So called Multiphonic sounds of string instruments (cello, violin, piano) and wind instruments (trombone, flute, oboe, clarinet, saxophone) have been analyzed and their power spectra (Fast Fourier transformation of sound files) have been compared. The power spectra of Multiphonic sounds of wind instruments exhibit a structure with a) Harmonics of the two tones (=pitches) T1 and T2 with different basic frequencies (Hz) and b) Complex tones with frequencies defined according to the formula: “m*T1+n*T2 (m, n are integer numbers <0>; m*T1 and n*T2 represent frequencies “f1” and “f2” in Hz of the respective Harmonics of T1 and T2). Power spectra of so called Multiphonic sounds of string instruments do not contain Complex tones, but Harmonics of only one basic frequency. In contrast to a regular monophonic sound, the first Harmonics of this so called Multiphonic sounds are massively damped so that the listener cannot identify the basic pitch, but will recognize several higher Harmonics sounding in parallel. This mimics a Multiphonic sound although it is a monophonic sound. It is proposed to name such sounds: “multiphonic sounding Harmonics” in contrast to real Multiphonic sounds generated by wind instruments containing Complex tones. The detailed analysis of the intensities of the Harmonics as well as the frequencies and intensities of the Complex tones of Multiphonic sounds leads to the conclusion that a pair of Complex tones with frequencies “f1+f2” and “f2-f1” is a result of an interaction between one Harmonic of each basic pitch T1 and T2 with respective frequencies f1 and f2. Based on this conclusion, a mathematical model is established which proposes the mechanism of a specific coupling of two standing waves within a wind instrument as the basic process to generate the Complex tones. A simulation of a real Multiphonic sound of a wind instrument using the mathematical model results in an acceptable correlation (R2 = 0.5) of the simulated and the real intensities of the Complex tones.

## Content

1. Introduction - What are Multiphonic sound

2. Material & Metho

3. Differences in the spectral characteristics of Multiphonic sounds generated by string instruments vs. wind instrument

4. Analysis of spectral characteristics of Multiphonic sounds generated with woodwind-instrumen

5. Model of the “specific interaction of standing waves in woodwind-instruments” as basic principle for the generation of the spectral characteristics of Multiphonic soun

## Summary

So called Multiphonic sounds of string instruments (cello, violin, piano) and wind instruments (trombone, flute, oboe, clarinet, saxophone) have been analyzed and their power spectra (Fast Fourier transformation of sound files) have been compared. The power spectra of Multiphonic sounds of wind instruments exhibit a structure with a) Harmonics of the two tones (=pitches) T1 and T2 with different basic frequencies (Hz) and b) Complex tones with frequencies defined according to the formula: “m*T1+n*T2 (m, n are integer numbers <0>; m*T1 and n*T2 represent frequencies “f1” and “f2” in Hz of the respective Harmonics of T1 and T2). Power spectra of so called Multiphonic sounds of string instruments do not contain Complex tones, but Harmonics of only one basic frequency. In contrast to a regular monophonic sound, the first Harmonics of this so called Multiphonic sounds are massively damped so that the listener cannot identify the basic pitch, but will recognize several higher Harmonics sounding in parallel. This mimics a Multiphonic sound although it is a monophonic sound. It is proposed to name such sounds: “multiphonic sounding Harmonics” in contrast to real Multiphonic sounds generated by wind instruments containing Complex tones. The detailed analysis of the intensities of the Harmonics as well as the frequencies and intensities of the Complex tones of Multiphonic sounds leads to the conclusion that a pair of Complex tones with frequencies “f1+f2” and “f2-f1” is a result of an interaction between one Harmonic of each basic pitch T1 and T2 with respective frequencies f1 and f2. Based on this conclusion, a mathematical model is established which proposes the mechanism of a specific coupling of two standing waves within a wind instrument as the basic process to generate the Complex tones. A simulation of a real Multiphonic sound of a wind instrument using the mathematical model results in an acceptable correlation (R[2] = 0.5) of the simulated and the real intensities of the Complex tone

## 1. Introduction – What are Multiphonic sounds?

Multiphonic sounds are “Complex sounds” which are a) increasingly used in musical compositions and b) generated by musicians of flutes, brass-, string- and woodwind-instruments as additional sounds to the classical “Single note-sounds” (often named as “pitches” or “monophonic sounds”) which are regularly classified according to the C, D, E, F, G, A, B-nomenclature. The basic characteristic of a Multiphonic sound is that at least two “Single-notes” (pitches) with different frequencies are generated by the same instrument and are clearly audible at the same time (Ref. 1). For the piano a Multiphonic sound is not meant to be the playing of two different notes by pushing down two different keys as this is a regular sound of a piano. A Multiphonic piano sound is generated with only one key of the piano and additional manual manipulation of the related string by the player, so that the impression is generated that tones of different non-harmonic frequencies occur at once. Consequently, a sound generated on a string instrument with the bow exciting more than 1 string at the same time is not considered to be a Multiphonic sound. In the literature, you may find the wording of “double-fingered sound” for such a sound – but it should not be considered as a Multiphonic sound under this definitio

Generating a “Single-note sound” or “monophonic sound” with an instrument also generates additional acoustic waves of different frequencies (Ref.1). But in contrast to a Multiphonic sound these frequencies are integer multiples named “Harmonics” of the basic frequency which classifies the pitch of the monophonic sound. A cacophony of different sounds or a noise sound will also not be classified “Multiphonic sound” as these sounds do not exhibit an audible structure which allows to recognizing at least two different dominating pitches (single tones). In some Multiphonic sounds up to 4 different “Single tones” (pitches) are audible and can be clearly differentiate

A wide range of Multiphonic sounds have been reported for string instruments played with a bow (e.g. violin, cello; Ref.2), for the Piano as well as for brass instruments (e.g. trombone), for flutes and especially for woodwind instruments like the clarinet (Ref.3), oboe (Ref.4) and the saxophone (Ref.5

For all instruments which allow the generation of Multiphonic sounds, it has been reported by musicians that a) it needs additional training and practice to play Multiphonic sounds and b) generating stable Multiphonic sounds is by far more difficult than playing stable regular monophonic sounds. Musicians have developed and published “fingering tables” for several instruments like Cello, Flute, Saxophone and Clarinet which help to generate certain Multiphonic sounds – in most cases these fingerings differ significantly from the fingerings used for playing regular monophonic sound

A spectral analysis of Multiphonic sounds should give more insight into the complex structure of these sounds. It has been reported by Benade (Ref.1) that the spectral analysis (power-spectrum) of a typical Multiphonic sound generated by a woodwind instrument exhibits two signals of relatively high intensity at different frequencies (=basic pitches or tones T1 and T2) plus the related Harmonics of these two pitches. Beside the signals at the two main frequencies (which are classified as T1 and T and are representing the pitch determining basic frequencies in Hz) and their related Harmonics, additional distinct signals can be identified at frequencies which are not integer multiples of T1 or T2. The signals at those frequencies are called “Complex-tones” in contrast to the pitch determining frequencies and their integer multiples (Harmonics). The frequencies of the Complex-tones can be described according to the following them

Where m & n are integer numbers <0>; T1, T2 represent the pitch determining basic frequencies of the two basic tones of the Multiphonic soun

An example: If T1 represents the basic pitch: 100 Hz and T2 represents the basic pitch: 230 Hz the related Harmonics of the tones T1 and T2 have the frequencies: …etc. /*…*etc. and can be detected in the power-spectrum after spectral analysis of the Multiphonic sound using the “Fast Fourier Transformation” The remaining signals in the power-spectrum are Complex-tones with the following frequencies: …et

This finding by Benade (Ref.1) is valid for “stable Multiphonic sounds” generated by woodwind instruments and can easily be reproduced. Data presented in this publication (see next section) as well as published data (Ref.6) demonstrate that stable Multiphonic sounds generated with brass-instruments exhibit the same characteristic

Recent publications on Multiphonic sounds of string instruments played with a bow (e.g. cello) and of piano Multiphonics (Ref.7, 8, 9) presented power-spectra which differ significantly from the power-spectra of Multiphonic sounds generated with woodwind or brass instrument

In this publication a) different power-spectra of Multiphonic sounds generated by different instruments will be presented, compared and discussed and b) a simple mathematical model and a corresponding simulation of a Multiphonic sound will be presented which may help to understand the characteristics and parameter of the complex power spectra of Multiphonic sounds generated by wind instrument

## 2. Material & Method

All recordings of sounds with various tenor saxophone players have been performed with the recording equipment and according to procedures already described elsewhere (Ref. 10, 11, 12

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, 13). Further mathematical procedures and calculations have been done using the commercially available software Microsoft/Exce

The calculation of “Resonance & Radiation” spectra have been done according to a recent publication (Ref. 14

Multiphonic sounds analyzed in this study of the saxophone have been downloaded as wav-files from various websites (Ref. 15, 16, 17

Download of Clarinet Multiphonics have been done from the following referenced websites (Ref. 18, 19

Sounds from other instruments which are analyzed and discussed in this study have been downloaded from various sources (Ref. 20, 21, 22, 23, 24, 25, 26, 2

## 3. Differences in the spectral characteristics of Multiphonic sounds generated by string instruments vs. wind instruments.

