Acoustic Pressure Distribution and Mode-Specific Analysis Along the Bore of the Alto Saxophone

Abstract

This study investigates the acoustic characteristics of the alto saxophone by analyzing the spectral content of sound pressure along its bore and examining the influence of register valves. A detailed in situ analysis is presented of internal sound pressure from the mouthpiece to the bell for notes ranging from D₃ to C#₅, using a thin probe microphone needle in the neck and a movable miniature microphone in the body of the saxophone. The findings reveal that the cut-off frequency for lower notes in the first register is located near the third mode, whereas for higher notes, it shifts closer to the fourth mode. This research investigated previous assumptions that the cut-off frequency lies near the sixth mode, instead demonstrating that it occurs at lower modes depending on the note played. In the second register, the cut-off frequency consistently aligns with the second mode for all notes. The results demonstrate that opening the register tone holes alters the sound pressure level (SPL) distribution and shifts the positions of sound pressure valleys, with the first register valve having a more pronounced effect on SPL and mode shape than the second register valve. For the fourth mode in the first register, the register valves exhibit a stronger influence on SPL distribution compared to mode 2.

1. Introduction

Unlike other woodwind instruments, such as the clarinet or flute, the saxophone did not evolve gradually but was purposefully designed to meet specific needs. In 1842, Belgian inventor Adolphe Sax introduced the saxophone [1], aiming to create a reed instrument with significantly higher loudness than the clarinet. The design proved highly successful and was rapidly adopted by the French army for its marching bands. Sax’s innovation included giving the saxophone a conical bore, in contrast to the cylindrical bore of the clarinet [2,3]. This design not only enhanced its acoustic power but also simplified its fingering system.

Structurally, the saxophone consists of several main components. A musician blows into a small slit between the mouthpiece and the reed. When a critical pressure is reached, the reed starts to vibrate, thus modulating the incoming airflow. The neck is the slightly curved metal part that connects the mouthpiece to the body of the saxophone. The first register valve is located here. The majority of tone holes are located on the body of the instrument. The opening and closing of different combinations of tone holes allow the player to control the effective length of the vibrating air column and thus change the note that is being played. For instance, when playing D₃, nearly all tone holes remain closed except for the last two on the saxophone’s body and one located at the instrument’s curvature. In contrast, when playing C#₄, no keys are pressed to close any tone holes. However, despite the absence of key activation, three tone holes remain closed due to the saxophone’s default mechanism. The detailed configurations of open-and-closed tone holes for the investigated notes are presented and illustrated in the following sections.

The saxophone’s fingering system is particularly intuitive compared to other woodwinds [4,5]. For most notes, a single register key, operated with the left thumb, allows players to transition between registers using the same fingering. Specifically, notes from D₄ to C#₅ in the upper register can be played with the same fingering as their lower register counterparts D₃ to C#₄ [6]. Only the lowest notes (C₃ to Bb₂) and the highest notes above C#₅ require different fingering. Modern saxophones use two register valves—one on the neck and another on the body—that are activated based on the note being played. A mechanical system opens the body register valve for notes D₄ to G#₄ while keeping the neck valve closed, and conversely opens the neck valve for notes A₄ to C#₅, leaving the body valve closed. The idea of the register key is rather simple. Actuating the key opens a small tone hole near the position of the pressure node of the first harmonic of the note in the lower register. Opening this hole reduces the pressure of the fundamental at that location, thus favoring the energy buildup in the second harmonic, which then becomes the fundamental of the octave note.

The acoustics of the saxophone have been studied extensively in terms of its external sound radiation [7,8,9,10,11,12]. However, the internal acoustic spectrum of the instrument—particularly the sound pressure distribution along its bore—has received less attention. Internal microphone systems have been explored primarily by musicians, aiming to capture the saxophone’s sound from within the instrument. Such systems can mitigate issues like variability in sound pressure caused by tone hole configurations or feedback problems during amplification. Yet, detailed studies of internal acoustics with high spatial resolution do not exist.

