A new model is proposed which offers the opportunity to describe and characterize the sound of a musician playing a monophonic instrument in such a way that audible differences can be expressed by two main parameters: 1) “virtual power spectrum” and 2) “Resonance & Radiation” spectrum of the analyzed sound. The “virtual power spectrum” can be calculated by using data of a Fast Fourier transformation (FFT) –analysis of the recorded sound for a mathematical trend analysis delivering a logarithmic function. For each pitch played with different intensities, a “virtual power spectrum” and the related logarithmic function can be calculated and can function as a parameter to describe certain characteristics of the sound. The “Resonance & Radiation” spectrum can be calculated by comparing the dB-values of the Harmonics of a sound (determined through FFT-analysis) with the calculated “virtual power spectrum”. The “Resonance & Radiation” spectrum is independent of a) the pitch played and b) the playing intensity, and is therefore a further parameter characterizing the sound of a playing system (musician & instrument). The model is valid for several monophonic wind instruments as well as for the Cello. Further research may show whether this model can be used for other monophonic instruments as well.

## Summary

A new model is proposed which offers the opportunity to describe and characterize the sound of a musician playing a monophonic instrument in such a way that audible differences can be expressed by two main parameters: 1) “virtual power spectrum” and 2) “Resonance & Radiation” spectrum of the analyzed sound. The “virtual power spectrum” can be calculated by using data of a Fast Fourier transformation (FFT) –analysis of the recorded sound for a mathematical trend analysis delivering a logarithmic function. For each pitch played with different intensities, a “virtual power spectrum” and the related logarithmic function can be calculated and can function as a parameter to describe certain characteristics of the sound. The “Resonance & Radiation” spectrum can be calculated by comparing the dB-values of the Harmonics of a sound (determined through FFT-analysis) with the calculated “virtual power spectrum”. The “Resonance & Radiation” spectrum is independent of a) the pitch played and b) the playing intensity, and is therefore a further parameter characterizing the sound of a playing system (musician & instrument). The model is valid for several monophonic wind instruments as well as for the Cello. Further research may show whether this model can be used for other monophonic instruments as well.

## 1) Introduction

Many professional saxophone players develop an individual and specific sound which is audible in a way, so the audience can distinguish between different saxophone players just by the differences in their individual sound. It has been shown previously (Ref. 1, 2) that power spectra (derived from Fast Fourier Transformation analysis = FFT) of certain tones (pitches) played on the Tenor saxophone by different professional players show significant differences in the intensity of the expressed Harmonics. These differences can be attributed to or even cause the individual sound of the player and his set-up (Remark: set-up means the type of T-sax, the used S-bow and mouth piece as well as the used reed). The power spectra of pitches played by the musician (with his/her instrument) contain so-called “Formants” and the calculation of a “Formant-spectrum” has been introduced as a parameter to describe the individual sound-characteristics of each player (Ref. 2). Differences in the sound of saxophone players can be attributed to a large extent to the differences in the calculated Formant spectra. The typical Formant spectra of saxophone players exhibit several formant signals in the audible frequency range (Ref. 3). Further, it has been demonstrated that professional saxophone players are able to keep their Formant spectra fairly stable over a wide range of played pitches (Ref. 2). The analysis of impedance spectra of saxophones (Ref. 4, 5) was able to demonstrate that various factors will influence the frequency dependent impedance of a saxophone which may also cause the expression of Formants in the power spectra

The basic assumption and procedure to calculate a Format spectrum was the definition of a virtual decay-curve defining the decrease in intensity of the Harmonics of a played Single tone (pitch) with increasing frequency (Ref.1). The decay curve was calibrated in such a way that the first Harmonics with the highest intensity show ideally no deviation or only minor deviations from the respective points on the decay curve. The result of this procedure was that with increasing frequency the db-values of Harmonics showed a positive deviation vs. the decay curve and these values were used to plot the Formant spectrum (Ref. 1, 2).