For several string instruments played with a bow (e.g. violin, cello) as well as for the piano it has been reported that a variety of Multiphonic sounds can be generated (Ref. 22, 23, 24). In this publication we will have a closer look at the Cello Multiphonics as these findings can be transferred to other classical string instruments played with a bow. An analysis of Multiphonic sounds generated by the piano is more difficult as due to the construction of the piano the generated sound shows a strong intensity dynamic so that no stable Multiphonic sound (stable intensity with minor audible sound variation) can be generated as it is the case with string instruments played with a bow. A recent publication of Vesikkala from 2016 (Ref. 7) reports in detail a) how to generate Multiphonics with the piano and b) the spectral characteristics of these Multiphonic piano sound

As already mentioned, the generation of a Multiphonic piano sound requires a) the push of a certain key and b) the manual manipulation of the related string in parallel. An example of such a Multiphonic piano sound generated with the C1-string manipulated at a certain point of the string is given in Fig.

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Figure 1: Power-spectrum (Fast Fourier Transformation) of a Multiphonic generated with the C-string of a piano - reproduced from Vesikkala 2016, page 108 – see Ref

In this example the “manipulation point” is at 9.1% of the length of the C-string which refers to the length of a string of this type to produce the pitch D1. The two most prominent signals in the given power-spectrum are at 362Hz and 724Hz, which refer to the 10th and 20th Harmonic of the tone D1. Peaks of less intensity are grouped around this prominent peaks and a third group of peaks is also visible with the prominent peak at the 31st Harmonic of pitch D1. All clearly visible peaks in the power-spectrum show a difference of around 36Hz to the next peak, which reflects the frequency of pitch D

Based on the findings by Vesikkala (Ref. 7) the following statements can be made on Multiphonic piano sound

1) A Multiphonic sound generated by pushing one key of the piano and an additional manual manipulation of the related string generates a series of Harmonics with a certain basic frequency and a characteristic intensity patter for certain groups of the Harmonics

2) The basic frequency of the series of Harmonics depends on the point of manipulation at the related string used to produce the Multiphonic sound

3) Due to the manual manipulation, the signal of the basic frequency as well as other Harmonics are massively damped. As neither the 1.Harmonic nor the 2.Harmonic (which define the pitch of a tone) are audible due to the massive damping a listener will not recognize the real pitch of the sound – in the example given in Fig.1 the real pitch would be D1 at approx. 36HZ

4) A listener may recognize two dominating pitches (in the given example in Fig.1 at approx.362Hz and 724Hz which refer to the 10th and 20th Harmonic of D1) which could be interpreted as a 1.-and 2.Harmonic of a given harmonic sound (in the given example, the listener may identify 362Hz as the basic frequency). But the listener will also recognize a number of non-harmonic signals clustered around these two signals, and additional signals at even higher frequencies. This may lead to the impression of a sound with various pitches which are not harmonically related – so this sound might be interpreted by the listener as a “Multiphonic piano soun

Following the definition for a Multiphonic sound (see Introduction) a piano sound of such type cannot be considered as a Multiphonic sound. Referring to the “real basic frequency” (in the example in Fig. 1 the basic frequency is D1 at approx. 36Hz) all visible signals show frequencies which are integer multiples of the basic frequency and therefore have to be considered as real Harmonics of D1. The selective damping of certain Harmonics generated through the manual manipulation of the exited string generates a sound which in the ear of the listener may be interpreted as a real Multiphonic sound. But in terms of the definition it is not a real Multiphonic sound but a harmonic sound with selectively damped groups of Harmonic

Looking at publications and published sound examples for so called Multiphonic sounds played on string instruments with a bow (Ref. 24), similarities of these sounds and the so called Multiphonic piano sounds are getting obvious. In Figure 2 a typical power-spectrum of a “Multiphonic cello sound” is displayed. In this case, the C-string of the cello has been exited with a bow while a certain manipulation of the string at the expected location of a node of the standing wave related to the 7th Harmonic of the tone C has been performed by the player. Based on this manipulation, the 7th Harmonic of C2 becomes the dominant signal of this generated sound with a frequency of 460Hz reflecting the pitch Bb

As already seen in the power spectrum of the piano (see Figure 1; Ref.7) additional signals grouped around the highest peak could be observed in the power spectrum of this cello sound. The peaks of the highest intensity within these groups show frequencies which are related to the 13th, 20th, 26th, 33rd, 39th and 46th Harmonic of C2. The slight deviation from the expected peaks related to the 14th, 21st, 28th 35th 42nd and 49th Harmonics of C2 are caused by the inaccuracy of the manual manipulation, as the manipulation point is close to the node of the standing wave of the 7th Harmonic but slightly shifted towards the node of the standing wave of the 6th Harmonic of C

It can be easily demonstrated that the point or location on the C-string where the manual manipulation takes place determines those Harmonics of C2 which are massively damped. This influences the audible sound accordingly, and the respective power spectrum displays this effect. In Figure 3 the C-string of the cello is manipulated at a point on the C-string which is located in the middle of the expected nodes of the standing waves for the 5th and 6th Harmonic of C2. As expected, the dominating signals in the power spectrum of this sound are at the frequencies of the 5th, 6th and 11th Harmonic of C2 (see Figure 3). The distance in Hz between the visible peaks in the power spectrum are identical and have the value of the first Harmonic of C2 (basic frequency). A listener to this sound may recognize 3 different pitches at once which do not seem to have a harmonic relationship: a) 5th Harmonic of C2 at 327.5Hz (E4); b) 6th Harmonic of C2 at 393.6Hz (G4) and c) 11th Harmonic of C2 at 721Hz (F5/Gb5). Such audible characteristics may lead to the audible impression of a Multiphonic sound. Based on the definition for a Multiphonic sound as given above (see Introduction) such a sound may sound like a Multiphonic but in fact is a harmonic sound with the basic frequency of C2 with selected massive damping of certain Harmonics including the first and second Harmonic which regularly define the audible pitch. Analysis of so called Multiphonic sounds played on the violin exhibit the same characteristics as displayed for the cello. These data are not shown here in order to keep the length of this publication reasonabl

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Figure 2: Power spectrum (intensity in dB (Y-axis) vs. frequency in Hz (X-axis)) of a Cello sound generated with the C-string and a manual manipulation of the C-string around the expected node of the standing wave of the 7th Harmonic of the pitch C2. The highest peak of the first group of peaks has the frequency of the 7th-Harmonic of the pitch C2, the highest peak in the second group is related to the 13th Harmonic and the highest peak in the third group represents the 20th Harmonic of C2. It is obvious that standing waves of the Harmonics of C2 which have their related nodes far from the node of the 7th Harmonic are massively damped. The closer the nodes of the standing wave of the Harmonics of C2 are located to the node of the 7th Harmonic, the less is the damping of the standing wave and the stronger is the radiated signal displayed in the power spectru