The study by Petersen et al. explores the tone hole lattice cut-off frequency in conical resonators, with a particular focus on the saxophone [13]. While the cut-off frequency of cylindrical woodwind instruments like the clarinet has been extensively studied, the complex geometry of conical instruments introduces additional challenges that remain underexplored. This research generalizes the theoretical framework used for cylindrical lattices to conical resonators, defining a local cut-off frequency and investigating its impact on acoustic wave propagation. By analyzing both acoustically regular and irregular tone hole lattices, the authors developed methods to estimate the cut-off frequency based on input impedance and tone hole geometry.

Recent work measured the internal acoustic spectrum of the tenor saxophone at 21 discrete locations between the bell and the body register valve using white noise as input [14]. The study investigated the acoustic modal behavior using experimental acoustic modal analysis, a method traditionally applied to solid structures but adapted here to analyze air column resonance within the instrument. By measuring frequency response functions with an array of microphones positioned along the saxophone body, the authors extracted modal parameters such as resonance frequencies, loss factors, and mode shapes. This approach provided a spatially detailed characterization of the instrument’s resonator, revealing how different tone hole configurations, the presence of the mouthpiece, and register holes affect the instrument’s modal response. While their findings provided useful insights, the spatial resolution was insufficient for analyzing higher-order mode shapes, and critical regions between the mouthpiece and the body register valve were not measured. The study was further limited to the lowest notes with closed tone holes.

It was proposed that the saxophone’s cut-off frequency is located near its sixth resonance frequency. However, the authors emphasized that higher spatial resolution measurements are required to confirm this finding.

This paper aims to address these gaps by systematically investigating the sound pressure distribution along the bore of an alto saxophone across different registers and notes in realistic playing conditions. Using high-resolution measurements at 5 mm intervals, this study explores the mode shapes and cut-off frequencies of various notes in the first and second registers. Throughout the study, written notation is used, noting that the alto saxophone is a transposing instrument, sounding a major sixth lower than written.

2. Materials and Methods

For the measurements, a Selmer Series II alto saxophone was used with a classical mouthpiece (Concept, Henri Selmer, Paris, France), which has an overall length of about 105 cm, and a bell diameter of 12 cm. The alto saxophone has a curved bell and neck. All positions reported in the Section 3 are measured along the center bore of the instrument, following the curves (in other words: as if the instrument would be straight). To avoid the effects of dehydration and reed inter-specimen variability, all measurements were performed with concert-quality synthetic reeds (Légère Signature series), which had been played for 10 h. Previous experiments have shown that after this period, the behavior of these reeds remains stable for many hours [15]. The airflow in the instrument is generated by a feedback-loop controlled blowing machine which keeps blowing pressure constant to a precision of 100 Pa. The mouthpiece was mounted in an artificial mouth, equipped with an artificial lip made of soft silicone material. The lip is positioned using a stepper motor-activated translation stage and a force transducer monitors lip force on the reed. The keys to close the tone holes are actuated by solenoids, and the whole setup is computer-controlled so that key tone hole settings, lip force, and blowing pressure can all be automatically adapted and recorded during the measurement. Details on the measurement setup have been previously published [16,17]. Previous work investigated the range of lip forces and blowing pressures, thus determining the so-called playability range for each note. For the current measurements, the A-weighted sound pressure level (SPL) was set at 103 ± 1 dB(A) for all the notes.

To measure sound pressures in the neck of the saxophone, a small hole was drilled, which tightly fitted around the needle of a probe microphone (Type 4182, Bruel and Kjaer, Nærum, Denmark). Figure 1 shows how the needle was positioned in the saxophone neck. On the outside of the neck, a thick layer of two-component silicone rubber was applied so that a good airtight fit was obtained around the needle to prevent any escape of sound pressure. The probe microphone was mounted on a stepper motor-actuated translation table, which moved the microphone with an accuracy greater than 0.1 mm. The needle was inserted in the neck so that the needle tip was positioned 30 mm from the tip of the mouthpiece. Putting the needle even deeper would not be possible as it would influence the space between the reed and the mouthpiece, thus impeding reed vibration. Therefore, the first measurement point is obtained at 30 mm from the mouthpiece tip.

