© The Author(s), Published by EDP Sciences, 2024

1 Introduction

The action of a piano key aims at propelling the hammer head towards the strings by briefly deepening the side of the key lever facing the pianist, or key front. Its design complies with numerous requirements, including the necessity to impart a sufficiently high speed to the hammer head at the end of the so-called “acceleration phase” [1], so that enough energy is transferred to the strings. Additionally, it ensures that the hammer leaves the strings immediately after striking them, preventing damping of their vibration. For this reason, the components of a piano action mechanism form a complex mechanical system, whose detailed physical description is challenging [2] and prompts intriguing questions regarding the extent of the control exerted by pianists.

The new millennium has witnessed the emergence in literature of multi-body models, relying on previous pioneering works, aimed at reproducing the motion of a piano action. The developed models formulate a system of non linear ordinary differential equations that requires numerical integration based on input variables reproducing the external excitation. The unknowns in these equations correspond to the coordinates of the components comprising the piano action model that has been adopted. These components are assumed to be rigid in most of the reviewed studies, except for two [3, 4], where flexibility of the hammer shank is considered. The model’s degrees of freedom vary depending on its complexity, ranging from one [5] or two [6, 7] to five in the realistic scenario of a modern grand piano action [4, 8, 9], with an additional degree for the damper spoon [10, 11].

The external excitation is incorporated into the equations by introducing a force profile applied at the key front, implicitly aiming to simulate the action performed by a real pianist, considered as a separate mechanical system. The primary objective of these studies is to perform real-time calculations of the realistic force opposed by the key during depression, aiming to develop electronic devices capable of inducing the same haptic feedback as a piano [6, 7, 10, 11]. In general, the reliability of the proposed models is demonstrated through comparisons of calculated outputs with measured ones under identical inputs. In nearly all cases, except for one [9], the virtual system operates on a force-driven basis, where the same external force profile is input in the model and applied to the real key through an external actuator, allowing subsequent comparison of kinematics. In this context, it has been shown [5] that even remarkably simple models prove effective in estimating the displacement of piano action components, while the inverse estimation of the force profile from key motion presents a significant challenge, as it is highly sensitive to both the type and parameters of the model.

Conceptually, shifting the perspective on piano action from an isolated system to a mechanical interface occurs when the exerted force, which concurrently drives haptic feedback, is regarded as the system output. Such a perspective highlights the fact that, particularly during the acceleration phase, the piano action and the pianist’s body are mechanically coupled through contact at the key front. This suggests an alternative perspective, considering the pianist as an integral part of the mechanical system under investigation. Refraining from treating the applied force as external would allow to question its nature. Particularly, it would be possible to understand the circumstances in which the force opposed by the action, carrying information about the action behavior, effectively influences the force at the key. This influence potentially holds a pivotal role in assessing model validity, but its full extent has not been thoroughly examined.

The omission of the exciter in piano action studies, despite its significance in framing the research scope, may stem from a limited comprehension of its interaction with the pianist’s body, notwithstanding the abundance of scientific literature on pianistic touch. Originating from pioneering studies conducted in the early 20th century [12], these investigations predominantly center on the variability of key depression profiles [13] or force applied by the pianist at the key [14] and explore their correlation with the resulting tone [15]. Some studies inquire about the effect of the type of key excitation on the internal behavior of the action and its effects on string excitation, as in the case of studies on the controllability of hammer shank vibration [13, 16], which possibly affects the longitudinal vibration of the strings. The conceptual framework supporting this kind of approaches underscores the notion that analyzing the pianist-key interaction involves treating the pianist as an external agent with respect to the system under examination. This framework is challenged by the fact that a piano keyboard is a haptic interface, whose behavior plays a prominent role in controlling the instrument.

Furthermore, the incorporation of a unifying perspective appears to align the physical modeling of piano key excitation more closely with prevailing concepts in piano technique. For example, pianists assertively maintain their ability to consciously manipulate the employed mass. This belief is exemplified by the commonly held notion that the historical evolution of piano technique has involved an increasing emphasis on employing keystroke strategies that mobilize the entire arm. Inspecting primary sources on piano pedagogy allows to infer that such a conception holds some truth. Indeed, the notion that utilizing arm movements to depress keys is seldom appropriate pervades the methods redacted across Europe during the first half of the 19th century, by educators and cosmopolitan virtuosi of the instrument, intermittently committed to teaching their artistry through the written word. As an illustration, the French Louis Adam (1758–1848) deemed arm engagement detrimental to tone quality ([17], p. 148), while the German Frédéric Kalkbrenner (1785–1849), instructed in Paris, regarded it as a sign of excessive rigidity ([18], p. 2). On the contrary, beginning in the late 19th century, the utilization of arm engagement has gained widespread recognition as beneficial, symbolizing a profound shift in paradigm and discourse within piano pedagogical literature. Rudolf M. Breithaupt’s (1873–1945) writings serve as a representative example of this evolution, advocating for arm involvement to prevent fatigue and enhance tone quality ([19], pp. 66–67, 342).