Based on the presented results for the cello and the piano (plus the highly similar characteristics of Multiphonic sounds of cello and violin; Ref. 22, 23, 24) it can be assumed that string instruments played with a bow as well as pianos are able to generate sounds which may be interpreted by the listener as a Multiphonic sound since the listener may identify two or even more different non-harmonic pitches of comparable intensity. The detailed analysis of the power spectra of such sounds gives arguments for a different interpretation and understanding of this so called Multiphonic sounds. In fact only a single basic tone (basic frequency) with several Harmonics (multiples of the basic frequency) is generated but the pitch determining first and second Harmonics as well as other Harmonics are massively damped. So the listener will not be able to identify the real basic frequency of the sound but instead will recognize different signals of not damped Harmonics showing frequencies with obviously no harmonic relationshi

Reflecting the given definition for a Multiphonic sound (see Introduction) it might be reasonable to use the expression: “multiphonic sounding Harmonics” for the so called Multiphonic sounds generated with string instruments. In the following it will be shown that Multiphonic sounds generated with flutes, brass- and woodwind instruments are in line with the definition of a Multiphonic sound and have different spectral characteristics than the “multiphonic sounding Harmonics” of string instrument

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Figure3: Power spectrum of a Cello sound generated with the C-string and a manual manipulation of the C-string in the middle of the expected nodes of the standing waves for the 5th and 6th Harmonic of C2. The dominating signals in the power spectrum of this sound are at the frequencies of the 5th, 6th and 11th Harmonic of C

At this point, an additional aspect might be of interest. Beginners in playing a string instrument like a violin or a cello must practice intensively to develop their sound from a scratchy and dissonant sound to a solid harmonic sound. In comparing the power-spectrum of a scratchy beginner sound with a nice violin sound (played by a professional violinist) it can be demonstrated that the “beginner sound” is showing some characteristics of the power spectrum of a “multiphonic sounding Harmonic

As shown in Figure 4 the beginner sound shows the expected strong signal at the 1.Harmonic of the played pitch, but the next strong and therefore clearly audible signals are that of the 7th and 12th Harmonics with a group of further Harmonics surrounding these two – a characteristic feature also seen in power spectra of “multiphonic sounding Harmonics” (see Figure 1-3). In contrast, the 1st and 2nd Harmonics are the dominating signals in the power spectrum of the “nice” sound (played by a professional musician) with remaining Harmonics having less intensity. So the listener of the “nice” sound can easily identify the pitch as 1st and 2nd Harmonics are clearly dominating the sound and the remaining Harmonics just influence the sound impression but do not disturb the pitch impression. In the beginner sound the 1st, 7th and 12th Harmonics are clearly audible as they have relatively high intensity and so the listener may interpret these signals as 3 different pitches sounding in parallel – similar to the impression of listening to a “multiphonic sounding Harmonic

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Figure 4: Power spectra (intensity in dB vs. frequency in Hz) of the pitch D4 of a violin played by a beginner (beginner sound = red curve) or a professional musician (nice sound = black curve). The 7th and 12th Harmonic of high intensity in the beginner sound refer to the pitches C7 and A7. The frequency scale of the beginner sound has been shifted by -50Hz to allow a better comparison of the intensity (dB) of the signals at the same frequency (Hz

As already mentioned in the introduction, it has been reported that Multiphonic sounds generated by woodwind instruments show a different structure of the related power spectra (Ref. 28, 29, 30). Benade reports that in Multiphonics of woodwind instruments two different pitches (T1 and T2, with T1<T2 in Hz) which are not harmonically related dominate the audible sound (Ref. 1). The power spectra of those sounds are showing Harmonics related to T1 and T2, but also signals with frequencies not being integer multiples of T1 and T2. These signals are called “Complex-tones” and their frequencies can be calculated according to the formula mentioned in the Introduction (see above

In the Figures 5-9 power spectra of Multiphonic sounds generated with Trombone, Flute, Oboe, Clarinet and Saxophone demonstrate the more complex structure of the Multiphonic sounds generated with wind instruments vs. the “multiphonic sounding Harmonics” of string instruments. All Multiphonic sounds of wind instruments investigated in this study exhibit power spectra with a structure as described by Benade (see Introduction, Ref. 1

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Figure 5: Power spectrum of a Multiphonic sound generated with a Trombone. The two main pitches of this Multiphonic sound are Bb2 (T1) and F3 (T2). The Trombone player is singing the Bb2 pitch while he is playing the pitch F3 on the Trombone in parallel. Based on the chosen interval of a fifth between T1 and T2, the power spectrum of this Multiphonic sound mimics some features of a “multiphonic sounding Harmonic” of a string instrument. For further details see tex

It is important to mention that brass instrument players can generate the Multiphonic sounds differently as players of other wind instrument (Ref. 6). To generate a brass Multiphonic a certain pitch is played with the instrument while the player sings another pitch at the same time into the instrument. Especially famous jazz trombonists have been known for using Multiphonic sounds (e.g, Albert Mangelsdorff, Nils Landgreen). If the interval between the two different pitches T1 and T2 have been well selected and balanced correctly the frequencies of certain Complex tones and the regular Harmonics of T1 and T2 will overlap so that these pitches will be pronounced more than others and therefore three or more different frequencies (pitches) are audible at once. A power spectrum of such a well-balanced Trombone Multiphonic is shown in Figure 5. In this example, T1 and T2 have frequencies of 116.5Hz (pitch Bb2) and 176.5Hz (pitch F3). The well-balanced interval between these two frequencies is a fifth, which could be interpreted that both frequencies would have an imaginary root or basic frequency around 60Hz (Bb1) and have common Harmonics at around 350Hz (F4) and 700Hz (F5). Concerning the Complex tones, the power spectrum of this Multiphonic sound exhibits a strong signal around 410Hz which is a combination of the two Complex tones with frequencies defined by: a) 2*T1+T2 and b) 3*T2-T1. In addition, the Complex tone with the frequency at approx. 237Hz (2*T2-T1) is overlapping and supporting the 2.Harmonic of T1 at 233Hz. With these features based on the interval of a fifth between T1 and T2, the power spectrum of this Multiphonic sound mimics a harmonic sound with the basic frequency at around 60Hz (Bb1) but missing this basic tone. This is a similarity to the power spectra of “multiphonic sounding Harmonics” of string instruments. It has been impressively demonstrated by Velut et al. (Ref. 6) that if the intervals between T1 and T2 are not balanced in such a unique way, the power spectra will show additional clearly distinguishable peaks (related to Complex tones) with frequencies being in line with calculations using the formula mentioned in the Introduction. It can therefore be concluded that Trombone Multiphonics generated by singing and playing different pitches in parallel are matching the definition of a Multiphonic sound given in the introduction, although their power spectrum may look under certain circumstances (certain intervals between T1 and T2) similar to the power spectrum of a “multiphonic sounding Harmonic” of a string instrumen

For instruments like the Flute or woodwind instruments like the Oboe, Clarinet (Ref. 31) or Saxophone the player must follow a different procedure in order to create a stable Multiphonic sound than with brass instruments like Trombone or Trumpet. In most cases, it is relevant to have a special fingering and some training to create stable Multiphonic sounds in flutes and woodwind instruments. The two basic tones of such a Multiphonic sound are generated within the instrument due to the right blowing technique of the player and a fingering which allows a periodic self-oscillation (standing wave) of at least two different frequencies within the tube of the instrument in parallel. The power spectra of such Multiphonic sounds regularly show two prominent signals at frequencies of no harmonic relationship (representing the basic pitches T1 and T2) plus additional peaks of lower intensity (see Figure 6a as an example for a Multiphonic sound generated with a Flute). These signals are either signals representing Harmonics of T1 or T2 or are signals of the “Complex tones” (see also Introduction