The sound pressure level in the saxophone bore was measured using a high-quality miniature microphone (4062, DPA, Copenhagen, Denmark). The microphone has a diameter of just 5.4 mm, and the type used for the experiment has an extended dynamic range of over 150 dB SPL. The microphone was mounted in a 3D-printed spherical cage made out of four thin plastic supports. This cage keeps the microphone at least 1 cm away from the instrument walls during its travel through the instrument bore. A thin pulling wire was connected to the cage on one end and to a linear translation stage at the other end so that the translation stage pulls the cage and the microphone through the saxophone bore. The stage was driven by a computer-controlled stepper motor. In the straight part of the instrument, the position of the microphone in the instrument can be measured with a precision of better than 1 mm. In the curved part near the bell, the positioning of the microphone is somewhat less reliable, but results will show that in this region, sound pressure levels only change very gradually, so exact microphone positioning is less crucial.

Both microphones have a dynamic range of over 150 dB SPL. The frequency response of the probe microphone is flat to within 2 dB over the entire used frequency range of 50 Hz to 4 kHz. The frequency response of the miniature microphone is flat to within 1 dB in this frequency range. Signals were digitized at a sampling frequency of 44.2 kHz using a low-noise 24-bit audio digitizing unit (Focusrite, Scarlett 4i4). The signal level of the miniature microphone is not calibrated. Therefore, calibration was obtained in situ using the RMS signal of the probe microphone when its tip was positioned very near the location of the miniature microphone. In this way, the miniature microphone measurements can also be reported in dB sound pressure level. A marker was placed on the cable of the miniature microphone to ensure precise positioning. The microphone was then adjusted until the marker aligned exactly with the upper edge of the saxophone’s body. Subsequently, the saxophone body was connected to the neck, positioning the miniature microphone inside the neck at the same location as the position of the probe microphone’s needle.

To assess the uncertainty associated with the microphone positioning, a measurement was conducted inside the saxophone. The miniature microphone was displaced by plus and minus 10 mm relative to the probe microphone’s needle and sound pressure was recorded. The results indicated that the sound pressure level at the fundamental frequency remained within a 1 dB variation. However, for higher modes, SPL variations increased up to 5 dB. The radiated sound pressure generated by the instrument was measured using a microphone (Bruel and Kjaer Type 2669) positioned in front of the instrument bell at 15 cm.

Both translation stages and the microphone A/D systems were controlled by the same computer so that the measurement could be fully automated. For each tested note, both microphones were moved in steps of 5 mm, and at each position, the sound signal was recorded for 2 s. To enhance the accuracy of amplitude estimations in the measured signals, zero-padding was applied prior to computing the discrete Fourier transform. Following the transformation, the peak amplitudes of the modes were identified using the findpeaks function in MATLAB R2021a. The probe microphone needle was moved first from the mouthpiece tip to the starting position of the miniature microphone. Its total travel distance was 175 mm. Then, the miniature microphone was moved from its position in the neck up to the front of the bell.

In the Section 3 and Section 4, R1 and R2 refer to the first and second register valves, respectively, while the term “modes” is used to describe the sound pressure level patterns of the fundamental frequency and its harmonics, with mode 1 representing the fundamental frequency.

The automated actuation system allowed us to measure the following notes: D₃, E₃, F₃, F#₃, G₃, G#₃, A₃, B₃, C₄, C#₄, D₄, E₄, F₄, F#₄, G₄, G#₄, A₄, B₄, C₅, C#₅.

