Helmholtz Motion Research Articles

Experimental assessment of playability limits during bowed string attacks

The playability limits in the bow force and bow acceleration parameter space (usually visualized as a Guettler diagram) define the conditions for establishing Helmholtz motion during bowed attacks in bowed-string instruments. Despite few theoretical and numerical studies, there is little empirical validation of these limits. The experimental scanning of a Guettler diagram is a tedious process, as it requires the use of a bowing machine to control the bowing parameters. This study proposes a method for collecting and analyzing data to produce measured Guettler diagrams. Experiments were conducted using a cello string in a monochord arrangement with fixed terminations. The string was excited using a robotic arm that allows control of the playing conditions by changing the bow force and bow acceleration. Each set of measurements contains more than 900 data points to obtain high-resolution Guettler diagrams. In addition, the measurements are repeated in reverse order and after dismounting and remounting the string. The results are compared with existing literature, with particular attention to theoretical limits. The experimental results suggest that the static and dynamic friction coefficients vary with bow force and acceleration, modifying the playability limits. [Work supported by the Austrian Science Fund (FWF) (P34852-N).]

Assessing playability limits of bowed-string transients using experimental measurements

Understanding the dynamics of bowed-string attacks involves exploring the relationship between bow acceleration, bow force, and the generation of Helmholtz motion during transients. This study addresses the following research question: How do theoretical limits of “playability” predict these parameters? Motivated by the need for experimental evidence in this domain, we present a comprehensive investigation into bowed-string transients within the bow acceleration and bow force parameter space, known as the Guettler diagram. This study exclusively employs an experimental methodology. The setup, employing a robotic arm, permits the collection of transient data under varying bowing conditions. Analysis of the bridge force waveform allows for the extraction of pre-Helmholtz transient times. Our results reveal a triangular playable region in the Guettler diagram, consistent with theoretical predictions and previous experimental findings. However, Guettler’s analytical limits for playable regions during transients show limitations. We investigate the role of friction, a key parameter idealized in the model used for obtaining these limits. Measured friction coefficients from transients reveal discrepancies with prior experimental studies, highlighting the need for further investigations in this direction.

Two-dimensional playability maps for single-reed woodwind instruments

One method to analyze musical instrument playability is by visualizing a two-dimensional subspace of the musician's control parameters. This hasbeen widely used in the form of Schelleng diagrams for bowed string instruments. There, it is possible to identify regions within the bow force—bowing position subspace where Helmholtz motion is achieved. Such diagrams may be populated either on the basis of experimental measurements, or via physical modeling. In fact, physical modeling is particularly suited to this task, since playing parameters can be directly controlled. It has been recently suggested (Woodhouse, in Proc. SMAC 2023) that similar diagrams may be used for analyzing wind instrument playability. This study aims at exploring this direction using a physical model which has been previously validated against experimental measurements. Initially, an informed decision is made on which parameters should be chosen for plotting an equivalent playability diagram. Subsequently, various diagram versions are generated, focusing on different aspects of the generated tones. As already pointed out by Woodhouse, similarities appear between the diagrams for bowed-string and woodwind instruments. It remains to examine how valuable information regarding woodwind playability may be extracted from such studies.

A mathematical study about the sustaining phenomenon of overtone in Flageolet harmonics on bowed string instruments

Flageolet harmonics is a playing technique, in which a player lightly touches a nodal point on a string with their finger. Previous studies have reported that the harmonic sound sustains for a short time after the finger is removed from the string while flageolet harmonics is performed on bowed string instruments. However, the mechanism of this harmonics-sustaining phenomenon and the parameter dependency of its sustaining time remains unclear. The purpose of this study is to mathematically investigate the dependence of sustaining time on parameters related to violin playing and string characteristics, and thereby, to elucidate the mechanism of the harmonics-sustaining phenomenon. To this end, a mathematical model was devised by incorporating the effects of bowing and touching into a one-dimensional wave equation. Subsequently, numerical simulation were performed to analyze the behavior of the model. The devised model successfully reproduced the harmonics-sustaining phenomenon in which the parameter dependence of sustaining time was qualitatively consistent with the author’s empirical observations. It was found that the parameter dependence of sustaining time follows the power law. Furthermore, dimensional analysis was performed, yielding a formula that expresses the relationship between the sustaining time and the maximum and minimum bow force required to generate Helmholtz motion.