In Figure 6b, the clearly distinguishable peaks of Figure 6a have been plotted and marked in a way so that their relationship with T1 or T2 gets obvious. Black bubbles belong to the Harmonics of T1, red bubbles represent the Harmonics of T2 and blue bubbles are signals from “Complex tones”. It is important to recognize that the Harmonics of each main frequency (T1 or T2) show a similar intensity-decrease with increasing frequency, as this can be observed for the Harmonics of regular monophonic sounds (Ref. 1, 11, 12, 14). Such a significant decrease of intensity with increasing frequency cannot be observed for the Complex tones of a Multiphonic sound (see also Figures 8-1

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Figure 6a – Power spectrum of a Multiphonic sound generated with a Flute. The two peaks of the highest intensity represent the signals T1 (416Hz; pitch Ab4) and T2 (495Hz; pitch B4). The remaining clearly distinguishable peaks are either Harmonics of T1 or T2 or are “Complex tones” with frequencies according to the formula : Frequency (Hz) of Complex-tone (i) = n* T2 + m*T1 (see details in Introduction

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Figure 6b – Data of the power spectrum of the Flute Multiphonic sound from Figure 6a. The large red and black bubbles represent the peaks of the main two pitches (T1=black pitch: Ab4; T2=red pitch: B4). The smaller red bubbles represent the peaks of the Harmonics of T2; the smaller black bubbles represent the Harmonics of T1. The blue bubbles represent the peaks of the “Complex tones

The Oboe is known to generate a sound with overtones (Harmonics of higher frequency) of higher intensity vs. the Flute. So it could be expected that the Complex tones of an Oboe Multiphonic sound are more pronounced. Figure 7 confirms this assumption by displaying the peaks of a power spectrum of a typical Oboe Multiphonic soun

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Figure 7 – Peaks of the power spectrum of a typical Multiphonic sound from an Oboe. The large red and black bubbles represent the peaks of the main two pitches (T1=black 1077Hz; T2=red 1527Hz). The smaller red bubbles represent the peaks of the Harmonics of T2; the smaller black bubbles represent the Harmonics of T1. The blue bubbles represent the peaks of the “Complex tones

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Figure 8 – Peaks of the power spectrum of a typical Multiphonic sound from a Clarinet. The large red and black bubbles represent the peaks of the main two pitches (T1=black 715Hz; T2=red 1039Hz). The smaller red bubbles represent the peaks of the Harmonics of T2; the smaller black bubbles represent the Harmonics of T1. The blue bubbles represent the peaks of the “Complex tones

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Figure 9 – Peaks of the power spectrum of a typical Multiphonic sound from a Tenor Saxophone. The large red and black bubbles represent the peaks of the main two pitches (T1=black 191Hz; T2=red 598Hz). The small red bubble represents the peak of the 2nd Harmonic of T2; the smaller black bubbles represent the Harmonics of T1. The blue bubbles represent the peaks of the “Complex tones

As already demonstrated in the power spectrum of the Flute Multiphonic sound (see Figure 6b) the Harmonics of T1 and T2 show reduced intensities (dB-values) with increasing frequency (Hz). In contrast, the intensities of the Complex tones do not show such a trend. In fact, there is no obvious correlation between intensity and frequency of the Complex tone

The power spectrum of a Clarinet Multiphonic sound show the same characteristics concerning the Harmonics and the Complex tones, although the intensity of the signals at the basic frequencies of T1 (715Hz) and T2 (1039Hz) show large differences. It is further interesting to note that the Complex tones with the frequencies a) T2-T1 (325Hz) and b) 2*T2-T1 (1366Hz) show a slightly higher dB-value than the main peak of T1. Again, there is no obvious correlation of intensity and frequency of the Complex tone

As example for a Multiphonic sound generated by a Saxophone, a corresponding power spectrum is displayed in Figure

As conclusions of the analysis of the power spectra of the “so called” Multiphonic sounds generated by string-, brass- and woodwind instruments the following statements can be mad

1) The so called Multiphonic sounds generated with string instruments, differ to Multiphonic sounds of wind instruments in such a way that these sounds consist of Harmonics of one single basic pitch only but with massive damping of some Harmonics and/or amplifying of other Harmonics so that the audible impression is that of a Multiphonic sound. Therefore, it is proposed to name such sounds: “Multiphonic sounding Harmonics” to separate from real “Multiphonic” sounds generated with wind instruments

2) The Harmonics of the basic two pitches T1 and T2 of a real Multiphonic sound show an inverse correlation of intensity (dB) and frequency (Hz) as it can be observed for Harmonics of monophonic sounds. No such correlation could be observed for the Complex tones of Multiphonic sounds

3) If the intensities (dB) of the two basic pitches of a Multiphonic sound (T1 and T2) are in the same range, the Complex tones have significantly lower intensities than the first Harmonics of T1 or T2. In case the first Harmonic of one basic pitch shows a significantly lower intensity than the first Harmonic of the other basic pitch, it can be observed that some Complex tones have an even slightly higher intensity than the first Harmonic of the basic pitch of low intensity (see Figure 8

## 4. Analysis of spectral characteristics of Multiphonic sounds generated with woodwind-instruments

It has been reported previously that professional saxophone players create their specific sound by expressing certain formants which can be identified in the power spectra of the played pitch (monophonic sound) and which define the player`s typical audible sound (Ref. 32, 33). Differences in the sound of saxophone players can be attributed to a large extent to the differences in the expression of formants when playing monophonic sounds. Based on power spectra of monophonic sounds, “formant spectra” could be calculated which exhibit various formant signals in the audible frequency range (Ref.11). It has been demonstrated that a saxophone player is keeping his “formant spectrum” fairly stable for a wide range of played pitches (Ref. 32). Recently a “Resonance& Radiation” spectrum (RR-spectrum) has been proposed which could be calculated using intensity-data of the Harmonics of a monophonic sound Ref. 14). This RR-spectrum is interpreted as a sound determining parameter of a playing system consisting of the musician and his/her instrument. As a RR-spectrum of a Tenor-saxophone (Figure 10) displays significant changes of the “Resonance & Radiation” factor (RR-factor) with changing frequency (Hz) it can be concluded that in case of a Multiphonic sound, not only the intensities of the Harmonics of the basic frequencies T1 and T2 are influenced by the RR-factor, but also the intensities of the Complex tones. So it is of importance to analyze not only the frequencies of the Complex tones within a Multiphonic sound but also their intensities in relation to the intensities of the Harmonics of T1 and T

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Figure 10: Calculated “Resonance & Radiation” spectrum of a Tenor saxophone – reproduced from Rehm (Ref. 14

It is obvious from the presented power spectra (see Figures 6b, 7, 8, 9) that the intensities (dB) of the Complex tones do not show a correlation with the frequency (Hz). This is clearly demonstrated in Figures 11 and 12 which show intensities (dB) of the Complex tones of Multiphonics generated with the Tenor saxophone (Fig.11) and the Oboe (Fig.12). There are no indications that there is any kind of linear or logarithmic correlation of the intensity and the frequency of Complex tones of a Multiphonic sound. As the frequencies of Complex tones follow the equation: n*T2+m*T1 (where “n” and “m” are integer numbers <0> and T1, T2 are representing the pitch determining basic frequencies (Hz) with T2>T1) it can be assumed that an interaction of the acoustic waves of the Harmonics of T1 and T2 create the Complex tones. It could be expected that e.g. the 3rd Harmonic of T1 and the 1st Harmonic of T2 create the Complex tones with frequencies of “1*T2+3*T1” and “3*T1-1*T2”. Looking at another example: the 2nd Harmonic of T1 and the 3rd Harmonic of T2 may create the Complex tones with frequencies at “3*T2-2*T1” and “3*T2+2*T1”. If the intensity of the Complex tones depends on the intensity of the two Harmonics of T1 and T2 creating such Complex tones, it should be possible to find a correlation of these parameters. A first indication of a correlation should be seen by plotting the intensity (dB) of the Complex tones vs. the sum of the factors ∣n∣ and ∣m∣ (“n+ m”) given by the equations to define the frequencies of the Complex tones (see above and Introduction). As in general the intensities of Harmonics of T1 and T2 decrease with increasing frequency (see Figures 6-9; Ref. 13), it could be expected that Complex tones created by certain Harmonics of T1 and T2 should have a decreasing intensity with an increasing value for the sum of ∣n∣ and ∣m∣ (“n+m”