3. Results

3.1. Acoustic Characteristics of the Instrument Bore Across Modes in the First Register

3.1.1. Lower Modes (Modes 1–3)

Figure 2 illustrates the sound pressure level distribution along the bore of the instrument for notes in the first register. The upper row highlights the positions of tone holes that are open or closed during the production of each note. Only tone holes actively used (i.e., opened or closed) to play the investigated notes are displayed. Tone holes not depicted in the figure remain closed throughout. The tone holes are represented by discs, proportionally scaled to reflect their size. Each note is assigned a distinct color. Closed tone holes are indicated by colored discs. The first two dots in the figures correspond to the register valves R1 and R2. R2 is utilized for playing notes from D₄ to G#₄, while R1 is employed for notes ranging from A₄ to C#₅. SPL data within the first ~175 mm of the bore were acquired using a probe microphone, while data beyond this region were recorded with a miniature microphone.

The subfigures in the columns of Figure 2 present the SPL distributions for various modes. The vertical axis represents SPL amplitude, while the horizontal axis corresponds to the position along the bore. Vertical dashed lines mark the tone hole positions, and individual notes are color-coded for clarity.

In mode 1, the SPL peaks at the mouthpiece, with the highest level observed for C#₄ (160 dB SPL). At the first measurement point, SPL values for all notes range from 154 dB SPL to 160 dB SPL. For the first register, both register tone holes remain closed, and the first valve opening occurs at approximately 375 mm for C#₄, leading to a noticeable decrease in SPL. For instance, at 375 mm, the SPL decreases to 149 dB SPL for D₃ while for C#₄ the sound pressure level dropped to 133 dB SPL. The SPL along the bore decreases consistently for notes D₃ through G#₃. However, for A₃, B₃, C₄, and C#₄, the minimum SPL is not observed at the instrument’s end but at approximately 760 mm, with values of 107, 96, 97, and 100 dB SPL, respectively. At the bell, the SPL rises to 106, 98, 99, and 103 dB SPL for these notes.

In mode 2, distinct peaks and valleys become evident in the SPL along the bore. A pronounced valley is observed for all notes between approximately the first and second register tone holes, with the position of the valley varying by note. For D₃, the valley occurs closer to the second register valve, while for C#₄, it aligns more closely with the first register valve. Between 550 mm and 1000 mm, D₃ consistently exhibits the highest SPL in modes 1 and 2. However, in mode 3, this pattern shifts, with F₃ for instance surpassing D₃ at 400 mm (128 dB SPL for F₃ vs. 124 dB SPL for D₃). While peaks and valleys remain prominent for notes D₃ to G#₃, they become less distinct from A₃ onwards. For C₄ and C#₄, a pronounced valley reappears around 760 mm.

In mode 3, the spacing between SPL peaks and valleys is non-uniform. For example, in G#₃, the interval between the first and second valleys is 220 mm, whereas the distance between the second and third valleys extends to 380 mm. Similar irregularities are observed across other notes.

3.1.2. Higher Modes (Modes 4–6)

In mode 4, SPL attenuation along the bore varies across notes. For most notes, such as G₃ and C#₄, the SPL reduction is minimal; the SPL at the first peak is 138 dB SPL for G₃ and 130 dB SPL for C#₄, decreasing to 135 dB SPL and 125 dB SPL, respectively, at the final peak. However, notes like A₃ and B₃ exhibit substantial attenuation, with A₃ decreasing from 120 dB SPL at the first peak to 70 dB SPL near the bell. For modes 5 and 6, SPL peaks and valleys are evident throughout the bore despite open tone holes for certain notes. The contributions of these modes at the bell differ across notes. For instance, in mode 5, the lowest SPL at the bell is observed for D₃, whereas in mode 6, C#₄ exhibits the lowest SPL.

3.2. Acoustic Characteristics of the Instrument Bore Across Modes in the Second Register

Figure 3 illustrates the sound pressure level (SPL) distribution along the instrument’s bore for notes in the second register, produced using the first or second register valve. The conventions follow the same format as those outlined in Figure 2.

Mode 1 in Figure 3 reveals similar trends as observed in the first register’s mode 2 in Figure 2. The opening of tone holes modifies the SPL distribution within the bore, with a distinct valley appearing between the first and second register tone holes. This correspondence arises because the first mode in Figure 3 aligns with mode 2 in Figure 2. The most notable difference between the two figures lies in the SPL magnitude: higher modes in the second register exhibit lower SPLs.

