodwind instruments is the complexity of these instruments as such, combined with the complexity of the parameters influencing the sound. Fortunately, it has been reported recently that “Wine glasses” which are often used by musicians as parts of glasharps can generate sounds containing all three types of tone-groups which are characteristic for multiphonic sounds of woodwinds (Ref. 11). As wine glasses have a less complex structure than woodwind instruments, we have chosen wine glasses as “simple tube-instruments” to investigate the principle of origin of complex tones as part of multiphonic sounds. Especially detailed analysis of the kinetics and phases of the oscillations of the two basic frequencies and the complex tones as their derivatives, should help to clarify the basic mechanism behind the formation of complex tones.

## Material & Methods

Recordings of sounds (wav-files) of wine glasses have been performed as described previously (Ref. 11). Analysis of wav-sound files and calculation of power-spectra (Fast Fourier Transformation= FFT) of those sounds have been done with the software Praat (Ref. 12). Praat has the function to extract the oscillations at certain frequencies via a reverse FFT from the power spectra. For this extraction method, it is crucial to set the right bandwidth in Hz in order to gain a good signal/noise ratio and a good time resolution. For the analysis presented in this study, the bandwidth has been set in most cases to 20Hz, as this setting has shown to give the best results. In some cases also bandwidth of 40Hz or even 60Hz have been used. Reducing the bandwidth below 20Hz will have the effect that fast transients of the intensity of the oscillation at the selected frequency may not be reflected correctly in the extracted wave.

Multiphonic sounds generated with a King tenor saxophone have been downloaded from Ref. 13 and other multiphonic sounds of woodwind instruments from different sources (Ref. 7, 14, 15) have been used for the analysis with Praat (Ref. 12).

The method described by Denninger (Ref. 16) to compare the intensities of the two peaks of an Eigenfrequenz (=natural frequency) in a power spectrum of a wine glass after a hit-excitation has been used to localize the points on the rim of the bowl of the wine glass with minimum or maximum relative movement during the production of an audible sound.

By using the integrated functions of Praat software, amplitude values of sine-like oscillations could be calculated from dB-values of these oscillations and vice versa. Also, the “power/sec” and the “energy” of sine-like oscillations could be calculated, as the power correlates with the 2.power-value of the amplitude – if the amplitude of an oscillation is given in Pascal (Pa), the power of this wave is given in Pa2.

In this study, we introduce the “power-profile” of two parallel sine-like oscillations at frequencies F1 and F2 as the result of the multiplication of the amplitude-values of these oscillations at a certain time (ti). The equation of this mathematical calculation is:

Amp-value-F1(ti) * Amp-value-F2(ti) = Power-pointF1/F2(ti)

Amp-value-F1(ti) and Amp-value-F2(ti) are the amplitude-values of the waves with the frequencies F1 and F2 (Hz) at the time (ti).

Plotting the values of the “Power-pointsF1/F2” in the time domain will give the “power-profile” of the parallel oscillations with the frequencies F1 and F2 “.

## Results

### a) Analysis of sounds of wine glasses

Although each single wine glass is generating a specific and “individual” sound after an excitation through a hit (hit-excitation) against the bowl (due to its specific parameters like shape, diameters, thickness of the glass and its overall structure; Ref. 17) the generated sounds show common spectral characteristics. The power spectra are dominated by the Eigenfrequenzen (natural frequencies) of the wine glass, where several signals of the Eigenfrequenzen consist of a double-peak with only a few Hz difference (Ref. 11, 16-20). It has been demonstrated recently that the power spectra of sounds of wine glasses contain signals – beside the Eigenfrequenzen – which a) can be classified as harmonics of one of the Eigenfrequenzen and also signals which b) can be classified as complex tones as the frequencies of their dB-maxima are given by the equation : F-Ct(Hz) = F-Ei(Hz) + F-Ey(Hz) – where F-Ct is the frequency of the dB-maximum of the Complex tone and F-Ei & F-Ey are the frequencies of the Eigenfrequenzen Ei and Ey in Hz (Ref. 11). It has further been described that the intensity of the Eigenfrequenzen show a typical decay kinetic resulting in a linear decline of the dB-value vs. time with a negative slope-value given in -dB/sec. For some wine glasses, the negative slope-values and the frequencies of the various Eigenfrequenzen show a linear correlation as well (Ref. 11).

