Anders Askenfelt

Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods

The first attempt to generate musical sounds by solving the equations of vibrating strings by means of finite difference methods (FDM) was made by Hiller and Ruiz [J. Audio Eng. Soc. 19, 462–472 (1971)]. It is shown here how this numerical approach and the underlying physical model can be improved in order to simulate the motion of the piano string with a high degree of realism. Starting from the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, as well as the force acting on the bridge, are given by the same scheme. The performance of the model is illustrated by a few examples of simulated string waveforms. A brief discussion of the aspects of numerical stability and dispersion with reference to the proper choice of sampling parameters is also included.

Measurement of the bowing parameters in violin playing. II: Bow–bridge distance, dynamic range, and limits of bow force

A method is described that allows measurement of the bowing parameters in violin playing under normal conditions. The measured parameters include bow position, bow velocity, bow–bridge distance, and bow force. The use of the method is illustrated by registrations of typical bowing patterns (sustained notes, scales, crescendo–diminuendo, sforzando, and saltellato). The examples are analyzed with a focus on the players’ use of the bow–bridge distance and bow velocity in controlling the dynamic level. The correspondence between predicted and observed changes in the dynamic level is discussed, as well as the safety margins in bow force. The measurement method is a development of equipment earlier described [A. Askenfelt, J. Acoust. Soc. Am. 80. 1007–1015 (1986)].

Measurement of bow motion and bow force in violin playing

A simple piece of equipment which makes it possible to simultaneously register the motion of the violin bow and the downward force on the string (‘‘bow pressure’’) under normal playing conditions is described. Registrations of various bowing gestures in violin playing obtained with the equipment are presented and discussed.

Measuring the rate of change of voice fundamental frequency in fluent speech during mental depression

A method of measuring the rate of change of fundamental frequency has been developed in an effort to find acoustic voice parameters that could be useful in psychiatric research. A minicomputer program was used to extract seven parameters from the fundamental frequency contour of tape-recorded speech samples: (1) the average rate of change of the fundamental frequency and (2) its standard deviation, (3) the absolute rate of fundamental frequency change, (4) the total reading time, (5) the percent pause time of the total reading time, (6) the mean, and (7) the standard deviation of the fundamental frequency distribution. The method is demonstrated on (a) a material consisting of synthetic speech and (b) voice recordings of depressed patients who were examined during depression and after improvement.

On Vibration Sensation and Finger Touch in Stringed Instrument Playing

The vibration levels in four traditional stringed instruments during playing have been investigated, including the double bass, violin, guitar, and the piano. The vibration levels, which were measured at several positions and at different dynamic levels, were evaluated with respect to reported thresholds for detection of vibrotactile stimuli. The results show that the vibration levels are well above threshold for almost all positions on the instruments in normal playing. It is concluded that the perceived vibrations may be of some assistance with regard to intonation in ensemble playing, in particular for the bass instruments. The finger forces exerted when playing the bowed strings, as well as the touch forces in piano playing were studied briefly. It was concluded that the kinesthetic forces perceived in playing may assist the timing in performance.

From touch to string vibrations. I: Timing in the grand piano action

This article describes an experimental study of the timing in the grand piano action. The function of the action is described by timetables for the motions of the moving parts. Important timing properties included are the relation between key bottom contact and hammer–string contact, the interval of free hammer motion before the impact on the string, and the hammer–string contact duration. The influence of the regulation and dynamic level on these timing properties is analyzed. The results of the measurements are discussed, with a focus on the implications for piano playing, regulation, and design.

Numerical simulations of piano strings. II. Comparisons with measurements and systematic exploration of some hammer‐string parameters

A physical model of the piano string, using finite difference methods, has recently been developed. [Chaigne and Askenfelt, J. Acoust. Soc. Am. 95, 1112–1118 (1994)]. The model is based on the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, from which a numerical finite difference scheme is derived. In the present study, the performance of the model is evaluated by systematic comparisons between measured and simulated piano tones. After a verification of the accuracy of the method, the model is used as a tool for systematically exploring the influence of string stiffness, relative striking position, and hammer‐string mass ratio on string waveforms and spectra.

From touch to string vibrations. II: The motion of the key and hammer

This article describes an experimental study of the motion of the grand piano key and hammer. The motions were registered by means of an optical position measuring system. The measurements include typical key and hammer motions at different dynamic levels and for different types of ‘‘touch.’’ Also, the accelerating force on the hammer before release (‘‘let‐off’’), and the following free motion of the hammer were investigated. Resonances in the hammer were detected and examined by modal analysis. A possible influence of the pianist’s touch on the string vibrations via the hammer resonances is discussed.

Measurement of the bowing parameters in violin playing

A method is presented that allows the measurement of the bowing parameters in violin playing, without interfering with normal playing conditions. The measured parameters include the force between bow and string (“bow pressure”), the position of a contact point between bow and string along the bow (“bow position”), and the distance between the contact point and the bridge (“bow‐bridge distance”). Typical registrations for a sample of bowing patterns will be presented, and the violinists use of the bowing parameters in controlling the dynamic level will be discussed.

Piano string excitation V: Spectra for real hammers and strings

New measurements of piano hammer properties are reported, as well as hammer accelerations during impact and resulting string motion on a grand piano. Comparison with extant theory on string–hammer interaction requires discussion of limitations due to: (1) nonlinear mode coupling for finite amplitude, (2) string stiffness and the resulting dispersion, (3) soundboard admittance, (4) finite hammer width, and (5) nonlinearity in the hammer. Items (1) and (4) are shown to be relatively unimportant for measurements made promptly after striking, while (3) is significant mainly for the uppermost treble octave. Item (2) becomes important above roughly 1 kHz for notes at the bass end, 5 kHz in midrange, and 10 kHz at the treble end. Item (5) is the most important limitation, and leads us to consider either using an ‘‘effective compliance’’ to obtain an approximate spectrum or performing numerical integrations to handle the nonlinearity. With these limitations in mind, an examination is made of how well the theory c...