3.3. Effect of the Register Hole

To investigate the impact of register valves on mode shapes, Figure 4 compares mode 2 from the first register (Figure 2) to the corresponding mode when the register valve is open for the different notes. This means that mode 2 in the first register is mode 1 in the second register. Mode 4 in the first register becomes mode 2 in the second register. Mode 6 in the first register becomes mode 4 in the second register. The mode shape of the first register is shown in orange whereas the same mode shape in the second register is shown in green. The SPL is plotted on the vertical axis, with the bore position along the horizontal axis. Vertical dashed lines represent open tone hole positions, while solid vertical lines indicate closed tone holes. The used register tone hole position (R1 or R2) is marked by a red vertical line.

For mode 2 of D₃, a valley in SPL occurs at 320 mm. When the second register valve is open, the valley shifts to 335 mm. Generally, the influence of R2 on mode shape is minimal, with SPL values increasing by approximately 2 dB when R2 is engaged. The valley’s position shifts less than 15 mm across notes when opening R2, and its proximity to the mouthpiece increases from D₃ to G#₃. Notably, the distance between the SPL valley and the second register valve is smallest for D₃ and largest for G#₃. For notes A₃ to C#₄, the first register valve is used to overblow the first register notes. Figure 4 demonstrates that R1 has a more pronounced effect on mode shape compared to R2. SPL increases significantly beyond the first register valve when R1 is engaged. A discontinuity at 250 mm is observed for note B for R1, likely due to data acquisition errors. R1’s influence is particularly notable for C#₄. The SPL valley deepens when R1 is opened, with values decreasing from 102 dB SPL in mode 2 to 88 dB SPL when using R1 at 595 mm.

Figure 5 examines the impact of register valves on mode 4 of the first register notes. As in Figure 4, the SPL is plotted along the bore, with open-and-closed tone hole positions represented by dashed and solid vertical lines, respectively. The register valve positions are again marked in red. For mode 4, register valves exert a more substantial influence on SPL distribution compared to mode 2. Most notes using R2 exhibit lower SPLs in mode 4 when the valve is open, contrary to the observations in mode 2 (Figure 4). Conversely, notes using R1 generally display higher SPLs in mode 4 compared to the corresponding mode in the first register, which also contrasts with the trends observed for R1 in mode 2. A notable exception is observed for A₄, where the mode using R1 exhibits higher SPLs than mode 4 in the first register.

Figure 6 presents mode 6 for the first register notes from Figure 2 and the corresponding mode when the register tone hole is open for the various notes. The data representation follows the same format as in Figure 4 and Figure 5, highlighting the influence of register tone holes on the mode shape. For mode 6, multiple peaks and valleys are distinctly visible, with their spacing being generally equidistant. The deviations in peak and valley positions are minimal, with a maximum observed variation of 20 mm, indicating a high degree of regularity in the mode structure.

3.4. Internal and External Acoustic Spectra

Figure 7 illustrates the acoustic spectra measured inside the mouthpiece and outside the instrument (radiated spectrum) for D₃ and C#₄, focusing on the first 14 modes. The figure shows SPL as a function of frequency on the left y-axis, while the right y-axis depicts the difference in SPL (Attenuation) between the SPL in the mouthpiece and the radiated sound. For D₃, the most pronounced SPL difference occurs at the fundamental frequency, where the SPL outside the mouthpiece is 72 dB lower. A general trend emerges where the SPL difference diminishes with increasing mode frequency, indicating greater acoustic energy transmission at higher modes. This trend is less significant for C#₄, where the SPL difference for the fundamental frequency is 57 dB—15 dB lower than for D₃.