Investigation of the Helmholtz Motion of a Violin String: A Finite Element Approach

Abstract In the context of this work, a violin string motion is examined using a finite element approach. The string is formulated via ideal string elements and is bowed at one point on the string; hence, there is a nodal contact between the bow and the string. The bow movement induces the stick-slip effect, which is the cause for the violin string sound. The present paper aims at the investigation of the stick-slip phenomenon of bowed strings, considering well-known bowed string effects like the Helmholtz corner modulation, the Schelleng ripples, and the flattening effect. One key element that is used in this work is the Schelleng diagram, which indicates the “perfect” bow force depending on the bowing position. Within these parameters, the Helmholtz motion is carried out. Additionally, different friction characteristic curves are applied in order to study the impact of the rosin on the string motion.

Parabolic curves in a Helmholtz solution for a bowed string

If one is not familiar with the physics of the violin, it is not easy to guess, even for an experimental physicist, that the so-called Helmholtz motion can be obtained as a solution to the one-dimensional wave equation for the motion of a bowed violin string. It is worth visualising this aspect from a graphical perspective without recourse to ordinary Fourier analysis, as has customarily been done. We show in this paper how to obtain the shape of the Helmholtz trajectory, that is, two mirror-symmetric parabolas, in the ideal case of no losses from internal dissipation and no viscous drag from the air and the non-rigid end supports. We also show that the velocity profile of the Helmholtz motion is also a solution of the one-dimensional wave equation. Finally, we again derive the parabolic shape of the Helmholtz trajectory by applying the principle of energy conservation to a violin string.

Role of the Resonator Geometry on the Pressure Spectrum of Reed Conical Instruments

Spectra of musical instruments exhibit for-mants or anti-formants which are important characteristics of the sounds produced. In the present paper, it is shown that anti-formants exist in the spectrum of the mouthpiece pressure of saxophones. Their frequencies are not far but slightly higher than the natural frequencies of the missing part of the cone. To determine these frequencies, a first step is the numerical determination of the playing frequency by using a simple oscillation model. An analytical analysis exhibits the role of the inharmonic-ity due to the cone truncation and the mouthpiece. A second step is the study of the input impedance values at the harmonics of the playing frequency. As a result, the consideration of the playing frequency for each note explains why the anti-formants are wider than those resulting from a Helmholtz motion observed for a bowed string. Finally numerical results for the mouthpiece spectrum are compared to experiments for three saxophones (soprano, alto and baritone). It is shown that when scaled by the length of the missing cone, the anti-formants frequencies in the mouthpiece are very similar for the three instruments and correctly predicted by the model. Finally, the numerical computation shows that anti-formants and for-mants might be found in the radiated pressure.

Complementary empirical data on minimum bow force.

The minimum bow force Fmin necessary to maintain Helmholtz motion on stringed instruments was widely believed to be proportional to the bow velocity vB, to the reciprocal of the bridge resistance R, and to the reciprocal of the relative bow-bridge distance β squared [Schelleng, J. Acoust. Soc. Am. 53, 26-41 (1973)]. More recently, a study reported independence from vB and an overproportional reciprocal effect of R while confirming the 1/β2 proportionality [Schoonderwaldt, Acta Acust. Acust. 94, 604-622 (2008)]. Here, a bowing pendulum is used which facilitates precise control and measurement of related parameters. The string excitation at the contact point is recorded to instantly classify Helmholtz motion (HM) versus non-Helmholtz motion with one (nHM-1) or more additional slips during the stick phase. Monitoring of the classification supports the control of bowing parameters during measurement. These are directed towards the regions of transition between HM and nHM-1 to reveal the parameters related to Fmin. The empirical data gained from cello strings suggest that HM requires a baseline force Fmin,0 even for very low vB. Fmin,0 depends on R and ß.

Minimum bow force measured with a high-precision bowing pendulum

The minimum bow force Fmin necessary to establish Helmholtz motion on stringed instruments was widely believed to be proportional to the bow velocity vB, to the reciprocal of the bridge resistance R, and to the reciprocal of the relative bow-bridge distance ß squared [J. Schelleng, J. Acoust. Soc. Am. 53, 26-41 (1973)]. More recently, a study reported independence from vB and an overproportional reciprocal effect of R while confirming the 1/ß² proportionality [E. Schoonderwaldt, Acta Acustica United with Acustica 94 604-622 (2008)]. Here, a bowing pendulum is used which facilitates precise control and measurement of related parameters. The string excitation at the contact point is recorded to instantly classify Helmholtz motion (HM) versus non-Helmholtz motion with one (nHM-1) or more slips during the stick phase. Monitoring of the classification supports the control of bowing parameters during measurement. These are directed towards the regions of transition between HM and nHM-1 to reveal the parameters related to Fmin. The empirical data gained from cello strings suggest that HM requires a baseline force Fmin,0 even for very low vB. Fmin,0 depends on R and ß. Fmin, then grows proportionally with vB and with 1/ß rather than with 1/ß².

