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Session 03 - Pianos

Modeling the longitudinal vibration of piano strings
B Bank, L Sujbert
Budapest University of Technology and Economics, Department of Measurement and Information Systems, Budapest, Hungary

This paper is about the physical modeling of the longitudinal string vibrations in the piano. Informal listening tests show that the longitudinal vibrations play an important role in attack of the sound, and are responsible for the metallic character of low notes. However, their modeling has not yet been considered in the literature.
First, a simple mathematical model is developed for qualitative understanding. The longitudinal vibration is made up of the free vibration of the longitudinal modes, and a forced vibration coming from a second-order nonlinearity. For the latter, a closed-form solution is given in the case of sinusoidal transversal displacement with non-rigid termination. To investigate how these effects develop in more natural circumstances, finite difference string and hammer models are used, with parameters taken from real pianos. The results are compared to piano measurements.
The aim of this study, besides a better insight to the underlying physics, is to develop sound synthesis methods. Therefore, an efficient modeling approach is presented, which can be used with digital waveguide string models. The approximation is based on neglecting those effects, which have less importance from the perceptual point of view. Sound examples will be presented as a part of the talk at the SMAC2003 Conference.

Quality of treble piano tones
A Galembo
Sechenov Institute of Evolutionary Physiology and Biochemistry, Russian Academy of Sciences, Lab 32, St. Petersburg, Russian Federation

In this study it is shown that the quality of a treble tone of the piano is defined primarily by the temporal dynamics of the perceptually competing impact noise ("knock") and the string-born tonal element (the partials responsible for pitch sensation). The relation between the two elements depends primarily on the parameters of the hammer-string contact, and not so much on the soundboard properties. The knock component is "colored" by the metal frame, in particular by the choice of material. Two ways (mechanical and electronic) of experimentally separating the knock and the tonal element in a real piano tone for listening tests and demonstrations are shown. (The paper presents results of previous industrial research conducted in the acoustic laboratory of the Leningrad Piano Factory, Russia to English readers.)

Phase randomisation in piano bass tones
A Galembo¹, A Askenfelt²
¹Sechenov Institute of Evolutionary Physiology and Biochemistry, Russian Academy of Sciences, Lab 32, St. Petersburg, Russian Federation; ²KTH, Dept. of Speech, Music and Hearing, Stockholm, Russian Federation

It has been shown that the timbre of synthesized multi-component harmonic bass tones with steady magnitude spectrum depends on the phase spectrum, particularly on the randomness of the relative phases between partials [Galembo et al, JASA 110, 1649-1666]. For inharmonic spectra (as in the piano), the waveform will be non-stationary due to the temporal evolution of the phase relations. This creates audible attack transients which are characteristic for the particular starting phases. The relevance of these findings for the perception of real piano tones has been unclear. A piano tone shows a complex dynamic evolution of the spectrum during attack and decay, which besides inharmonicity, is influenced by the properties of the sound board and room reverberation. In this pilot study it is shown that frequency sweeps and associated pitch glides due to phase rotation introduced by string inharmonicity are present in real piano bass tones. The evidence is clear in the string and bridge velocities, and also in the radiated sound close to the piano, although to a lesser extent.

Measurement and reproduction accuracy of computer-controlled grand pianos
W Goebl¹, R Bresin²
¹Austrian Research Institute for Artificial Intelligence, Vienna, Austria; ²Royal Institute of Technology (KTH), Department of Speech, Music, and Hearing (TMH), Stockholm, Sweden

The recording and reproducing capabilities of a Yamaha Disklavier grand piano and a Bösendorfer SE290 computer controlled grand piano were tested, with the goal of examining their usefulness for performance research. An experimental setup consisting of accelerometers and a calibrated microphone is used to capture key and hammer movements, as well as the acoustic signal. Five selected keys are played by pianists with two types of touch ('staccato - legato'). Timing and dynamic differences between the original performance, the corresponding MIDI file recorded by the computer-controlled pianos, and its reproduction are analyzed. The two devices performed quite differently with respect to timing and dynamic accuracy. The Disklavier's onset capturing is slightly more precise (+/-10 ms) than its reproduction (-20 to +30 ms), the Bösendorfer performs generally better, but its timing accuracy is slightly less precise for recording (-10 to 3 ms) than for reproduction (+/-2 ms). Both devices exhibit a systematic (linear) error in recording over time. In the dynamic dimension, the Bösendorfer shows higher consistency over the whole dynamic range, while the Disklavier performs well only in a wide middle range. Neither device is able to capture or reproduce different types of touch.