In this publication, we change the assumption by using the signals of all Harmonics to calculate a “virtual decay curve” instead of a decay curve defined by only the first 2-4 Harmonics (see Ref. 1, 2 for the procedure to determine a Formant spectrum). The basic idea for this changed assumption is the following:

Player and instrument generate a Single tone sound, where the following parameters can be used to mimic the power spectrum of the played sound:

a) The blowing intensity drives the generation of a monophonic sounds where the intensities of the Harmonics of this sound (defined in dB-values) tend to follow a logarithmic function of the frequency given by Formula I.

Formula I.: ** virtual dB of Harmonic (fi) = - m * Ln(fi) + b **

“** fi **” is the frequency at which the Harmonic(i) shows maximum intensity (Remark:

**is an integer multiple of the basic frequency of the played pitch). According to the definition of dB as a logarithmic value of the intensity, we can write the following equation:**

*fi*Formula II.: ** virtual dB of Harmonic(fi) = log10(virtual intensity of Harmonic(fi)) **

Plotting the virtual dB-values of all Harmonics vs. the frequency will result in the “virtual power spectrum” of the played pitch. The factor “** -m **” defines the (negative) slope of the spectrum with increasing frequency;

**is a constant with the dimension dB. It is assumed that for wind instruments, the value for “**

*“b”***” is mainly determined by the blowing pressure of the player. An increase of the blowing pressure will result in an increase of “**

*–m***” and the slope of the virtual power spectrum will flatten with increasing frequency of the Harmonics. Consequently, reducing the blowing pressure will lead to a decrease of “**

*-m***” and the slope in the virtual power spectrum gets steeper with increasing frequency (Hz).**

*-m*b) As the playing system (player and instrument) is not an ideal acoustical system with several interactions (Ref. 6) it can be assumed that this playing system will show differences in resonance and radiation efficiency with changing frequency of played Harmonics. As a consequence, the measurable intensity of a** Harmonic(fi) **can be interpreted to be determined by the blowing pressure according to Formula I. and II. undergoing a modification by the “Resonance & Radiation” efficiency of the playing system at the frequency

**.**

*fi*c) So the measurable db-value of a** Harmonic(fi) **could be described with the following formula:

Formula III.: ** measurable dB of Harmonic(fi) = log10(virtual intensity of Harmonic(fi) * RR(fi)) **

** RR(fi) **is a factor representing the combined relative effects of the resonance and radiation efficiency of the playing system at the frequency

**.**

*fi*d) Although the measurable dB-values of the Harmonics of a played pitch (which form the power spectrum) will deviate from the virtual dB-values, it will be possible to estimate the logarithmic-function which determines the virtual dB-power spectrum through mathematical trend analysis. The basic idea of this assumption is that the deviations will be equalized by a mathematical trend analysis using a sufficient number of Harmonic signals.

It will be demonstrated in this publication that a) by using the assumptions above it is possible to calculate “Resonance & Radiation” spectra of playing systems and that b) such spectra can function to describe and interpret the sound characteristics a musician and his/her instrument (playing system) is generating.

## 2) Material & Methods

All recordings of sounds with various tenor saxophone players have been performed with the recording equipment and according to procedures already described elsewhere (Ref. 1, 2).

For recordings of tenor saxophone sounds played with different blowing pressure, the professional musicians have been asked to play a defined pitch with 4 different blowing pressures. The highest blowing pressure to be used should be the blowing pressure the musician is regularly using to play the certain pitch and is named “Regular”. The opposite point should be the lowest blowing pressure which still allows a stable sound production – named “Low”. The musician has been asked to play the same pitch with two additional but different blowing pressures. One pressure should be slightly higher than the “Low” level – named “Low+1” – and the other should be lower than the “Regular” level – named “Low+2”. The professional musicians showed a high degree of reproducibility concerning the blowing pressures. With all professional musicians playing various pitches, the acoustic energy is reliably increasing from “Low” to “Low+1” and further to “Low+2” with the highest acoustic energy at “Regular” (data not shown).