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Figure 11- dB-values (Y-axis) of the Complex Tones of a Multiphonic sound generated with a Tenor saxophone are plotted against the frequencies of the Complex tones. The solid line represents the logarithmic regression function, the dotted line represents the linear regression function. The regression functions with the respective regression coefficients (R[2]) are given in the grap

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Figure 12- dB-values (Y-axis) of the Complex Tones of a Multiphonic sound generated with an Oboe are plotted against the frequencies of the Complex tones. The solid line represents the logarithmic regression function, the dotted line represents the linear regression function. The regression functions with the respective regression coefficients (R[2]) are given in the grap

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Figure 13: dB-values (Y-axis) of the Complex Tones of the Multiphonic sound displayed in Figure 11 plotted against the absolute values of the sum of ∣n∣ and ∣m∣ (“n+m”) from the equation defining the frequencies of the Complex tones. The solid line represents the logarithmic regression function, the dotted line represents the linear regression function. The regression functions with the respective regression coefficients (R[2]) are given in the grap

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Figure 14: dB-values (Y-axis) of the Complex Tones of the Multiphonic sound displayed in Figure 12 plotted against the absolute values of the sum of ∣n∣ and ∣m∣ (“n+m”) from the equation defining the frequencies of the Complex tones. The solid line represents the logarithmic regression function, the dotted line represents the linear regression function. The regression functions with the respective regression coefficients (R[2]) are given in the grap

Figure 13 and 14 show for the Tenor saxophone and the Oboe a plot of the intensities (dB) of the Complex tones vs. the respective absolute values of the sum of ∣n∣ and ∣m∣ (“n+m”) and the related linear and logarithmic regression-functions. The regression coefficients (R[2]) of these linear and logarithmic functions (range of 0.44-0.51, see Fig. 13, 14) are significantly higher than those of the linear and logarithmic functions of intensity vs. frequency (range of 0.016-0.059; see Fig. 11, 12). This can be interpreted as an indication that an interaction of certain Harmonics of T1 and T2 are responsible for the creation of Complex tones. It can be assumed that not only the frequencies of the created Complex tones are determined by the frequencies of the respective Harmonics, but also the intensities of the Complex tones seem to be determined to some extent by the intensities of the respective interacting Harmonic

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Figure 15: Power spectra showing the Harmonics of basic frequencies T1 (418Hz, black dots) and T2 (552Hz; red dots) of the Multiphonic sound: “2-mono-v3” downloaded from Ref. 16. The dotted lines represent the corresponding logarithmic regressions functions. The regression function and the respective R[2] value for T1 is printed in black letters, for T2 it is printed in red letter

Most of the Multiphonic sounds of woodwind instruments examined in this study show a certain characteristic concerning the Harmonics of the basic frequencies T1 and T2. The example displayed in Figure 15 shows the power spectrum of the Harmonics of T1 (basic frequency: 418Hz) and T2 (basic frequency: 552Hz) of a Multiphonic sound generated with a Tenor saxophone. The Complex tones of this Multiphonic are not displayed. It is obvious that the intensity of the 1st Harmonics of T1 and T2 deviate only slightly and the logarithmic regression curves are similar, which means that the sound of the Harmonics of T1 and T2 as part of the overall sound of the Multiphonic have nearly the same quality and intensity. It could further by recognized that the value for the factor “-m” of the logarithmic regression functions of the Harmonics of T1 and T2 of a Multiphonic sound (format of the regression function: dB = -m*ln(fi) +b) was in all cases in the range of or even below “-m” values of regression functions obtained with monophonic sounds of very low intensity (Ref. 13). This is in contrast to the impression of the musicians generating the Multiphonic sounds, as they state to use “mostly medium to regular and in some cases even higher blowing pressure to produce Multiphonics sounds”. With regular blowing pressure the “-m” values of the logarithmic regression function of the monophonic sounds generated with a Tenor saxophone are in the range of “-5” to “-14” (depending on the pitch of the monophonic sound played) and even at very low blowing pressure the values of “-m” are in the range of “-20” to “-24” (Ref. 14). So it could be assumed that a significant amount of the blowing energy does not end up in the Harmonics of the two basic frequencies of a Multiphonic sound, but may contribute to the intensities of the Complex tone

For the Multiphonic sound presented in Figure 15, a “Resonance & Radiation” spectrum can be calculated according to the method described by Rehm (Ref. 14) by using the data of the Harmonics of T1 and T2 (see Figure 16). This “Resonance& Radiation” spectrum shows similar characteristics concerning the position of minima and maxima as the “Resonance & Radiation” spectra calculated from monophonic sounds generated on a Tenor saxophone (compare Fig.10). So it seems that the intensities of the Harmonics of the two frequencies of a Multiphonic sound (T1; T2) are influenced in the same way by the “Resonance & Radiation” characteristics of a playing system (in this case a Tenor saxophone) as the Harmonics of a monophonic soun

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Figure 16: “Resonance&Radiation” spectrum of Multiphonic: “2-mono-v3” (see Harmonics in Figure 15) calculated according to Rehm (Ref 13) with data of Harmonics of T1 (black dots) and of Harmonics of T2 (red dots

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Figure 17: dB values of a) the Harmonics of T1 (418Hz) of Multiphonic: “2-mono-v3” (see Figure 15), = black curve and of b) the Complex tones with frequencies (Hz) given by the function: “n*T1+T2”; = blue & brown curves (n = integer number>0

Although the data presented in Figures 13 and 14 indicate that the Complex tones with frequencies of “x*T1+y*T2” and “x*T1-y*T2” (where x and y are integer numbers >0 and x*T1 > y*T2) are resulting from the interaction of the specific Harmonics of T1 and T2 with frequencies of x*T1 and y*T2, it is necessary to find further arguments to substantiate this assumption. The data presented in Figure 17 and 18 may deliver such arguments. In both Figures, the dB-values of the Harmonics of the basic frequencies T1 (Figure 17) and T2 (Figure 18) are compared with the dB-values of the Complex tones with the frequencies a) n*T1+T2 and b) n*T1-T2 (see Figure 17; n = integer number >0) and with the dB-values of the Complex tones with the frequencies a) n*T2+T1 + b) n*T2-T1 (see Figure 18). In Figure 17 the displayed Complex tones might result from an interaction of the Harmonics of T1 with the 1st Harmonic of T2, whereas in Figure 18 the displayed Complex tones might result from an interaction of the Harmonics of T2 with the 1st Harmonic of T1. In both cases, the 1st Harmonic of T1 or T2 would represent a constant factor which interacts with several Harmonics of the other basic frequency of the Multiphonic sound. In case of “n*T1+T2” the frequencies of the resulting complex tones would be shifted towards higher values vs. the respective Harmonic of T1 and at “n*T1-T2” the frequency of the resulting Complex tones are shifted to lower frequencies compared to the respective Harmonic of T1. So even if the intensities (dB) of the Complex tones are influenced by the intensities of the respective Harmonics of T1, a shift of the resulting curves of the Complex tones to either higher (n*T1+T2) or lower frequencies (n*T1-T2) vs. the curve of the Harmonic of T1 should be expected. Such a shift of the curves of the Complex tones can be identified in Figure 17. With T1 as the constant factor (Figure 18) a clear shift of the curves of the Complex tones is only partly visible. The similarity of the curves of the Complex tones vs. the Harmonics of T1 and T2 might be understood as an indication that the intensities of the Harmonics of T1 and T2 with the frequencies (Hz) at f1 and f2 influence the intensities of the Complex tones with frequencies at f1+f2 and f2-f1 (f2 > f1). The sum of the presented data support the conclusion that the Complex tones with frequencies of f1+f2 and f2-f1 are resulting from an interaction of two acoustic waves with the frequencies at f1 and f

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Figure 18: dB values of a) the Harmonics of T2 (552Hz) of Multiphonic: “2-mono-v3” (see Figure 15), = red curve and of b) the Complex tones with frequencies (Hz) given by the function: “n*T2+T1”; = blue & brown curves (n = integer number >0