A typical result of the analysis of the decay kinetics of the various signals of a sound generated by a wine glass after hit-excitation is presented in figures 1A-1F. Figure 1A shows the amplitude-values vs. time of the sound generated by a hit against the bowl of the wine glass, and Figure 1B displays the power spectrum of this sound. The Peaks within the power spectrum can be classified as a) “Eigenfrequenzen”, b) “Harmonics of the Eigenfrequenzen” and as c) “Complex tones” (see Material & Methods and Ref. 11) which is demonstrated in Figure 1C. Most of the Eigenfrequenzen show the expected double-peaks (Ref. 17-20) whereas the maxima of the Eigenfrequenzen F3, F6 and F7 show a single dB-maximum only. For F6 and F7, a double peak can also be generated by changing the position of the hit against the bowl of the glass slightly (data not shown). In case of F3, this will not result in a double peak. Based on the knowledge that acoustic signals generated by the oscillation of the bowl and rim will have a double-peak as a consequence of the two possible oscillation-modes (Ref. 17) it can be concluded that the oscillation responsible for the Eigenfrequenz F3 is not generated by the bowl, but by other parts of the wine glass. It is worth to note that in all wine glasses examined (with different shapes and sizes) signals classified as complex notes could be detected. Most of the complex tones had frequencies which were a sum of the frequencies of two Eigenfrequenzen – only in a few cases also complex tones with frequencies resulting from a difference of two Eigenfrequenzen could be detected. Figure 1D displays the decay kinetic of the intensity (dB) of the Eigenfrequenzen F1 and F2, as well as of the complex tone with the frequency: F1A+F2A. As expected the decay of the Eigenfrequenz of F1 is slower than the decay of F2, but both show a typical kinetic with a period of a linear reduction of the intensity (dB) vs. time (sec). The intensity of the complex tone F1A+F2A is significantly lower compared to the Eigenfrequenzen and shows a faster decay-kinetic. Although the determination of the decay-kinetic of the complex tones is more difficult and - due to the lower intensity of these signals - results in a lower R2-value of the linear regression, it could be observed that the slope of the complex tone is highly similar with the sum of the slope-values of the two Eigenfrequenzen. This is demonstrated in figure 1E where the slope of the complex tone is determined to -25,2 (dB/sec) and the sum of the slope-values of F1A and F1B result in a value of -26,5 (dB/sec). The dB-values of the multiplication of the amplitude-values of the Eigenfrequenzen F1A and F2A and a further multiplication by the constant: 1.2 is shown in figure 1E. The determined slope of this multiplication of amplitudes is -26,1 (dB/sec) which is close to the expected value of -26,5 (dB/sec). In this case, the factor 1.2 was chosen in order to give similar amplitude values in the linear dB-decay phase as the complex tone. It is worth to note that the chosen multiplication-factor has no influence on the dB-decay-kinetic and on the slope-value. Data on the decay of the Eigenfrequenzen F1, F2 and F4 (F3 excluded as this is not an oscillation of the bowl) are plotted in figure 1F and confirm the already published linear correlation of the frequency and the decay-kinetic of the Eigenfrequenzen caused by the bowl (Ref. 11). Further, it is shown that the 1st Harmonic of F2A and the complex tone (F1A+F2A) have a decay kinetic which do not match the linear regression curve of the Eigenfrequenzen.

Using a different wine glass of the same type will give similar, but not identical results. The power spectrum of the sound generated through hit-excitation (figure 2A) is similar to the spectrum in figure 1B, but shows significant differences, especially in the intensities of the Eigenfrequenzen. Figure 2B shows the decay-kinetic of F2A and F4A and of the complex tone (F2A+F4A). The slope of the dB-decay of the complex tone (F2A+F4A) is -43,7 (dB/sec) which is close to the value of -44,3 (dB/sec) for the sum of the slope-values of F2A (-17 db/sec) and F4A (-27,3 dB/sec) and the value of the multiplication of the amplitudes of F2A and F4A at – 44,1 (dB/sec) – see figure 2C. This glass also shows the linear correlation of the frequency and the decay-kinetic of the Eigenfrequenzen as well as the deviation from this correlation by the decay-kinetics of the 1st Harmonic of F2 and the complex-tone (F2A+F4A) - see figure 2D.

Based on the given examples (figures 1 & 2) and similar analysis using other types of wine glasses, it can be concluded that the amplitude of complex tones fulfill the following equation:

Amp-complex tone at frequency (F(i)+F(y)) = Amp-F(i) * Amp-(Fy) * E-factor-Ct(F(i)+F(y))

Short version: Amp-Ct(F(i)+F(y)) = Amp-F(i) * Amp-F(y) * E-Ct(F(i)+F(y)) with: F(i)>(Fy) in Hz

E-factor-(Ct) is a positive constant specific for the respective complex tone (at the certain frequency F(i)+F(y)) interpreted as “Efficiency-factor”. The higher the value for the E-factor, the larger is the amplitude of the Complex tone vs. the result of the multiplication of the amplitudes of the two respective sounds F(i) and F(y). This could be interpreted as a higher efficiency of the sounding system to create the complex tone with the frequency “F(i)+F(y)”.

It has already been proposed that a specific coupling of two standing waves of different frequency within a woodwind instrument causes the complex tones, and that this yet not understood mechanism can be described by the multiplication of the amplitude-values of the two oscillations (Ref.3). This is similar to the equation above for wine glasses. The sound of wine glasses do not result from standing waves inside the bowl, but from oscillations mainly of the bowl itself (Ref. 16, 17). So it is unlikely that the underlying mechanism to generate complex tones is a mechanism happening in the medium air but in the material glass. This means that the wine glass itself should develop an oscillation with the frequency of a complex tone F(i)+F(y) initiated by the combination of the two oscillations with the frequencies F(i) and F(y). The software developed by the University of Munich “TUM” (Ref. 17) has been used to study and simulate the oscillations of the rim and bowl of wine glasses.