The piano action as the performer's interface: Timing properties, dynamic behaviour and the performer's possibilities
W Goebl¹, R Bresin², A Galembo²
¹Austrian Research Institute for Artificial Intelligence (ÖFAI), Vienna, Austria; ²Royal Institute of Technology (KTH), Department of Speech, Music and Hearing (TMH), Stockholm, Sweden

A concert pianist is able to produce every imaginable nuance of expression by handling the 88 keys of his/her piano, no one of which travelling through a greater distance than one centimeter. In this study, we investigate the temporal behaviour of grand piano actions by different manufacturers with different types of touch ('legato' versus 'staccato'). An experimental setup consisting of accelerometers and a calibrated microphone was used to capture key and hammer movements, as well as the acoustic signal. Five selected keys were played by pianists with the two types of touch. The analysis of the three-channel data was automated by computer software. Discrete measurements (e.g., finger-key, hammer-string, key bottom contact times, hammer velocity) were extracted for each of the 2563 recorded tones in order to study several temporal relations. Travel times of the hammer (from finger-key to hammer-string) as a function of hammer velocity varied clearly between the two types of touch, but only slightly between the pianos. A travel time versus hammer velocity function found in earlier work derived from a computer-controlled piano could be replicated. Key bottom contact times exhibit larger variability between types of touch and pianos. However, no effect of touch type was found in the peak sound level (in dB as a function of hammer velocity). This finding raises once again the question whether other factors than hammer velocity influence the tone of the piano.

The pretransient of the harpsichord sound
T Gäumann
Ecole Polytechnique Fédérale, Institut denChimie Physique Moléculaire, Lausanne, Switzerland

The plectrum lifts the string of the harpsichord until the string bounces off and jumps downward. In contrary to the piano, the string is simultaneously excited to a vertical and a horizontal vibration and reaches its maximum amplitude in the beginning. The movement of the string is a pulse with very steep rise and decay times that travels to the bridge. The harmonics of the string vibration extend into the ultrasound frequency region with high initial amplitudes. The bridge transfers the signal to the soundboard where it is radiated into the air. The emission of the sound transient is delayed compared with its formation. On the mikrophone a precursor can be observed: Although of smaller amplitude, it is important for the "color" of the sound. We call it the pretransient. With the help of laser trigonometry, accelerometers, and mikrophones the movement of jack, plectrum, string and bridge have been measured. It allows the attribution of some of the signals. E.g. the first sound to be observed corresponds to the vibration of the plectrum when it bounces off. It lasts a few milliseconds and is in the frequency range of 3 to 8kHz. The origin and significance of some of the pretransient signals is discussed.

Realization of an electric clavichord
J Knif
Sibelius-Academy, Music Technology, Helsinki, Finland

The author has developed and built an electric (e-) clavichord with plans to continue the development. The construction of the e-clavichord markedly differs from that of the acoustical clavichords. Clavichords were common keyboard instruments from the 15th to the early 19th century. Among keyboard instruments the action of the clavichord is the simplest possible, yet the most expressive, yielding to a wide dynamic range, vibrato, and pitch bending possibilities. The dynamic level of the acoustical clavichord is, however, very low. There have been attempts to overcome this limitation by amplifying its sound, picked up by either acoustic or piezoelectric microphones. To the authors' knowledge there have not been attempts to use magnetic pick-ups. This approach was chosen here, because the aim was not to produce a typical clavichord sound, but to create a new type of instrument. The e-clavichord has a solid body, no soundboard, and a compass of only two octaves to reduce costs and make it portable. Because of the magnetic pick-up, iron strings had to be used instead of the normal brass. The resulting instrument is playable with satisfaction. Certain problems in string damping mechanics and handling noises have to be addressed and more pick-up types tested. Because of certain lack of character in the tone, a semi-acoustic type should also be tried.

Piano string vibration and problems of its numerical modeling
A Raffaj
Petrof, spol. s r.o., Research dept., Hradec Kralove, Czech Republic

The specifities of a piano string vibration and their influence on a final piano sound timbre are summarized. The numerical model of a piano string was created by means of the finite difference methods (FDM).
As piano hammer model which excites virtual piano string was used non-linear hereditary (hysteretic) hammer model introduced by Stulov.
The dynamical force-compression characteristics of piano hammer are measured on the special measuring apparatus, which will be described and compared to the Stulov's apparatus. The measured results is compared to piano hammer properties measurements which are done usually by hammer manufacturers.
A brief discussion of hammer properties influence on final piano timbre is presented. For the simulation of reality the real soundboard was excited by shaker which is supplied by computed signal. This signal is proportional to force transmitted from string to the bridge. In this way is aurally judged the influence of string qualities, way of hammer manufacturing and quality changes during the voicing.
At the conclusion the possibilities of the numerical model improving by addition of the longitudinal modes simulation and non-linearity transfer of string vibration to the bridge is presented.

Sonological analyses of Harpsichord Sounds
A Schneider, A Beurmann
Universität Hamburg, Musikwissenschaftliches Institut, Hamburg, Germany

Historical harpsichords build by famous instrument makers such as the Ruckers family (Antwerpen), the German/Dutch Johannes Dulcken, or Italian masters such as Giovanni Celestini, Antonio Migliai or Guiseppe Mondini are said to be perfect in regard of craftman-ship, playability, and sound quality. With respect to sound characteristics, one often reads verbal descriptions of professional musicians whereby, for example, the sound of certain Italian instruments is regarded as being "dark" and "full" whereas the sound of well-known Flemish harpsichords in comparison has been judged to be "bright" and "transparent". Our study presents sonological analyses of sounds recorded from some historical harpsichords (Ruckers, Mondini, Tasquin, Kirckman) of one of the authors. Sonological analyses that are part of organological documentation aim at an objective description of temporal and spectral sound characteristics (such as attack transients, spectral energy distribution relative to time and trajectories of spectral centroid, inharmonicity of partials, etc.) for each single instrument. Further, the effect of different tunings which in instruments producing sustained notes (such as pipe organs) yet also in harpsichords clearly influences the spectral composition of the radiated sound, is taken into account. Our analyses use short-time FFT, LTAS, Phase Vocoder, Wigner-Ville-Transforms, LPC, wavelet filter banks (Gammatone, Gauss), as well as autocorrelation and phase space techniques.