On Minimum Bow Force for Bowed Strings

A famous theoretical prediction of the minimum bow force to maintain Helmholtz motion of a bowed string is re-examined to take account of effects associated with resonances of the instrument body. Starting from a more robust assumption of an ideal stick-slip velocity waveform at the bowing point rather than a perfect sawtoothshaped excitation force at the bridge, the analysis predicts that the minimum bow force, and the force waveform exciting the instrument bridge, can depend in a complicated way on the position of the bow on the string. Also, the frequency of maximum wolfiness of an instrument like a cello is predicted to shift away from that of the strong body resonance causing a wolf note. Simulations are used to evaluate the new formulation. For the simple case in which the string vibrates only in a single polarisation, the results are accurately confirmed. However, simulation also reveals that string vibration in the second polarisation can change the detailed response. Further simulations are used to investigate the influence on minimum bow force of some physical details of the model, especially torsional string motion and the presence of sympathetic strings.

Maximum bow force revisited.

Schelleng [J. Acoust. Soc. Am. 53, 26-41 (1973)], Askenfelt [J. Acoust. Soc. Am. 86, 503-516 (1989)], Schumacher [J. Acoust. Soc. Am. 96, 1985-1998 (1994)], and Schoonderwaldt, Guettler, and Askenfelt [Acta Acust. Acust. 94, 604-622 (2008)] formulated-in different ways-how the maximum bow force relates to bow velocity, bow-bridge distance, string impedance, and friction coefficients. Issues of uncertainty are how to account for friction or for the rotational admittance of the strings. Related measurements at the respective transitions between regimes of Helmholtz motion and non-Helmholtz motion employ a variety of bowing machines and stringed instruments. The related findings include all necessary parameters except the friction coefficients, leaving the underlying models unconfirmed. Here, a bowing pendulum has been constructed which allows precise measurement of relevant bowing parameters, including the friction coefficients. Two cellos are measured across all strings for three different bow-bridge distances. The empirical data suggest that-taking the diverse elements of existing models as options-Schelleng's model combined with Schumacher's velocity term yields the best fit. Furthermore, the pendulum employs a bow driving mechanism with adaptive impedance which discloses that mentioned regimes are stable and transitions between them sometimes require a hysteresis on related parameters.

Playability of a bowed string physical model including finite-width thermal friction and hair dynamics

We report on the playability of a bowed-string physical model that combines digital waveguide and finite-difference time-domain frameworks. We extend previous approaches by combining a finite-width bow-string interaction model with a dynamic friction model based on simulating heat diffusion along the width of the bow. Bow hair dynamics are incorporated in the bow-string interaction, which includes two transversal string polarizations. The bridge termination is realized using a digital reflectance matrix model obtained from fitting two-dimensional driving-point admittance measurements. We present preliminary results from a playability study in which we explore the establishment of Helmholtz motion in static and dynamic bowing conditions.

Vibration Characteristic of Violin Bowstring and Experimental Research on Helmholtz Motion of String

The sound source of violin is string vibration. By the stick–slip friction action between bow and string of violin, string and bridge form a complicated vibration system. The interaction mechanism of bowstring is very intricacy. In this paper, through theoretical analysis and experiment, the vibration behavior of violin string is investigated. We measured the vibration images of bowing and plucking violin string on a high-speed photography optical platform and got two different character of string vibration, including vibration displacement, velocity and space trajectory of the mark point on string. By comparing the displacement curves of the different points among on string, the variety of the ratio of positive process time to negative one of vibration sawtooth waveform are studied, and also the formation process of string vibration enveloping curve. On this basis, the Helmholtz motion of string vibration which is presented at 1862 is studied.

Observation of Motion of Bowed Strings and Resonant Strings in Violin Performances

The motion of a bowed string and a resonant string of a violin were simultaneously observed for the first time. The results of the direct observation of string motion in double stops and harmonics are also presented. The importance of the resonance was experimentally demonstrated from these observations. It is suggested that players should take account of the resonance and ideal Helmholtz motion in violin performances.