## 5. Model of the “specific interaction of standing waves in woodwind-instruments” as basic principle for the generation of the spectral characteristics of Multiphonic sounds

The following assumptions are made based on published data (Ref. 28, 29, 30, 31, 34, 35) and the data presented in this stud

1) In case of a Multiphonic sound with two distinct main frequencies (T1 and T2) being generated by a musician playing a woodwind instrument (playing system), the two basic tones with their related Harmonics are subject to the same conditions as the Harmonics of monophonic sounds generated with the same playing system

2) The Complex tones with the frequencies “m*T1+n*T2” (m and n being integer numbers >1) are the result of a specific interaction of standing waves generated within the instrument with frequencies of m*T1 (mth-Harmonic of T1) and n*T2 (nth Harmonic of T2). This specific interaction of two standing waves with the result of two Complex tones is named: “Coupling of standing waves

3) A stable Multiphonic sound represents a steady state where all physical processes are in an equilibrium. Changing one factor will generate a transition phase with three possible endpoints: 1) no stable sound can be generated; b) a stable monophonic sound will be generated b) a new stable Multiphonic sound will be generated which differs in multiple acoustic parameters from the former Multiphonic sound

4) A significant part of the blowing pressure (energy input) does not contribute to the audible and measurable intensities of the Harmonics of T1 and T2, but to the intensities of the Complex tones. Thus, would lead to a simple equatio

*Input-energy = (Energy Harmonics T1/T2) + (Energy Complex tones) + (Energy-loss)
*

*5) We can further assume that there is an energetic equilibrium between the intensities ( Int) of the audible Harmonics of T1 and T2 with frequencies f1 and f2 and the intensities of the Complex tones (Ct) with the frequencies (f1+f2) and (f2-f1); with f2>f1 and m*T1=f1; n*T2=f2*

6) A coupling of two overlapping standing waves of different frequency requires the same energy of each of the standing waves. So we can define the following equatio

Using these assumptions, the following equations are proposed as base for a simple mathematical model, to simulate the acoustic characteristics of a Multiphonic soun

1) For a monophonic sound the audible intensity of a Harmonic at frequency fi can be written a radAmpaw(fi) is “the radiated (and therefore audible) Amplitude of an acoustic wave at frequency fi in Hz”, Ampsw(fi) is “the Amplitude of the standing wave at frequency fi” and Rad(fi) is the “Radiation factor of the playing system at frequency fi

2) Based on the assumption 6 above, we can write the following equation for the coupling of standing waves of the basic tones T1 and T2 of a Multiphonic soun AmpswT1(fx)coupled is the “Amplitude of the standing wave of T1 at frequency fx coupled with a standing wave of T2 at frequency fy”; AmpswT2(fy)coupled is the “Amplitude of the standing wave of T2 at frequency fy coupled with a standing wave of T1 at frequency fx

3) Combining equations 1 and 2 and assuming that the energy of the coupled standing waves of T1 and T2 is equally distributed to the two Complex tones we can writ radAmpCT(fx+fy) is the “radiated Amplitude of the Complex tone at frequency fx+fy”; Rad(fx+fy) is “the Radiation factor of the playing system at the frequency fx+fy Consequently, we can further defin radAmpCT(fy-fx) is the “radiated Amplitude of the Complex tone at frequency fy-fx” with fy>fx; Rad(fy-fx) is the “Radiation factor of the playing system at the frequency fy-fx

4) Using assumption 4 the following can be define AmpswT1(fi)coupled is the “part of the Amplitude of the standing wave of T1 at frequency fi which is coupled to Harmonics of T2”; AmpswT1(fi)real is the “part of the Amplitude of the standing wave of T1 at frequency (fi) which feeds the radiation of an acoustic wave at frequency fi”; AmpswT1(fi)generated is the “Amplitude of a standing wave of T1 at frequency fi as a result of the input energy = blowing pressure Consequently, we can write the following with respect to T

5) From equation 4 the following definition of a “Coupling efficiency (CoEff) of a standing wave of T1 or T2 at frequency fi” can be derive

The experimental set-up in this study does not allow measuring or calculating values for CoEff of the standing waves T1 and T2 and their Harmonics. The differences in the values of “-m” of the logarithmic regression curves for the Harmonics of T1 and T2 of a Multiphonic sound (range of “-m” is -20 to -30, see Figure 15) vs. the “-m”-values of a monophonic sound (range of “-m” is -5 to -14 at regular blowing pressure, Ref: 14) indicate that with increasing frequency a larger portion of the input energy (blowing pressure) is transferred to the Complex tones and the values for CoEff-T1(fi) and CoEff-T2(Fi) are increasing. At the hypothetical point of CoEff(fi) = 1, the entire energy of the standing wave at the frequency fi would be transferred to the Complex tones and no acoustic wave at the frequency fi will be radiated. At this frequency, equation 4 (see above) would change t

Based on the detailed analysis of the frequencies of the Complex tone, we can conclude that “AmpswT1(fi)coupled” forms a coupling with all Harmonics of the standing waves of T2. For the purpose of simplicity and as we have no other indication, we can assume that the energy of AmpswT1(fi)coupled is equally distributed among the Harmonics of the standing waves of T

Using the assumptions and related equations above, we can set up a mathematical model which enables us to simulate certain acoustic parameters of a Multiphonic sound generated with a woodwind instrument. The simulation presented in the following has basic frequencies of: T1=418Hz and T2=552Hz in order to simulate the Multiphonic sound presented in Figures 15-18. For simplicity, the number of Harmonics of T1 and T2 are restricted to 10. For the 10th Harmonic of T1, the following definition is made concerning the Coupling efficiency: CoEffT1(4180Hz)=0,95. This simplification still generates in total 200 Complex tones vs. 20 Harmonics of T1 and T

As we assume that the intensity of the generated Complex tones is partly defined by the Radiation factor at the respective frequencies of the Complex tone (see equations under point 3 above) it would be necessary to know the Radiation-factor (Rad) of the playing system at frequencies “fx+fy” and “fy-fx” in order to simulate the radiated intensities of Ct(fx+fy) and Ct(fy-fx). Unfortunately the experimental set-up does not allow measuring**Rad**(fi) directly but the intensities of the Complex tones of n*T1+T2 and n*T2+T1 displayed in Figures 17 and 18 might be used to get an idea of the Radiation-factors at the frequencies “fx+fy” and “fy-fx”. Taking the data of the Complex tones displayed in Figure 17 and 18 and using the procedure to calculate a “Resonance & Radiation” spectrum (Ref 14) might deliver the Radiation-factors influencing the intensities of the Complex tones. Data from Figure 17 deliver calculated Radiation-factors displayed in Figure 19; Figure 20 shows the results of the calculation using the data from Figure 1

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Figure 19: Relative Radiation factors calculated according to Rehm (Ref. 14) using the measured intensities of the Complex tones displayed in Figure 17 plotted against the frequencies of the Complex tones (X-axis). The red dots represent the Complex tones with frequencies “n*T1-T2” and the black dots represent Complex tones with frequencies “n*T1+T2”. The Y-axis has a logarithmic scal

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Figure 20: Relative Radiation factors calculated according to Rehm (Ref. 14) using the measured intensities of the Complex tones displayed in Figure 18 plotted against the frequencies of the Complex tones (X-axis). The red dots represent the Complex tones with frequencies “n*T2-T1” and the black dots represent Complex tones with frequencies “n*T2+T1”. The Y-axis has a logarithmic scal

The results of both calculations (Figures 19 and 20) show some similarities but also significant differences. The data in Figure 19 exhibit some typical characteristics of the “Resonance & Radiation” spectrum (see Figure 16) which would be expected as the Radiation-factor is supposed to be an integral part of the combined “Radiation & Resonance” spectrum. The data in Figure 20 show some significant differences to the data displayed in Figure 19 and to the “Radiation& Resonance” spectrum as well. This leads to the conclusion that the inconsistency of the calculated Radiation-factors does not allow defining a pure “Radiation”-spectrum at this stag