Simplified projections of the first and second oscillation-modes of the bowl of a wine glass are shown in figures 3A and 3B. The left parts of the figures focus on the areas of maximum (large arrows) and minimum (small arrows) movement of the rim, whereas the right parts demonstrate the two possible types of oscillation for each oscillation-mode, which are responsible for the observed double-peaks in the power spectra. The audible sound is mainly generated by the changing air pressure within the bowl due to the changing volume of the bowl as a result of the oscillation of the rim and the walls of the bowl (Ref. 16, 17). If the 1st mode (being responsible for the signal of the first Eigenfrequenz) and the 2nd mode (responsible for the second Eigenfrequenz) oscillate in parallel with their two oscillation-types, we expect a complex oscillation pattern of the wall and rim of the bowl - see simplified projection in the left part of figure 4. The projection in figure 4 is based on an experiment where the points of the maximum and minimum movement of the rim for the two oscillation modes have been determined (see Materials & Methods). It is obvious from the right part in figure 4 that the combined oscillation of the 1st and 2nd oscillation mode generate areas of the rim which show a maximum oscillating movement especially if the maxima of both modes are closely related but also areas which will show only a minor or even no oscillating movement. These “non-oscillating-areas” (no-areas) are caused by the overlapping of the 1st and 2ndoscillation mode. So these no-areas act like a stable wall against the oscillating air-pressure generated by the combined oscillations of the 1st and 2nd mode. This means that the power generated by the oscillating air-pressure will act fully against the no-areas, but no oscillation of this area at the frequency of the 1st and 2nd Eigenfrequenz can occur. The power of an acoustic wave correlates with the 2.power of its amplitude – so if the amplitude is given in “Pa” the power of the wave will be given in “Pa2”. It can be assumed that beside the power of the oscillations of the 1st and 2nd Eigenfrequenz also a combined power of both amplitudes may act against the no-areas. This additional “power-profile” could be calculated as a multiplication of the amplitudes of both oscillations (resulting in a Pa2-value). It has been shown that the multiplication of two sine-waves with the frequencies F1 and F2 in Hz (with F2>F1) result in an oscillation consisting of two waves with the frequencies F2-F1 and F1+F2 (Ref.7). So we may interpret the “power-profile” of the 1st and 2nd Eigenfrequenz as a power being able to initiate an oscillation of the no-areas. The no-areas will neither oscillate at the frequencies of the 1st - nor of the 2nd Eigenfrequenz, as explained above. But as the power-profile contains components with frequencies of “F2-F1” and “F1+F2”, this impulse might initiate a corresponding oscillation of the no-areas. By comparing the phases of the oscillations of the complex tones with the phases of the 1st and 2nd Eigenfrequenzen and the phases of the oscillations within the power-profiles of both Eigenfrequenzen, it should be possible to find arguments for the ideas presented above. If the complex tones in wine glasses are the result of the power-profiles, the phase of the oscillation of the complex tone with the frequency F(ct), and the phase of the oscillation of the signal within the power-profile with the same frequency F(ct), should be identical.

The data presented in figures 5 and 6 for a wine glass filled with 50ml of water demonstrate the fulfillment of the above-mentioned expectation. As, it has been previously reported that wine glasses with a relatively large volume may generate sounds of higher intensity if filled with some water (Ref. 11), such a condition have been chosen for the experimental analysis. The data presented in figure 5 showing the decay kinetics of the Eigenfrequenzen F1 (633,3Hz) and F2 (1696Hz), the complex tone “F1+F2” (2329Hz) and of the “Multiplication of amplitude-values of F1 and F2”, generally confirm the findings displayed in figures 1 and 2 with high precision. Figure 6A shows the oscillation of the complex tone (red curve) at the frequency of F1+F2 (2329Hz) and the “power-profile” of the Eigenfrequenzen F1 and F2 (black curve) resulting from the multiplication of the amplitude-values of both oscillations. It is obvious from figure 6A that the power-profile is not an oscillation with one single frequency, but is a combination of two oscillations of different frequency. An FFT-analysis confirms the expected overlapping of two oscillations with frequencies at F1+F2 (2329Hz) and F2-F1 (1063Hz) as parts of the power-profile. In figure 6B, the two oscillations with frequencies at 1063Hz (brown curve) and 2329Hz (blue curve) forming the power-profile are also plotted into the graph. Therefore, figure 6B clearly demonstrates that the phase of the oscillation of the complex tone is identical to the phase of the oscillation within the power-profile having the frequency of the complex tone.

In another experiment with a similar wine glass, complex tones with the frequency F1+F2 and F2-F1 could be detected, although the signal at F2-F1 was weak. Figure 7A shows the oscillations of the Complex tones at 1063Hz (F2-F1) and 2329Hz (F1+F2). It was expected that the addition of both oscillations will result in an oscillation pattern which should be highly similar to the power-profile of the oscillations F1 and F2. This is not the case (see figure 7B) but if the phase of the oscillation at 1063Hz is inverted (which corresponds to a phase-shift of 50% of the respective wavelength) the sum of both oscillations shows a high similarity to the power-profile (figure 7C).

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