Indeed, the introduction of the so-called “weight technique” in piano treatises at the turn of the 20th century, such as those by the above-mentioned Breithaupt, coincides precisely with the advocacy of the abandonment of the “old” piano technique, deemed as finger-focused and often described as too “rigid” [20]. In a general context marked by positivism, the abandonment of the digital technique was generally attributed in the sources to the evolution of instrument construction [21]. One may resume the argument as follows: the advent of acoustically more powerful instruments, whose keyboards would be designed to capitalize on this power, necessitates the use of larger muscle chains than in the past. In fact, all hypothesized causal links justifying the above argument would deserve closer examination. To this respect, noting that the use of larger muscle chains implies in turn the involvement of greater bodily masses, the physical aspects underlying the link between the evolution of instrument construction and the involvement of larger masses by the pianist in key depression are of great interest.

Building upon the findings of a prior study [20], we herein conduct a comprehensive review of contemporary literature on piano action modeling and control by pianists, challenging the prevalent notion that the key can be studied in isolation (Sect. 2). The present study endeavors to incorporate the pianist into the analyzed system by empirically examining the impact on piano action motion of masses comparable in magnitude to those engaged by a pianist during keystrokes. We identify a distinct set of physical control parameters during piano keystrokes, underscoring the active participation of the pianist’s bodily mass. To gauge the inertia actively managed by the pianist during a keystroke, as it appears at the key front, we introduce a simplified model of the upper limb’s motion, and the impact of inertia on hammer kinetic energy, hence tone volume, is subsequently illustrated within a one-degree-of-freedom (1-DOF) representation of a piano action (Sect. 3). Experimental validation of these findings is conducted on three distinct keys of two pianos (Sects. 4 and 5). The one, dating from the 1820s, was crafted in the London workshop of William Stodart, a renowned builder of that time, and has been brought back to playing order for this study. The second is a modern counterpart, a semi-professional Kawai grand piano (RX-2, 1995). The experimental application to such disparate instruments as the Stodart, historically linked to the presumed “digital” technique, and the Kawai, whose constructive principles closely trace back to the rise of the “weight” technique, allows us to shed light on the assumed causal relationship between piano manufacture and playing technique. In the context of applying similar measures to heritage instruments, the proposed approach aims to develop a non-invasive protocol for characterizing piano touch-tone relationship properties. This protocol should be compatible with pianists’ actions and provide comprehensive results even when information on regulation and construction is limited or absent.

2 State of the art

The review that follows addresses the dynamics of piano action modeling and control by integrating the bodily presence of pianists, challenging the concept of the key as an isolated system. It emphasizes the importance of considering the inertia exerted by pianists at the key front, underscoring its influence on the key motion profile, hence the tone produced, and suggesting implications for piano touch control strategies.

2.1 The realism of mechanical measurements of piano action dynamics

Observations on the dynamical behavior of a piano action during a keystroke rely on the application of a force at the key front through mechanical excitation. When the compatibility of the excitation with what a human would produce is not a concern, various devices can be employed, such as weights (e.g., 1 kg in [4, 11]) placed on the key and released to initiate the motion. The force profile can be directly generated either by a human [5, 10] or by appropriately driving an actuator, ensuring repeatability [7]. In some cases, the force profile input is arbitrarily determined by the driver to replicate a human-made displacement [4], while in other instances, the actuator is commanded so to directly mimic the force profile produced by a human [3, 6, 8].