The piano soundboard behaviour in relation with its mechanical admittance
J S Skala
Petrof , spol s.r.o., Research, Hradec Kralove, Czech Republic

Piano soundboard input mechanical admittance was investigated in wide frequency range. In lower frequency range strong differences between peak and pit values have been observed. With the increasing frequency, the wrinkled character of soundboard admittance is changed to more smooth one, due to a coupling of modes. The idea was to compare piano soundboard behaviour upon driving by shaker supplied with simple harmonic force signal on near peak and pit (mode + antimode) frequency. The measurement was made on soundboard fixed to backframe and iron frame without strings. We expect that on antimode frequency the soundboard radiation will be very poor. The final perceived sound volume differs less than it has been expected. A discussion if rest of sound energy can be leakaged out more either on other frequency partials by soundboard or by other piano parts (iron frame, rim) is included. This effect is one (but not only one) reason for we can hear a piano tone with frequency in which level of the soundboard admittance is very low.

Experimental and theoretical studies of piano hammer
A Stulov
Institute of Cybernetics at Tallinn Technical University, Centre for Nonlinear Studies, Tallinn, Estonia

Based upon the large number of experimental data obtained using a special piano hammer testing device, it has been shown, that all the present-day piano hammers have as a quality the hysteretic type of the force-compression characteristics. This not a chance because such a hysteretic character has been developed step-by-step following the history of evolution of piano hammers since the instrument was created. The dynamical behaviour of the contemporary piano hammer can be described by different mathematical hysteretic models. In addition to the four-parameter nonlinear hysteretic model of piano hammer, another new a three-parameter hysteretic model was developed. This is very similar to the nonlinear Voigt model and permits a description of the dynamical hammer felt compression that is consistent with experiments. The both models are based on an assumption that the hammer felt made of wool is a microstructural material possessing history-dependent properties. The equivalence of these models is proved for any realistic values of hammer velocity. The continuous dependencies of the hammer parameters on the key number are obtained, which is the first known case of such an analysis. The application of hysteretic models to numerical simulation of the grand piano hammer-string interaction is demonstrated. The flexible string vibration spectra excited by different piano hammers are analysed. All that together, leads to a new method for piano stringing-scale design.

Analysis of piano tones with soft and hard touches
H Suzuki¹, J Kanno², S Mikimoto³
¹Information and Network Science Department, Chiba Institute of Technology, Japan; ²Paris, France; ³Shinjyuku, Japan

The relation between the touch and the piano tone is not fully understood yet. Keys for G3, G4, and G5 were played with two types of touches, i.e., hard (A) and soft (B) touches. A key was pressed and recorded with touch A for several times until the peak level was within a target range, for example 90 dB +/- 0.3dB. Next, the same was repeated with touch B. This combination was repeated for 10 times. Individual signals were transfered from a digital recorder onto a PC and edited so that all signals have the same onset time and duration. The signals were analyzed by use of FFT. A meaningful spectrum difference was observed only for note G5.

Deviations in perception of tuning of pianos by piano instructors and piano tuners.
M Zenker¹, E Castro Sierra², A Ramírez³
¹National School of Music UNAM, Music Instruments, Mexico City, Mexico; ²National School of Music UNAM, Neurosciences and Psychoacoustics Department, Mexico City, Mexico; ³CINVESTAV, Section of Bioelectronics, Electrical Engineering Department, Mexico City, Mexico

Two studies on accurate piano tuning are being undertaken at National Music School of UNAM. They take into account, among other factors, the necessity of mistuning the two or three strings of the same tone (Weinreich, 1979). First, an analysis of deviation in the perception of tuning by 20 piano instructors and 2 piano tuners with the AEP EEG method (Lopes Dasilva, 1999) and, second, an analysis of tuning of 15 of the best pianos at the Music School and of the private pianos of the piano instructors. The objective is to compare the amplitude and the latency of the P-300 wave evoked by infrequent tones with specific frequencies in a sequence, and relate the latencies and amplitudes of P-300, evoked by the infrequent tones, with the perception of the fundamental of the frequent tones by the subjects. We expect that the tuners, working under best conditions, and from the AEP results obtained, will be found to tune the instruments with a deviation of more than ±5¢, excepting the octaves. Additionally, employing an absolute tempered tuning at 440 Hz, the piano instructors should show a deviation in the perception of tuning, from memory, of more than ±5¢.

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