For D₃, the seventh mode (1223 Hz) lies closest in frequency to the fourth mode (1295 Hz) of C#₄. The SPL difference between the mouthpiece and the external environment is approximately 30 dB for both notes at these modes. Similarly, the eleventh mode (1921 Hz) of D₃ is nearest to the sixth mode (1942 Hz) of C#₄. At these frequencies, the SPL attenuation is 21 dB for D₃ and 13 dB for C#₄, reflecting a larger sound energy radiation for C#₄ in this mode.

4. Discussion

4.1. Mode Shapes and Sound Pressure Levels in the First and Second Register

Figure 2 illustrates the mode shapes for the first six modes of various notes in the first register of the saxophone, with sound pressure levels plotted along the bore. Inside the mouthpiece, SPLs as high as 160 dB were measured, consistent with known values for the mouthpiece cavity, necessitating the use of specialized microphones [2]. Lucchetta et al. previously measured sound pressure levels at 21 discrete locations along the bore, from the bell to the body register valve, using white noise as input [14]. While they proposed micro-electro-mechanical microphones for higher spatial resolution, these devices are unsuitable for realistic playing conditions due to their limited linear range (approximately 120 dB SPL). The present study achieved a spatial resolution of 5 mm and extended measurements to include the region between the mouthpiece and the second register valve.

For modes 1 and 2, the open tone holes significantly influenced SPL within the bore. Increasing the number of closed tone holes resulted in less attenuation along the bore. From mode 3 onward, higher SPLs were observed for higher notes in the first register, despite the increased number of open tone holes. Open tone holes equalize the internal pressure at their location towards atmospheric pressure, shortening the instrument’s acoustic length [18]. This phenomenon predominantly affects low frequencies, as higher frequencies are less influenced due to the air’s inertia [19]. At high frequencies, the acoustic energy is able to propagate past the first open tone hole [20]. The frequency at which this transition occurs is the cut-off frequency [21]. The cut-off frequency for the three lowest notes on a tenor saxophone was identified, suggesting it lies near the sixth mode [14]. Our results revealed clear standing wave patterns even for the third mode. Standing wave patterns were observed for notes D₃ to G#₃, but higher notes in the first register showed no standing wave patterns, suggesting that the cut-off frequency is dependent on the played note’s frequency. Petersen et al. provided analytical solutions for the cut-off frequency of notes in the first register of the alto saxophone [13]. For instance, they calculated a cut-off frequency range of 1.1–1.4 kHz for C#₄, aligning with our results, which place the cut-off frequency near the fourth mode (1.3 kHz). For D₃, our findings suggest a cut-off frequency near the third mode (531 Hz), consistent with Petersen et al.’s range of 500–750 Hz. Similarly, for all other notes in the first register, the measured cut-off frequencies matched analytical predictions. Additionally, this study confirmed that the cut-off frequency decreases from higher to lower notes within the first register.

Figure 3 depicts the mode shapes of the first six modes for notes in the second register. A discontinuity in SPL was identified for some notes, preventing reliable data analysis within these regions. Discontinuities were observed in the higher-order modes, particularly for notes A and G. However, these discontinuities remained within the 5 dB uncertainty margin attributed to mispositioning effects in higher modes. In contrast, the discontinuity observed for note B exceeded this uncertainty range and could not be solely attributed to positioning errors. Consequently, the results for this note should be interpreted with caution. Similar to the first register, clear standing wave patterns were visible from mode 2 onward, indicating that the cut-off frequency in the second register lies near the second mode. The cut-off frequency decreases from approximately 1314 Hz for C#₅ to 703 Hz for D₄, reflecting a downward trend analogous to that observed in the first register.

Further examination of Figure 2 revealed distinct mode shapes for G#₃ and A₃ in mode 4. Fourier spectra for these notes showed an additional peak near mode 4. For example, while the frequency of mode 4 for A₃ was 1037 Hz, a secondary peak was observed at 1096 Hz, likely corresponding to the reed’s resonance frequency. Mode frequencies near the reed’s resonance frequency appear to be influenced by this resonance, resulting in altered mode shapes for these notes.