Recordings of the analyzed Cello-sound have been downloaded as wav-files from the internet (Ref 7).

Analysis of the recordings and the calculation of power spectra of these sounds through Fast Fourier transformation have been done as already published using the software Praat (Ref. 1, 2, 8). Further mathematical procedures and calculations have been done using the commercially available software Microsoft/Excel.

The following formula derived from the data analysis within Praat have been used to transfer dB-values from the power spectra into relative Amplitude values of a sine wave and vice versa:

“Amplitude of sine wave” = 7.574*10-[5] * e (“dB-value”*[0].[1155])

“dB-value” = 8.6562 * Ln(“amplitude sine wave”) + 82.132

As the amplitude is linearly correlated with the intensity of a sine wave, the calculation of the factor** RR(fi) **for Harmonics at the frequency

**has been done according to the following formula:**

*fi*Formula IV.: ** RR(fi **) =

*(virtual Amplitude of Harmonic(fi)) / (real Amplitude of Harmonic(fi))*** RR(fi) **is a dimensionless factor which can be plotted against the frequencies of each

**of an analyzed recording to obtain a “Resonance & Radiation” spectrum of this sound. For displaying a “Resonance & Radiation” spectrum, it is proposed to use a linear Hz-scale for the X-axis and a logarithmic scale for the Y-axis. The logarithmic Y-axis has the advantage of a better visualization of the impact of the related**

*Harmonic(fi)***values on the audible sound at the frequency**

*RR(fi)***as a**

*fi***-value of 0.25 has the same but reciprocal impact on the intensity of the**

*RR(fi)***as a**

*Harmonic(fi)***-value of 4.**

*RR(fi)***3) Results of measurements and calculations**

A typical power spectrum of a Single Tone (pitch) played by a professional tenor saxophone player showing only the maximum db-values of the Harmonics plotted against the frequency is displayed in Figure 1. Using the data points of the Harmonics of this power spectrum, a mathematical trend analysis to calculate a logarithmic function can be obtained (see dotted line in Figure 1). The parameters and the regression coefficient of the logarithmic function obtained are displayed in the right upper corner of Figure 1.

Abbildung in dieser Leseprobe nicht enthalten

Figure 1 Maximum dB-values of the Harmonics of pitch B (220Hz) played on a tenor saxophone (black dots) plotted vs. the frequency (Hz) at which the maximum dB-value of the Harmonic occurs. The Y-axis shows dB-values and X-axis shows the frequency in Hz.

By following the procedure as described in the chapters “Introduction” and “Material & Methods” a “Resonance & Radiation” spectrum (RR-spectrum) for this specific sound generated by the musician playing the pitch B on his/her tenor saxophone can be calculated (see Figure 2).

Abbildung in dieser Leseprobe nicht enthalten

Figure 2. “Resonance & Radiation” spectrum calculated according to the procedure described in Material & Methods and using the data of Figure 1. The Y-axis has a logarithmic scale and no dimension, as it represents the value of** RR(fi) **(see Introduction and Material & Methods). The X-axis is showing the frequency (Hz) on a linear scale.

Abbildung in dieser Leseprobe nicht enthalten

Figure 3: Logarithmic functions derived from power spectra of pitch A played on a tenor saxophone with 4 different blowing pressures (see Material & Methods). The player was asked to increase the blowing pressure from the lowest possible pressure (black line = Low) in four steps to his regular blowing pressure (red line –Regular). The blowing pressure for the blue line (Low+1) was higher than for the black line (Low) but lower than for the golden line (Low+2). The related logarithmic functions are plotted close to the respective curves (see also Table 1). The Y-axis shows dB-values and X-axis shows the frequency (Hz).