Regarding the latter scenario, the task of accurately replicating force profiles through motor commands is briefly acknowledged as a challenging one, owing to factors like the significant difference between the compliant arm of a human and the comparatively stiffer arm of a rotary motor [8]. Nonetheless, the majority of the studies reviewed considers the piano action solely as an isolated mechanical system subject to external force, and the mechanical impedance of the actuator has been deemed irrelevant to the studies’ objectives, or sometimes entirely overlooked. The realism of such a framework is not evaluated, despite the inevitable presence of an actual mass that moves along with the key lever. This mass is often substantial, as in the above-mentioned case of excitation driven by a 1 kg weight. Concerning actuation by humans, the span of values of such a mass can be large indeed. In fact, the production of a single piano tone by a human involves motor abundance [22], signifying that the same fingertip movement can be achieved, even extending to the sole upper limb, through a potentially unlimited number of spatiotemporal motion profiles across all joints of the kinematic chain. As a consequence, the mass of the actuator could be either negligible, when experimenters strike the key while keeping their entire arm still except for one finger, or otherwise very significant.

Gillespie et al. [7] argue that to truly grasp the “feel” of a piano action, it is imperative to incorporate the actuator impedance into the mechanical system under examination. They introduce a linear model designed to scrutinize the operating point of a piano action, providing insights into instrument design by considering the “impedance match,” including inertia, between the piano action and the pianist [23]. Transitioning from instrument design to performance, there arises an intriguing proposition: pianists stand to gain from a deliberate impedance mismatch instead, by adjusting their own actuation impedance.

In fact, recognizing the potential consequences of differing impedance balances between the two subsystems is vital when addressing the “feel” of a piano action. When the actuator impedance significantly exceeds that of the excited mechanical system, the system operates in a purely motion-driven regime: the exciter can replicate any intended trajectory, regardless of the mechanical system under excitation. In the opposite scenario, the system operates in a purely force-driven manner: the external driving force is transmitted unchanged on the key. Interestingly, this also implies that, in a motion-driven regime, the force measured on the key is indeed sensitive to the force opposed by the action during the imposed trajectory, thereby capable of conveying information about its behavior. In a purely force-driven regime, conversely, the force feedback from the action sub-system is virtually null.

To accurately evaluate a piano action model by comparing measured and estimated forces at the key, one should ensure the force profile not to be compromised by inadequate control over the actuator impedance. Otherwise, the comparison could be misleading, as the force profile might lack relevant information about piano action dynamics. Moreover, the very relevancy of the comparison should rely on how force feedback effectively conveys information about the keyboard dynamics to pianists. This must depend on the impedance pianists put on the key while playing, and particularly the amount of inertia they engage.

2.2 The control parameters of piano touch

Otto Ortmann, a pioneer in the study of the physics of piano touch, is considered as the first to have attempted to ascertain the physical reality of beliefs transmitted in the field of the teaching of piano touch, starting from demystifying the physical counterparts of these beliefs and observing their effects on key movement. As a piano teacher and later the director of the Peabody Conservatory, his preliminary systematization of these beliefs constitutes a credible testament to the state-of-art piano technique discourse of his time. In his The physical basis of piano touch and tone (1925) [12], he observes that changes in what he calls the “playing unit” (finger, the whole hand, hand plus forearm, the whole arm) can effectively increase the speed of the key by increasing what he calls the applied “weight.” Moreover, beyond any intentional changes in the applied force, he conceptualizes touch as more or less rigid^(a)1 (i.e., whether wrist and elbow are tight or not) and percussive^(a) (i.e., whether the moving finger strikes the surface of the key or rests on it before descent).

Ortmann’s observations lead him to claim that the qualitative difference in key movement is solely influenced by percussiveness. A percussive^(a) touch is characterized by a distinct spike in velocity at the onset of key displacement, resulting from the momentum imparted by the striking finger to the key. Subsequent studies have confirmed Ortmann’s assertion, leading to the common categorization of piano touch into “struck” and “pressed” [24]. This distinction is perceptually significant, as the impact between finger and key affects the tone’s attack by adding a precursor to the sound that arises from strings vibration [15]. Additionally, for producing very loud tones, the “struck” touch has been claimed to be more efficient in terms of muscular effort [14].

Drawing upon Ortmann’s publications and other relevant sources in 20th-century piano pedagogy, McPherson and Kim [25] seek to provide a thorough understanding of key depression control for expressive purposes. They identify five parameters, three of which influence tone production before the escapement of the hammer:

• The velocity^(b) imparted to the key, controlling the speed of the hammer;

• Percussiveness^(b), referring to the momentum generated by the finger at the start of key depression;

• Rigidity^(b), categorized as low, medium, or high, depending on whether the force primarily originates from finger joints, the wrist, or the entire arm, with hand muscles held either loose or tight.

