Acoustic Identity of Restored Pianos of the Romantic Period
From ISMA95, Paris, France
Flavio Ponzi*, Alessandro Cocchi**, Massimo Garai**, Giovanni Semprini**, Lamberto Tronchin**
* ISPIRO - Music and Theatre Studies Department - University of Bologna - Italy
** Technical Physics Inst. - Faculty of Engineering - University of Bologna - Italy
This paper is intended as a contribution to defining the relationship between the physical changes usually encountered in the sound-boards of 19th century pianos and their sound efficiency. The knowledge and quantification of the extent to which the original physical conditions of the parts determining the sound efficiency, in this case the harmonic table and ribs system, have changed is in fact important for two reasons: detection of the physical change itself and opportunity to document the tone and appearance of romantic and post-romantic pianos. In fact, these instruments have become unusual (and hence cannot be readily identified) before actual physical changes, due to the obsolescence of the related aesthetic taste.
For this task, the effects were quantified of obsolescence and restoration on the sound-board of an Erard petit à queue (Paris 1853) which recently underwent a thorough restoration in terms of sound and appearance. Changes to the sound efficiency of the instrument were verified with spectrum and time analyses run on sound samples taken before and after restoration.
1 Physical Changes of Sound-Boards Due to Obsolescence
The physical changes related to obsolescence are due to the following main reasons:
1) the differences in the shrinkage coefficients of the wood, radially and axially, corresponding to changes in moisture content. Because of the way in which pianos are built, the transversal cohesion of the panels forming the sound-board is ensured by the ribs crossing the two anatomical directions.
In the wood of Picea abies karst., in an axial direction, the maximum possible shrinkage accompanying the change from saturation to anhydration of the tissues is no more than 0.03% (unless the wood be a "reaction" one, fact to be excluded for quality instruments), while radially, shrinkage increases to 3.3% and reaches 6.8% tangentially . A 1% variation in the equilibrium humidity causes a mean variation in size in the wood of Picea abies of 0.24% between axis and radius. These data suggest the remarkable tensions to which may be submitted, following variations in the humidity of the wood, the sound-board-rib system following the above-mentioned crossing, of the anatomical directions between the component parts. The sound-boards of 19th century pianos have mainly come down to us cracked at several points. This is due to the blocking of the shrinkage by strong fixing of the sides onto the stout wooden frame and by the points of intersection of the ribs which are also set into the wooden frame. Although it is wholly likely that the wood used was already in equilibrium conditions with regard to the moisture content (several ways for seasoning the wood were proposed in the first half of the nineteenth century ), it may readily be understood that the manufacturing process, involving strict production rates, was affected by a wide range of environmental conditions. Hence, sound-boards, even when made from well-seasoned "quarters", might have been assembled in non-optimal hygrometric conditions.
2) The viscous-elastic behaviour of the wood (emphasised by the cycles of increase-decrease in the moisture content of the material), which causes permanent deformations to the sound-board, even without loads which would cause plastic deformations in the true sense of the term.
These circumstances have two main consequences:
1) the sound-board-rib system is clearly lowered (3-6 mm and more at points of maximum width), at the same time cancelling the vertical component of the strength exerted by the strings on the bridges;
2) the deformation to which the sound-board-rib was subjected on stringing (due to the considerable vertical arrow), initially mainly elastic, stabilised as the reciprocal constraints settled and became almost permanent, also for not excessive loads.
Shrinkage and viscous compliance both contribute, albeit independently, to this lowering.
2 Methods of Restoration
The decrease in the strength exerted by the strings on the bridges is the most important measurable effect among the physical changes affecting the sound of historic pianos, apart from occasional acts of destruction. In the 1853 Erard piano restored on this occasion, this strength changed from approximately 390 N (± 20%) overall, measured before restoration, to over 1950 N (± 20%) after restoration (excluding in both cases the overspun strings). The restoration aimed at setting the sound-board in a plane configuration (with the instrument stringed), through filling the cracks, complete disassembling, detaching the ribs, and subsequent re-gluing of the original ribs.
In order to show the changes in sound correlated exclusively to the restoration of the strength produced by the strings to values close the original ones, and to the restoration of the sound-board to its initial mainly elastic behaviour, the following precautions were taken, before taking sound samples prior to restoration:
1) the action of the instrument (Erard double échappement) was brought back to its maximum efficiency;
2) the cracks caused by shrinkage were repaired.
Sound samples were then taken (at intervals of a fourth and a fifth), pressing the keys mechanically with four predetermined differing degrees of intensity: the recording was repeated on the same notes upon completion of restoration.