In general, it is necessary to deliver arguments for the assumptions made in “Introduction” that 1) a certain blowing pressure will generate a virtual power spectrum following a logarithmic function of the format** dB(fi) = -m*Ln(fi)+b **and that 2) an increasing or decreasing blowing pressure will have an impact on “

**”. Experiments with a professional saxophone player using his/her favorite set-up and playing certain pitches with different blowing pressures should deliver arguments pro or contra to this assumption. The logarithmic trend analysis of the power spectra of the pitch A played with four different blowing pressure (intensities) is shown in Figure 3. The formula for each trend analysis is also given in Figure 3, and it is obvious that the slope of the related logarithmic curves (defined by the factor “**

*-m***”) flattens with increasing blowing pressure. Similar results can be obtained with other pitches played – a summary of the related functions of the logarithmic trend analysis for selected pitches played with different blowing pressure is given in Table 1.**

*-m*Abbildung in dieser Leseprobe nicht enthalten

Table 1: List of the calculated logarithmic functions using the power spectra of the pitches A (196Hz), B (220Hz), Octave-A (395Hz) and Octave-B (440Hz) played on a tenor saxophone with 4 different blowing pressures from “lowest possible blowing pressure” (Low) to the “regularly used blowing pressure” (Regular). For further details, see Material & Methods.

It is obvious from the data listed in table 1 as well as from comparable data generated with other professional saxophone players (data not shown) that for a broad range of pitches (excluding flageolets) the slope of the calculated logarithmic function given by “** -m **” is flattening (= increasing value of “

**”) with increasing blowing pressure. This type of correlation could also be found for other wind instruments investigated like the Clarinet, the alto Flute, the Oboe, the trumpet and the trombone (data not shown). So, it could be concluded that for several wind instruments an increase in blowing pressure will result in a power spectrum with relatively increased intensities of the Harmonics at higher frequencies vs. the Harmonics at or close to the basic frequency of the played pitch. Although string instruments are not in the focus of this publication, a similar effect could be observed with Single Tones played by a professional violin player with increasing audible volume (data not shown). These findings deliver strong arguments to support the assumption made in “Introduction” that a logarithmic function calculated by mathematical trend analysis can be interpreted as a virtual power spectrum of the played Single tone describing relevant characteristics of the audible sound, which has already been proposed elsewhere (Ref. 9, 10).**

*-m*With the data obtained from the experiments with changed blowing pressures (logarithmic functions listed in Table 1) “Resonance & Radiation” spectra could be calculated for each played pitch and intensity. A comparison of some obtained “Radiation & Resonance” spectra of different pitches played with different blowing pressures are shown in Figures 4-6.

Abbildung in dieser Leseprobe nicht enthalten

Figure 4: Calculated “Resonance & Radiation” spectra of the pitch B (220Hz) played by a professional musician on a tenor saxophone with the lowest possible blowing pressure (B-Low; black line) and regular blowing pressure (B-Regular; red line). The Y-axis has a logarithmic scale and no dimension, as it represents the value of** RR(fi) **. The X-axis is showing the frequency (Hz) on a linear scale.

Abbildung in dieser Leseprobe nicht enthalten

Figure 5: Calculated “Resonance & Radiation” spectra of pitch B (220Hz; red line; “B-Regular”) and pitch A (196Hz; black line; “A-Regular”) played by the same professional musician on a tenor saxophone as in Figure 4 with regular blowing pressure. The Y-axis has a logarithmic scale and no dimension, as it represents the value of** RR(fi) **. The X-axis is showing the frequency (Hz) on a linear scale.

Abbildung in dieser Leseprobe nicht enthalten

Figure 6: Calculated “Resonance & Radiation” spectra of pitch A (196Hz; black line; “A-Regular”) and pitch octave-A (395Hz; red line; “OctA-Regular”) played by the same professional musician on a tenor saxophone as in Figure 4 -5 with regular blowing pressure. The Y-axis has a logarithmic scale and no dimension, as it represents the value of** RR(fi) **. The X-axis is showing the frequency (Hz) on a linear scale.

There are high similarities of the “Resonance & Radiation” spectra obtained with a playing system (player with his/her set-up and instrument) playing various pitches with different blowing pressures. By comparing “Resonance & Radiation” spectra of different playing systems (e.g., different players with their equipment), similarities can also be observed but show more variations. This might be caused by the individual differences among the players and their equipment (see spectra of different players playing the pitch B in Figure 2 and Figure 4).