The experimentally determined “Resonance & Radiation” spectrum (see Figure 16) delivers a combined relative value of the effects of Resonance and Radiation of the playing system, but not for the Radiation alone. Using the data of the “Resonance & Radiation” spectrum to simulate the intensities of the Complex tones generates a systematical error of unknown magnitude. As this issue cannot be solved at this stage, we compare two types of simulations: 1) the relative Radiation-factor for the Complex tones at any frequency is defined as 1.0 and 2) the interpolated values of the “Resonance& Radiation” spectrum experimentally determined (Figure 16) are used as Radiation-factors for the Complex tones. Although the data of both simulations will contain systematical errors of unknown magnitude, it might still be helpful to compare the data of these two scenarios with the measured data of the Multiphonic sound in order to get an idea of the general validity of the basic assumptions forming the mathematics of the simulatio

Figure 21 shows some data of the simulation with T1=418Hz and T2=552Hz. The simulated radiated Harmonics of T1 and T2 are highly similar to the respective measured Harmonics of T1 and T2 of the Multiphonic sound displayed in Figure 15. The values for “-m” of the logarithmic regression functions of the simulated Harmonics of T1 and T1 are close to the “-m” values of the original Multiphonic sound indicating that the intensity of the simulated Harmonics and the radiated Harmonics of T1 and T2 are highly similar. Also shown in Figure 21 are those intensities (dB-values) of the Harmonics of the standing waves of T1 and T2, which (according to the simulation) undergo a coupling to form the Complex tones. For the simulation of the “coupled dB-values”, logarithmic functions with a value for “-m”=-10 and an identical dB-value at 418Hz and 552Hz have been defined. Further, the hypothetical Radiation factors for the various frequencies of the Harmonics of these standing waves have been set to the value of 1.0 for demonstration purpose

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Figure 21: Simulated dB-values (Y-axis) of the radiated Harmonics of T1 (418Hz; black solid curve) and T2 (552Hz, red solid curve) and of the hypothetical dB-values of the Harmonics of the coupled standing waves of T1 (dashed red line) and T2 (dashed red line) plotted against the frequency (Hz). The logarithmic regression function and related R[2]-values of the radiated Harmonics are printed close to the regression curves (black dotted line and letters = T1; red dotted line and letters = T2). See text for details about the parameters of the simulatio

Figure 22 shows the simulation of the power spectra of the Harmonics of T1 and T2 if no coupling between the Harmonics takes place and all energy is radiated instead. The resulting hypothetical power spectra show typical characteristics (including the values for “-m” of the logarithmic regression functions) of power spectra of monophonic sounds generated with a Tenor saxophone at regular to medium blowing pressure (Ref. 14

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Figure 22: Simulated hypothetical power spectra (dB-values vs. frequency in Hz) of the basic frequencies T1 (418Hz; black solid curve) and T2 (552Hz; red solid curve) in case no coupling of standing waves of the Harmonics of T1 and T2 would take place and so all energy would be radiated as acoustic waves. The curves of the logarithmic regression (dotted lines) and the related functions of the power spectra including R[2]-values are printed in black for T1 and in red for T

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Figure 23: Simulated dB-values (Y-axis) for the radiated Harmonics of T1 (418Hz; black dots and curve) and the Complex tones with frequencies defined by “n*T1(418Hz)+T2(552Hz)” (blue dots and curve) and by “n*T1-T2” (yellow dots and curve) plotted against the frequency (Hz). In this simulated scenario, the Radiation-factors of the frequencies of the Complex tones have been set to the value Rad(fi) = 1.0. See text for more detail

To estimate the validity and accuracy of the simulation, the intensities (dB-values) of the Complex tones of the original Multiphonic sound should be compared with the intensities of the respective simulated Complex tones. This can be done with the data of certain Complex tones (n*T1+T2) of the original Multiphonic sound (presented in Figure: 17) and the corresponding Complex tones of the two simulation-scenarios displayed in Figures 23 and 24. For another set of Complex tones (n*T2+T1), such a comparison can be done with the data of the original Multiphonic sound shown in Figure 18 and of the two simulations-scenarios in Figures 25 and 2

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Figure 24: Simulated dB-values (Y-axis) for the radiated Harmonics of T1 (418Hz; black dots and curve) and the Complex tones with frequencies defined by “n*T1(418Hz)+T2(552Hz)” (blue dots and curve) and by “n*T1-T2” (yellow dots and curve) plotted against the frequency (Hz). In this simulated scenario interpolated values of the “Resonance & Radiation” spectrum displayed in Figure 16 have been used as relative Radiation-factors of the Complex tones. See text for more detail

Although the intensities of the Complex tones of the original Multiphonic sound show high similarities with the respective Complex tones derived from the two simulation-scenarios, differences can still be observed. As the Radiation-factors of the acoustic waves of the Complex tones are not known, it is unlikely that the two scenarios will deliver intensities of the Complex tones identical with the intensities of the respective Complex tones of the original Multiphonic sound. The best correlation (R[2] = 0.5) of the intensities of the measurable Complex tones of the Multiphonic sound and of the simulated Complex tones is achieved using the interpolated data of the “Resonance & Radiation” spectrum as replacement for the unknown Radiation-factors (compare Figures 24 and 26 of simulation with Figures 17 and 18 showing data of the Multiphonic sound

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Figure 25: Simulated dB-values (Y-axis) for the radiated Harmonics of T2 (552Hz; red dots and curve) and the Complex tones with frequencies defined by “n*T2(552Hz)+T1(418Hz)” (blue dots and curve) and by “n*T2-T1” (yellow dots and curve) plotted against the frequency (Hz). In this simulated scenario, the Radiation-factors of the frequencies of the Complex tones have been set to the value Rad(fi) = 1.0. See text for more detail

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Figure 26: Simulated dB-values (Y-axis) for the radiated Harmonics of T2 (552Hz; red dots and curve) and the Complex tones with frequencies defined by “n*T2(552Hz)+T1(418Hz)” (blue dots and curve) and by “n*T2-T1” (yellow dots and curve) plotted against the frequency (Hz). In this simulated scenario, interpolated values of the “Resonance & Radiation” spectrum displayed in Figure 16 have been used as relative Radiation-factors of the Complex tones. See text for more detail

It can be concluded that the presented idea of a “coupling of two standing waves of different frequencies” together with reasonable assumptions on the resonance and radiation characteristics of the playing system (musician and woodwind instrument) deliver a simple mathematical model which allows a fairly good simulation of the frequencies and intensities of the Complex tones and the Harmonics of T1 and T2 of a Multiphonic sound played on a woodwind instrumen

For the proposed coupling process (defined as:**[C]**) of two standing waves (Amp(i)sw) with different frequencies (fx and fy as integer multiples of T1; T2 in Hz) and the resulting radiation of the generated Complex tones (Amp(i)awCt) the following description can be mad

**[C]**is a synonym for the coupling process; Amp means amplitude, Ct means Complex tone, sw means standing wave; aw means acoustic wave; Rad means relative Radiation-facto

The classical interference of acoustic waves is described as a mathematical sum (+) of the functions of the two standing waves. It can be demonstrated that this procedure does not exhibit the observed characteristics, as we do not get waves with frequencies . This can be seen by taking two standing waves and with frequencies and , wavelengths and and the same amplitude . Notation was changed to simplify the following formulas and not cause confusion, as we now look at standing waves on a spatial axis with a time

Here the left and right sin term in and respectively are the two waves creating the standing wave, one going in positive direction (left) and one going in the opposite negative direction (right). We assume the reflection at a loose end is responsible for creating the standing wave, as this should best represent the sound generation in a wind instrumen

Now if**[C]**was the classical interference of two waves, we can add up the standing waves of the two different frequencie

We can now use the trigonometric identity twice to reformulate the classical interference ter

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We see that adding two standing waves as in the classical interference, does not provide trigonometric functions with frequencies of , but only multiplications of such. Further a Fourier transformation of this function leads to signals at the frequencies f1 and f2 but not at