3 Analysis of Deformations
The permanent deformations (strains) of sound-boards were measured on the piano before and after the restoration. They were expressed as displacements along the vertical axis with reference to a conventional zero-plane and were measured in several points, corresponding to the node of an ideal grid. With these data and a graphic software the equal-deformation curves have been produced (Figure 1). The strengths were computed starting from the Brook Taylor formula, widely employed by restorers . It gives the fundamental frequency f of a circular, homogeneous, harmonic string fixed at the extremities as a function of its length l, its diameter d, its density and of the applied strength T :
Knowing the pitch a string should generate and , and measuring l and d, it is possible to compute T. Due to the fact that the string makes an angle with the horizontal plane and using the power expansion of the trigonometric functions, the horizontal (Tx) and vertical (Ty) components of T were obtained as:
x and y being the projections of l on the horizontal plane and on the vertical axes, respectively. Measuring x and y before and after the restoration, Tx and Ty were computed in both cases.
Eq. (3) predicts rather small values for the total Ty: about 214 N after the restoration, and only 55 N before, when many strings are released. Then, measurements of the vertical strength were experimentally performed. The experimental values of Ty, although 5÷7 times greater than those predicted by eq. (3), are always small if compared with the horizontal strength exerted by the whole set of strings. From these data, it appears that the widely used Taylorís formula underestimates the vertical strength on the sound-board.
By an other approach, now under investigation, it is possible to compute the strength on the string from the bridge, using the elastic deformation of the string itself; in this case, the experimental values are strongly similar of the calculated ones.
4 Fourier Analysis
In order to evaluate the aesthetic importance of the performed sound-board restoration, a representative set of tones have been recorded, before and after the sound-board restoration, exciting one string at a time with four mechanically pre-defined intensity degrees. The time-domain decay wave forms have been submitted to a detailed spectral analysis, using an Ono Sokky CF 360 FFT analyser with a full scale frequency of 1 and 10 kHz, for a proper sampling of each sound. For example, Figure 2 is the amplitude characteristic of the first part of the transient of the LA 2 (220 Hz) before the restoration. Considering the energy content, the fundamental frequency is practically absent, the second harmonic is dominant, and the third and fourth harmonics are also present, although very weak. After the restoration the fundamental and the third and fourth harmonics are better balanced with the second harmonic; this emerging third harmonic, which corresponds to an interval of duodecima, is an important characterisation of the timbre of the string (Figure 3). The rate of decay of sound diminishes after the restoration, and at the end of the transient the energy is differently distributed the fundamental vanishes and higher harmonics become more and more important as the excitation intensity grows (Figure 4). Figures 3÷5 refer to the restored piano.
As a first step of an ongoing research on the aesthetic quality of romantic pianos, an 1853 Erard petit à queue piano was restored and the results were analysed using modern acoustics.
The sound-board deformations were recognised as the most important alterations to be restored, also because they affect the tuning and the timbre of the strings. It as been shown that the stresses on the sound-board cannot be predicted using the Taylorís formula like in common practice; now the research is dealing with the quantitative evaluations of the measurements errors and the effects of non-harmonicity. Spectral analysis revealed that many strings donít generates the fundamental note; instead they emit a set of harmonics that have to be balanced by a careful restoration in order to give the original timbre again. Moreover, the spectral content depends also on the analysis time interval, because the beginning of the transient is different from the tail, and on the intensity of the excitation. Therefore, quantitative analysis is needed to help the restorer in finding precise reference points. In this research a great degree of correlation between vertical strength of the strings on the bridge (i.e. permanent deformation of the sound-board) and sound quality was founded. The research is now going on with the aim of obtaining maps of the sound-board point impedance before and after the restoration, using a new technique under development.
 G. Giordano, Tecnologia del legno, Torino: Unione Tipografico-Editrice Torinese, 1981.
 R.E.M. Harding, The piano-forte, Cambridge: Cambridge University press, 1933.
 H. Junghanns, Der piano und flügelbau, Frankfurt am Main: Verlag Erwin Bochinsky, 1991.
 I. Nakamura ,"The vibration of a struck string (Acoustical research on the piano, Part 1)", J.A.S.J. (E), 13, pp. 311-21, 1992.
The authors wish to express their gratitude to Dr. G. Bonamini, Forest Sciences Inst., University of Florence, and Eng. A. Madini-Moretti, Engineering and Consulting, Milan, who always gave us precious information.