The resolution of the calculated “Resonance & Radiation” spectra depend on the pitch played. A low pitch generates more data points to calculate a spectrum than a higher pitch, as the lower pitch generates more Harmonics (= data points) in the same frequency-range vs. a higher pitch. Although the Cello is a string instrument, it might be ideal to test the assumptions for the calculation of “Resonance & Radiation” spectra using a Cello for two reasons: 1) The Cello can generate relatively low pitches with a high number of Harmonics in the frequency-range 0-4000Hz which means a high number of data points in this range and b) the power spectra of some Single tones played on the Cello show some similarities with the power spectra of the same pitches played on the tenor saxophone (data not shown).

In Figure 7 and 8 calculated “Resonance & Radiation” spectra obtained with the C- and D-String of the Cello are shown.

Abbildung in dieser Leseprobe nicht enthalten

Figure 7: Calculated “Resonance & Radiation” spectra of pitch C (66Hz; black line; “C on C-String”) and pitch octave-C (132Hz; red line; “Oct-C on C-String) played by a professional musician on the C-string of a Cello. The Y-axis has a logarithmic scale and no dimension, as it represents the value of** RR(fi) **. The X-axis is showing the frequency range of 0-4000Hz on a linear scale.

Although the resolution of the spectra shown in Figure 7 differ by a factor of 2 (due to the frequency-difference in the played pitch), both calculated “Resonance & Radiation” spectra show high similarities. This indicates that the relevant parameters which define the “Resonance & Radiation” spectrum of a playing system are independent of the pitch played and might be more instrument related instead. This idea is confirmed with the data displayed in Figure 8 as two different pitches played on two different strings of the Cello lead to highly similar “Resonance & Radiation” spectra.

Abbildung in dieser Leseprobe nicht enthalten

Figure 8: Calculated “Resonance & Radiation” spectra of octave-C (132Hz; red line; “Oct-C on C-String”) played on the C-string and of D (147Hz; black line; “D on D-String) played on the D-string by a professional musician. The Y-axis has a logarithmic scale and no dimension, as it represents the value of** RR(fi) **. The X-axis is showing the frequency range of 0-4000Hz on a linear scale.

At this point, it could be of importance to compare the obtained and calculated data with data of the measured Admittance “Y” of the strings close to the bridge of the Cello (Remark: Admittance “Y” is the reciprocal value of the Impedance Z: Y = 1/Z). The detailed work of H.Fleischer (Ref, 10, 11) has shown that the Admittance-spectra of the 4 strings of the Cello show high similarities, although the pitches played on these strings span a large frequency range. As an example, the measured Admittance-spectra for the C-string and D-string of a Cello are shown in Figures 9a and 9b.

Research by Ziegenhals (Ref. 13) has determined an averaged “response curve” for 15 Celli in the frequency-range of 0-5000Hz and demonstrated that the Cello is having several response-maxima and minima in this range (Figure 10).

Abbildung in dieser Leseprobe nicht enthalten

Figure 9a: Admittance (Y) of the C-string of a Cello in the frequency-range 0-1000Hz measured in tangential direction of the string. Figure reproduced from H.Fleischer (Ref. 11).

Abbildung in dieser Leseprobe nicht enthalten

Figure 9b: Admittance (Y) of the D-string of a Cello in the frequency-range 0-1000Hz measured in tangential direction of the string. Figure reproduced from H.Fleischer (Ref. 11).