Thompson previously reported that natural cane reeds exhibit a first resonance between 2 and 3 kHz [22]. However, this study used synthetic reeds from Légère, which exhibit slightly different resonance characteristics. The reed’s resonance frequency for the synthetic reeds used in this study is likely lower.

4.2. Register Tone Holes

Figure 4 demonstrates the impact of register tone holes on mode 2 in the first register. In general, opening the first or second register key led to an increase in the SPL of the second mode. This effect highlights the role of register keys in altering the acoustic impedance within the bore, thereby amplifying specific mode shapes [23]. In contrast, Figure 5 revealed that opening the first or second register key typically resulted in a decrease in the SPL of the fourth mode. This finding is in direct opposition to the trends observed in Figure 4, suggesting that the effect of register keys on SPL is mode-dependent and likely influenced by the interplay of bore resonance and tone hole positioning. Interestingly, exceptions were observed for G#₃ and A₃. For these notes, opening register keys R2 and R1, respectively, increased the SPL of the fourth mode. These anomalies may be attributed to the relatively large distance between the valley of the SPL for mode 2 (as shown in Figure 4) and the position of the register tone holes. In general, the opening of the register tone holes alters the SPL distribution and the position of valleys. R1 exhibited a stronger influence on SPL and mode shape than R2. This observation agrees with the findings by Lucchetta et al., who used white noise excitation to investigate the acoustics of a tenor saxophone [14]. Their study demonstrated that R1 strongly influences the first eight modes, whereas R2 predominantly affects only the first three modes.

4.3. Analysis of Internal and External Acoustic Spectra

Figure 7 presents the acoustic spectra for D₃ and C#₄, highlighting the differences in SPL inside the mouthpiece versus outside the instrument. A key observation was that for D₃, the lowest note in the first register, the SPL difference inside and outside the mouthpiece was more pronounced compared to C#₄ and other higher notes in the same register. This suggests that lower notes experience greater attenuation when transitioning from the mouthpiece to the free field.

When comparing modes with approximately similar frequencies across different notes, variations in the attenuation factor were observed. These differences indicate that the relationship between internal and external SPL is not uniform across the register. As a result, if an equalizer is used to replicate the free-field sound spectrum starting from the sound picked up by an internal microphone, the settings need to be adjusted on a note-dependent basis to account for the unique attenuation characteristics of each note.

Previous work proposed investigating the homogeneity of the radiated sound across different notes [13]. Specifically, they noted that for notes higher than F₃, the reed primarily interacts with two impedance peaks, whereas for lower notes, this interaction can increase to three or four impedance peaks. The implication is that lower notes may exhibit less variability in radiated sound due to more impedance peaks that may collaborate with the reed. However, our study did not provide sufficient evidence to draw conclusions regarding the homogeneity of radiated sound for notes below and above F₃.

5. Conclusions

This study investigated the acoustic pressure distribution along the bore of an alto saxophone, from the mouthpiece to the bell. The findings revealed that the first two modes in the first register are significantly influenced by the open tone hole configuration specific to each note. From mode 3 onward, distinct standing wave patterns were observed for lower notes, while these patterns were less pronounced for higher notes in the first register. This suggests that the cut-off frequency for lower register notes is near the third mode, whereas for higher notes, it lies closer to the fourth mode. These observations agree with the analytical predictions of Petersen et al. [13]. Moreover, the results of the present study indicate that the cut-off frequency occurs at lower modes and not at the sixth mode, as suggested in previous work [14].

For notes in the second register, the cut-off frequency was consistently found near the second mode across all notes. Additionally, the opening of the register tone holes was shown to alter the SPL distribution and shift the position of sound pressure level valleys, with R1 exerting a stronger influence on SPL and mode shape compared to R2.

An analysis of the acoustic spectra measured inside the mouthpiece and outside the bell demonstrated notable differences between the two. For lower notes in the first register, higher attenuation was observed when comparing the internal spectrum to the external spectrum. Additionally, the findings indicate that replicating the external spectrum of the saxophone starting from a signal picked up by an internal microphone requires note-specific adjustments of equalizer settings.