Therefore, the coupling process**[C]**cannot be classical interference. So we test whether the coupling process leading to the experimentally observed frequencies can mathematically be generated by a multiplication of the Harmonics of two standing waves with different frequencie

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We use to get the following for

We can merge the spatial cos parts and the frequency sin parts separately using the following identities and obtai

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When considering a fixed position we can simplify the spatial term to a constant and we see that the multiplication of two standing waves is generating two waves with frequencies that have the same intensity, as their amplitudes are equa

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This multiplication process shows the experimentally observed characteristics of the created Complex tones concerning the frequency but not concerning the intensity. This can be explained by looking at the bigger picture: The overall sound generated is not only generated by this coupling term, giving us the observed complex frequencies, but also consists of the standing waves themselves, giving us the radiated basic frequencies and Harmonics. Therefore, when describing the Multiphonic sound generation we do not know how much of the available total energy is going into the generation of the complex frequencies, so the coupling term has an unknown factor in fron

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To solve this unknown factor, we go back to the experimentally observed intensities of the coupled frequencies and propose the followin

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Note that we do not know whether the factor relies on the amplitudes of the standing waves, or whether the multiplication we propose as coupling process does only apply to the frequencies of the standing waves and their intensities follow a different coupling process, but the above formula is reflecting the experimentally observed data to a large exten

This would end up in the following equation which defines the intensities and frequencies of the acoustic waves forming the Complex tones as a result of the proposed coupling of two standing waves of different frequencies and

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*“Amp(i)”*is the amplitude of the two standing waves;*“(left)sin(fx)”*and*“(right)sin(fx)”*represent the two waves building the standing wave with frequency fx in Hz;*“(left)sin(fy)”*and*“(right)sin(fy)”*represent the two waves building the standing wave with frequency fy; “ ” and “ ” represent the generated acoustic waves with frequencies “*fy-fx*” and “*fx+fy*” of the Complex tones as the result of the coupling.*Rad(fx+fy)*and*Rad(fx-fy)*are the Radiation factors of the playing system at the frequency “*fx+fy*” and “*fx-fy*”, representing the factors the amplitude of the coupled waves are multiplied with coming from the amplitude of the initial standing waves with frequencies fx and f

The mathematical model proposed in this study and used to simulate some acoustic parameters of Multiphonic sounds generated with woodwind instruments has certain limitations due to the restriction of 10 Harmonics for T1 and T2 and the lacking of valid values for the Radiation-factors of the Complex tones, but despite this it delivers results which show a fairly good correlation with the measured parameters. As a next step, it seems necessary to develop a procedure to determine valid Radiation-factors of a playing system in order to further evaluate and eventually modify the proposed mode

## References

1) A. Benade; „Fundamentals of musical acoustics“; Second revised edition, Dover publications 1990, ISBN: 1397804862648

2) Website: https://cellomap.com/multiphonics-basics/; download of Cello Multiphonics; December 20

3) J. Liang; “Clarinet Multiphonics: A catalog and analysis of their production strategies”; Thesis Arizona State University; May 20

4) Website: https://commons.wikimedia.org/wiki/Category:Multiphonics; download of Oboe Multiphonic; November 20

5) Website: https://www.baerenreiter.com/materialien/weiss_netti/saxophon/mehrklang-auswahl.htm; Autoren: M. Weiss, G. Netti; downloaded March 20

6) L. Velut, C.Vergez, J.Gilbert; „Measurements and time-domaine simulations of multiphonics in the trombone”; Acoustical Society of America; 2016; pp 2876-28

7) J.Vesikalla; “Multiphonis of the grand piano – timbral compositions and performance with flageolets”; Thesis; University of Arts Helsinki Finnland; 20

8) I. Kubilay, J. Vesikalla, M. Pamies-Vila, T. Kuusi, V. Valimäki; “High-speed line-camera measurments of piano string multiphonics”; 22nd International congress on sound and vibration; Florence Italy; July 20

9) K. Guettler, H. Thelin; “Bowed-string multiphonics analyzed by use of impulse response and the Poisson summation formula”; Acoustical Society of America; 2012; pp 766-772 [DOI: 10.1121/1.365125

10) 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.

11) 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.

12) A.Rehm; „The magnitude of the frequency jitter of acoustic waves generated by wind instruments is of relevance for the live performance of music”;*Acoustics***2021**,*3*(2), pp 411-424; https://doi.org/10.3390/acoustics30200

13) References and details about software “Praat” see website: https://www.fon.hum.uva.nl/praat/ (status 1/202

14) A.Rehm; “Presentation of the „Resonance & Radiation” spectrum (RR-spectrum) as a parameter to describe the sound characteristics of musicians playing monophonic instruments”; 2/2022; ISBN: 9783346591654; Deutsche Nationalbibliothek; http://dnb.d-nb.

15) Website: https://tamingthesaxophone.com/saxophone-multiphonics; downloaded December 20

16) Website: https://quasar4.com/en/computerized-repertoire-multiphonics, downloaded April 20

17) Website: https://www.baerenreiter.com/materialien/weiss_netti/saxophon/mehrklang-auswahl.htm; Autoren: M. Weiss, G. Netti; downloaded March 20

18) Website University New South Wales with Clarinet Multiphonic Files: https://newt.phys.unsw.edu.au/music/clarinet/multiphonics/F4wA5.html; downloaded March 20

19) Website: https://heatherroche.net/2018/09/13/27-easy-bb-clarinet-multiphonics/; download of Clarinet Multiphonics September 20

20) Website: https://www.youtube.com/watch?v=r9lrjUatNA8; download of Trombone Multiphonic December 20

21) Website: https://en.wikipedia.org/wiki/Multiphonic; download of Trombone Multiphonic December 20

22) Website: http://extendedtechniques.blogspot.com/2012/05/multiphonics.html; download of Violin Multiphonic January 20

23) Website: https://www.youtube.com/watch?v=DHX3ggG88ng; download of Violin sounds January 20

24) Website: https://cellomap.com/multiphonics-basics/; download of Cello Multiphonics and fingering, December 20

25) Website: https://commons.wikimedia.org/wiki/Category:Multiphonics; download of Oboe Multiphonic November 20

26) Website: http://www.altoflute.co.uk/06-multiphonics/multiphonics.html; download of Flute Multiphonics November 20

27) Website: https://www.youtube.com/watch?v=mHxgpmb4bK0; download of Flute Multiphonics November 20

28) M. Proscia, P. Riera, M. Eguia; “Comparative study of saxophone multiphonic tones. A possible perceptual categorization”; Proceedings 12th international conference on Music Perception and Cognition; July 2012 Thessaloniki; pp 8

29) M. Proscia, P. Riera, M. Eguia; “A timbral and musical performance analysis of saxophone multiphonics morphing”; Procceedings international symposium on musical acoustics; June 2017 Montreal; pp 9-

30) P. Riera, M. Proscia, M. Eguia; “A comparative study of saxophone multiphonics: Musical, psychophysical and spectral analysis”; Journal of new Music research; 2014; DOI: 10.1080/09298215.2013.8609

31) S.Watts; “Spectral immersions: A comprehensive guide to the theory and practice of bass clarinet multiphonics”; Thesis; Keele University UK; 20

32) D. Gaebel, T. Lakatos, S. Weber, C. Valk, A. Rehm; „Schallwellenanalyse des Sounds professioneller TenorsaxophonspielerInnen. Teil 5: Variation i) der zeitlichen Ausprägung des Basistons und der Obertöne (Teiltöne eines Klangs) sowie ii) der Intensitäts- (Shimmer) und Frequenzschwankung (Jitter) der Teiltöne durch den Spieler zur Erzeugung des individuellen Sounds.“; ISBN: 9783668869998; Deutsche Nationalbibliothek; http://dnb.d-nb.

33) 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.

34) S.Linke, R.Bader, R. Mores; „Multiphonic modeling using Impulse Pattern Formation (IPF)”; arXiv: 2201.05452v1; January 20

35) D. Keefe, D. Laden; “Chaotic dynamics of woodwind multiphonics”; Journal Acoustic Society America Suppl.1, Vol.86, 118th meeting; 19

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