In acoustics, the value for the Admittance can be interpreted as an indication for the ability of an acoustic system to resonate at a certain frequency. The higher the Admittance of an acoustic system at a certain frequency, the higher is the ability of this system to resonate at that frequency. The Admittance spectra measured at all strings of the Cello display several peaks with three prominent areas at approx. a) 150-200Hz, b) 330-370Hz and c) 800-900Hz (see Figures 9a-b). It can be assumed that a Cello is able to shown strong resonance for signals in those frequency ranges. The three visible peaks in the “Resonance & Radiation” spectrum of the Cello measured using the C-string are located at approx. 330Hz, 520Hz and 720Hz and show some deviation from the frequency-areas with the highest Admittance. As the Admittance of a system correlates with the Resonance but not with the ability of an acoustic system to radiate acoustic energy, it could not be expected that the Admittance spectra of a Cello are highly similar to the “Resonance & Radiation” spectra. Further, it has to be considered that a) the resolution of the “Resonance & Radiation” spectrum is low compared to the Admittance-spectrum and b) the origin of the data and the used experimental set-up as well as the player and the used Cello are different. Despite these differences, the similarities between the Admittance-spectra and the “Resonance & Radiation” spectra are an indication that the Resonance spectrum of a Cello (qualitatively derived from the Admittance spectrum) is of relevance for the “Resonance & Radiation” spectrum. A pure Radiation spectrum of a Cello could theoretically be calculated as a difference of Resonance spectrum and “Resonance & Radiation” spectrum. But due to the low resolution of the latter and the differences in the set-up of both experiments, such a calculation is not feasible.

Although the “Resonance & Radiation” spectrum of the tenor saxophone in Figure 2 and the spectra shown in Figures 4-6 derive from 2 different professional tenor saxophone players, all spectra display four prominent maxima located at the same frequency-ranges: below 500Hz and around 1200Hz, 2000Hz and 3000Hz. This is an indication that the calculated “Resonance & Radiation” spectra reflect to a large extent the resonance and radiation characteristics of a tenor saxophone itself, combined with individual effects derived from the set-up and the musician.

Abbildung in dieser Leseprobe nicht enthalten

Figure 10: Averaged response defined by “acoustic level/force” (Y-axis in dB) vs. frequency (X-axis in Hz) for 15 Celli. Reproduced from Ziegenhals (Ref. 13)

This idea is confirmed by comparing the calculated “Resonance & Radiation” spectra of three different professional saxophone players using their own equipment and set-up while playing the same pitch (see Figure 11)

Abbildung in dieser Leseprobe nicht enthalten

Figure 11: Calculated “Resonance & Radiation” spectra of three different professional saxophone players (Player 1 – 3; blue-, red- and black lines) using their individual equipment and set-up while playing the pitch B (220Hz). The Y-axis has a logarithmic scale and no dimension, as it represents the value of** RR(fi) **. The X-axis is showing the frequency range of 0-4000Hz on a linear scale.

The “Resonance & Radiation” spectra of three tenor saxophone players (Figure 10) are similar concerning the number and frequency-position of maxima and minima, but also show significant differences in the relative values for** RR(fi) **at those maxima and minima. This could be interpreted as an argument that the frequency-dependent basic resonance and radiation characteristics of the played instrument (here tenor saxophone) are a key parameter to define the “Radiation & Resonance” spectrum of the playing system (= instrument & set-up & musician). The observable differences might be due to the individual parameters of player and set-up.

## 4) Discussion

For a simplified model to describe the characteristics of the sound of a Single tone (pitch) played by professional musicians on monophonic instruments, the following three parameters should be considered:

a) The playing intensity as the key factor determining the audible loudness of the played tone. (Remark: for wind instruments, the blowing pressure would be the key factor to increase the intensity)

b) The resonance-ability of the playing system at the frequencies of the Harmonics of the played pitch.

c) The ability of the playing system (mainly the instrument) to radiate acoustic energy at the frequencies of the Harmonics of the played pitch.

The model does not reflect that “individual variations of these parameters” of different playing systems (different musicians with their own equipment) may have an additional effect on the audible sound. So, the limitation of the model is that it will not sufficiently describe all audible differences in the sound of different musicians playing the same type but different instruments. To eliminate this “individual effect” as much as possible, only professional musicians have been chosen for the recordings, as it can be expected that professional musicians have a high capability to execute the experiments in a stable and reproducible way.