Real-time finite-difference string-bow interaction floating point gate array (FPGA) model coupled to a violin body

A string/bow interaction model as proposed by Bader, R.: Whole geometry Finite-Difference modelling of the violin. In: Proceedings of the Forum Acusticum 2005, 629-634, 2005 and a 3D violin geometry with top plate, back plate, and enclosed air was implemented in real-time using an FPGA hardware implementation. The bow/string interaction uses conditions for gluing and tear-off the string to/from the bow, where gluing happens with relative string/bow velocity below a threshold and tear-off with either the string curvature or the string tension at the bow point too high modelling the cases appearing with normal Helmholtz motion or the occurrence of subharmonics, respectively. The model can be played changing bow pressure and velocity, bow point on the string, and string length. The resulting sounds are highly realistic showing gradual timbre changes from normal Helmholtz motion, double slit motion, weakening of the fundamental, or noise with very low pressures. It appears that these interaction conditions can realistically be used to model the bow/string interaction.

Contribution to harmonic balance calculations of self-sustained periodic oscillations with focus on single-reed instruments

The harmonic balance method (HBM) was originally developed for finding periodic solutions of electronical and mechanical systems under a periodic force, but has been adapted to self-sustained musical instruments. Unlike time-domain methods, this frequency-domain method does not capture transients and so is not adapted for sound synthesis. However, its independence of time makes it very useful for studying any periodic solution, whether stable or unstable, without care of particular initial conditions in time. A computer program for solving general problems involving nonlinearly coupled exciter and resonator, HARMBAL, has been developed based on the HBM. The method as well as convergence improvements and continuation facilities are thoroughly presented and discussed in the present paper. Applications of the method are demonstrated, especially on problems with severe difficulties of convergence: the Helmholtz motion (square signals) of single-reed instruments when no losses are taken into account, the reed being modeled as a simple spring.

Cardiac video analysis using Hodge–Helmholtz field decomposition

The critical points (also known as phase singularities) in the heart reflect the pathological change of the heart tissue, and hence can be used to describe and analyze the dynamics of the cardiac electrical activity. As a result, the detection of these critical points can lead to correct understanding and effective therapy of the tachycardia. In this paper, we propose a novel approach to address this problem. The proposed approach includes four stages: image smoothing, motion estimation, motion decomposition, and detection of the critical points. In the image smoothing stage, the noisy cardiac optical data are smoothed using anisotropic diffusion equation. The conduction velocity fields of the cardiac electrical patterns can then be estimated from two consecutive smoothed images. Using the recently developed discrete Hodge–Helmholtz motion decomposition technique, the curl-free and divergence-free potential surfaces of an estimated velocity field are extracted. Finally, hierarchically searching the minima and maxima on the potential surfaces, the sources, sinks, and rotational centers are located with high accuracy. Experimental results with four real cardiac videos show that the proposed approach performs satisfactorily, especially for the cardiac electrical patterns with simple propagations.

On Transient Motion of a Bowed String.

The transient behavior of a bowed string from a Helmholtz motion to a double slip motion was observed using a simple apparatus consisting of nylon lines which served as a bow, and a string. As the bow speed increases, the motion transits from a Helmholtz motion to a double slip motion. Two Helmholtzian waves with different amplitudes and phases occur during the transition from a Helmholtz motion to a perfect double slip motion. This behavior is examined using numerical simulations, which provide good agreement with the experiments.

Modeling stiffness in virtual bowed-string instruments

Physical models of bowed-string instruments have achieved a degree of completeness that enables understanding and simulation of most of the phenomena that appear in real instruments. Moreover, improvements in hardware technology and the development of efficient signal processing algorithms enable the implementation of these models in real-time platforms, including also improvements and refinements which were not possible before. One of these elements is the stiffness of strings, whose main effect is to disperse the sharp corners that characterize the ideal Helmholtz motion. In this paper, a method is proposed for estimating numerical filters that model dispersion in strings. The phase dispersion of the string is modeled using allpass filters whose coefficients are obtained by minimizing the L-infinity norm of the error between the internal loop phase and its approximation by this filter cascade. This algorithm, which turns out to be a nonlinear version of the Remez exchange algorithm, can successfully design filters of low order which are suitable for real-time implementation, and also accurate enough to model strings of different materials.

The transient behaviour of models of bowed-string motion.

Theoretical models of the action of a bowed string may be able to shed light on differences of "playability" between different violins. Subjective judgements seem to be concerned, at least in part, with the robustness with which one particular oscillation regime of the string (the "Helmholtz motion") may be obtained under different bowing conditions. In this paper, after a review of bowed-string modelling, systematic simulation is used to obtain plots of the basin of attraction of the Helmholtz motion in a particular subspace of the player's control space. Variations in the size and structure of this basin of attraction are seen when parameters of the problem are varied, and some physical interpretation of these variations is given. Some parallels and contrasts are pointed out between the particular features of the bowed string as a nonlinear system, and the range of more familiar dynamical systems. (c) 1995 American Institute of Physics.