By knowing these three parameters (intensity, resonance; radiation) it should be possible to determine the general and relevant characteristics of a monophonic sound generated by a professional musician on a monophonic instrument.

Based on the data from the recordings with different saxophone players (playing various pitches with four different intensities = blowing pressures; see data in Figure 3 and Table 1 for one saxophone player) it can be concluded that the value of** “-m” **of the calculated logarithmic function (which describes the parameters of the virtual power spectrum; see Formula I.) is decreasing with increasing playing intensity. This statement is only valid for low to regular playing intensities, as playing intensities above an intensity-level defined by the players as “Regular” have not been investigated in this study.

This conclusion supports the assumptions a) that the playing intensity is not only the key factor determining the audible loudness of a monophonic sound but also determining the relative intensities of the Harmonics of this sound, b) that for each playing intensity the playing system is aiming to produce a sound where the intensities of the Harmonics (in dB) tend to define a logarithmic function of the frequency in Hz according to Formula I.(see “Introduction”) and c) that the measurable intensities of the Harmonics of a monophonic sound played with a certain intensity can be used to calculate a close estimate of the “virtual power spectrum” of this sound via mathematical trend analysis.

Interpreting the logarithmic function (calculated with the data of a measured power spectrum) as a good estimate of the “virtual power spectrum” would allow that this function can be used as reference to determine frequency dependent variations in the resonance- and radiation-ability of the playing system. Therefore, the ratio of the “measured amplitude of the Harmonic at the frequency** fi **” and the “amplitude of the Harmonic at

**calculated from the logarithmic function representing the virtual power spectrum” can be understood as the relative “Resonance & Radiation” factor at frequency**

*fi***(=**

*fi***) of the playing system. This conclusion offers the possibility to draw a “Resonance & Radiation” spectrum for an analyzed monophonic sound by plotting the values for**

*RR(fi)***of the Harmonics of this sound against their frequencies**

*RR(fi)***. This implies that the calculated “Resonance & Radiation” spectrum has an intrinsic error, resulting from the fact that the logarithmic function used for the calculations is just an estimate of the virtual power spectrum. Unfortunately, the magnitude of error can neither be estimated nor measured, which is an obvious limitation of the proposed model.**

*fi*Beside the limitations and simplifications of the model, the similarities of the “Resonance & Radiation” spectra obtained with different played pitches and different playing intensities (see Figures 4-8) indicate that for each playing system a unique “Resonance & Radiation” spectrum can be defined and is valid for a wide range of played pitches. The data in figure 11 demonstrate that similar playing systems (3 professional musicians playing their tenor saxophones) will result in similar “Resonance & Radiation” spectra. The playing systems compared in Figure 11 must be judged as similar as these are professional musicians, so they have some common technique of playing and all use a tenor saxophone. So, the three resulting “Resonance & Radiation” spectra (Figure 11) show common features of such a playing system (=professional musician playing a tenor saxophone) but also show differences which are due to the individuality of the players and differences in the equipment used. As it can be seen by comparing data from Cello and tenor saxophone (see Figures 7, 8, 11) it can be expected that different playing systems may generate „Resonance & Radiation“ spectra which show less similarities. In case of the Cello, it could further be demonstrated that measurable parameters which are of relevance for the sound like the Admittance (Figure 9a, 9b; Ref. 11) and the instrument-response (Figure 10; Ref. 13) show similarities as well as differences to the “Resonance & Radiation” spectra of this study. As there is a close correlation between the Admittance and the Resonance of a system and as the measured response curve is mainly due to the Resonance ability of the playing system, similarities of these parameters with** RR(fi) **could be expected. The observable differences may be due to the radiation ability of the playing system, which also forms the parameter

**but not the other two parameters.**

*RR(fi)*The data presented in this publication support the idea to use the “virtual power spectrum” and the “Resonance & Radiation” spectrum to describe the sound characteristics of a played pitch on a monophonic instrument – therefore supporting the introduced model. The mathematical trend analysis of the data of a power spectrum seems to generate a logarithmic function which is a good estimate of the key parameters of the “virtual power spectrum”. It has further been demonstrated that the presented model is easy to use and delivers valuable “Resonance & Radiation” spectra of a playing system. Further research might clarify whether this model can be transferred to other monophonic instruments as a reliable method to describe the sound characteristics of a playing system.

## 5) References

1) A.Rehm; L.Rehm; „Schallwellenanalyse des Sounds professioneller TenorsaxophonspielerInnen. Teil 1: Akustische Komponenten der Schallwelle, die vom Spieler generiert und reguliert werden und den Sound beeinflussen“; ISBN: 9783668712768; Deutsche Nationalbibliothek; http://dnb.d-nb.de

2) A. Rehm; „Schallwellenanalyse des Sounds professioneller TenorsaxophonspielerInnen. Teil 2: Methodik zur Bestimmung und Analyse von Formantenspektren und Formantenbändern aus mittels Fourieranalyse errechneten frequenzabhängigen Intensitätsspektren“; ISBN: 9783668777590; Deutsche Nationalbibliothek; http://dnb.d-nb.de

3) A.Rehm; „Schallwellenanalyse des Sounds professioneller TenorsaxophonspielerInnen. Teil 3; Vergleichende Analyse von Formanten gesprochener Vokale und Tenorsaxophontönen zur Bestimmung der Herkunft bzw. des Generierungsortes der Formantenbänder des Tenorsaxophonspiel im Frequenzbereich 0-10.000Hz“; ISBN 9783668815902; Deutsche Nationalbibliothek; http://dnb.d-nb.de

4) J.Chen, J.Smith, J.Wolfe; „Saxophone Acoustics: Introducing a Compendium of impedance and sound spectra”; Acoustics Australia; Vol.37; April 2009; pages 18-22.

5) Website der University of New South Wales / Australia: http://newt.phys.unsw.edu.au/jw/saxacoustics.html; downloads in January 2022

6) J.Wolfe, N.H.Fletcher, J.Smith; “ The interactions between wind instrument and their players”; Acta Acustica united with Acustica; 2015 Vol.101; pages: 211-233; doi: 10.3813/AAA.918820

7) Source of Cello sound files: Website: “https://cellomap.com/multiphonics-basics/“; files of Cello sounds (wav-format) downloaded in January 2022 from website.

8) References and details about software “Praat” see website: https://www.fon.hum.uva.nl/praat/ (status 1/2022)

9) M.Keidel, A.Rehm; „Schallwellenanalyse des Sounds professioneller TenorsaxophonspielerInnen Teil 7; Darstellung und Quantifizierung der Soundcharakteristika verschiedener Tenorsaxophone (Selmer & Keilwerth) und deren Bedeutung für den individuellen Sound von Saxophonspielern.“; ISBN: 9783346044570; Deutsche Nationalbibliothek; http://dnb.d-nb.de

10) A.Rehm, D.Gaebel, M.Keidel, T.Lakatos, C.Valk, S.Weber, L.Rehm; “Schallwellenanalyse des Sounds professioneller TenorsaxophonspielerInnen Teil 8; Methode zur Bestimmung von Spieler-typischen Soundspektren beim Saxophon“; ISBN: 9783346116826; Deutsche Nationalbibliothek; http://dnb.d-nb.de

11) H.Fleischer; “Admittanzmessungen an einem Cello“; Heft 03/09 der Reihe Beiträge zur Vibro-und Psychoakustik; 2009; ISSN 1430-936X.

12) H.Fleischer; „Vibro-acoustic measurements on the Violoncello”; Proceedings: Analysis and description of music instruments using engineering methods; Halle/Saale Germany, 12-13 may 2011; pp. 115-124.

13) G.Ziegenhals: „Subjektive und objektive Beurteilung von Musikinstrumenten. Eine Untersuchung anhand von Fallstudien“. Thesis TU Dresden 2010. Studientexte zur Sprachkommunikation Band 51 TUD press 2010 (ISBN 978-3-941298-71-2)

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