The authors note that the impact of varying rigidity^(b) is not discernible in key motion resulting from a non-percussive touch. While this means that the effect of rigidity^(b) is not visible solely from kinematics, it does not necessarily imply that the resulting tone from a “pressed” (non-percussive^(b)) touch is solely controlled through the velocity^(b) imparted to the key.

Ortmann notes that, regardless of percussiveness^(a), employing a rigid^(a) touch with curved fingers can produce louder sounds compared to a relaxed touch with flat fingers, even when pianists aim for the same volume. This implies that pianists, possibly unconsciously, link distinct bodily movements while playing with different outcomes in tone production. While this observation does not definitively prove that any alteration in mechanical impedance of the actuator directly affects piano tone, it raises the question of whether changes in mechanical impedance lead to adjustments in key control strategies, potentially influenced by shifts in haptic feedback. Neither key front kinematics nor force at the key front, when considered in isolation, can adequately explore this potential influence.

When considering the physical presence of pianists within the examined system, the level of rigidity^(b), particularly associated with the level of inertia engaged by pianists at the key front, emerges as a significant control parameter inherent in all types of key motion profiles. The impact of pianists’ engaged inertia has only been briefly mentioned regarding “struck” touch, suggesting that the momentum of the pianist initiating key depression results from the multiplication of finger impact velocity by the engaged inertia as observed at the key front [25]. However, this aspect has yet to be thoroughly investigated in the context of “pressed” touch.

3 Model

In the ensuing section, we construct a simplistic biomechanical model depicting the pianist’s activity while depressing a key, with a focus on identifying and quantifying the pertinent control parameters, while accounting for the pianist’s bodily presence. Subsequently, an introductory description of a real piano action serves to justify, following Thorin et al. [5], the introduction of a linear 1-DOF model, allowing to determine the kinematics of key during the acceleration phase, in response to a “pressed” keystroke. The model assumes that crucial elements of piano action physics, notably compliance and friction at component interfaces, have a negligible effect on hammer kinematics, within the defined conditions of excitation and during the acceleration phase. On the other hand, this framework facilitates the elucidation of the interplay between the control parameters and the ensuing velocity of the hammer, thereby influencing the resultant tone volume. Although broadly applicable to various types of piano actions, including the contemporary double escapement mechanism, our model is here tailored to the English escapement action [26], with which the Stodart pianoforte, considered in the present study, is equipped (see Fig. 1). Characterized by a pushing system, this action design enjoyed significant popularity in France and England for several decades spanning from the late 18th century to the latter half of the 19th century. The motivation behind this study stems from inquiries into the perceived “lightness” of touch, encompassing both the keyboard and the associated playing technique, exhibited by this specific piano specimen when compared to modern instruments.

Figure 1 View of an English escapement action (Stodart) during the acceleration phase (damper not reported). The lever arms, indicated by dashed arrows, determine the mechanical coupling between the key lever body and the hammer lever, through the escapement hopper.

3.1 The engaged bodily inertia of the pianist

The inertia involved in striking a piano key varies significantly depending on the fraction of the performer’s body engaged in the keystroke, referred to as the “playing unit” by Ortmann, whether it be solely the finger, the entire hand, the hand and forearm, or the entire arm. A basic model of the human limb enables estimation of the magnitude of the inertia engaged by the performer in each scenario. This model represents the arm as a two-dimensional chain of levers, referred to as “segments” herein, with joints serving as alternative rotation centers corresponding to the limb articulations. When analyzing rotation around a joint, friction is disregarded, and more distal joints are assumed to exhibit perfect rigidity, while more proximal segments remain immobile. Despite observations on pianists’ motion while striking a key revealing the significance of relative movements of the limb segments [27], it is reasonable to neglect it within the approximation of this study, considering the short duration of the acceleration phase of a key (which never surpasses 150 ms in conducted measurements). For a comprehensive explanation of the model’s development, interested readers are referred to a prior publication [20].

One of the primary aims of this paper is to examine a key assumption implied by the proponents of the weight technique: that the advent of more powerful pianos necessitates the engagement of larger bodily masses (see Sect. 1). Consequently, this study intentionally excludes source compliance to focus strictly on the mechanical effects of involving larger bodily masses in tone production. The proposed model therefore disregards muscular activity, despite its significance in the biomechanics of piano playing. Specifically, the torque responsible for any joint rotation in the arm arises from two main contributions: one of a strictly muscular nature, given by the contraction of the agonist muscles of the rotation, and one of gravitational nature, allowed by the relaxation of the antagonist muscles of the rotation. Pianists appear to benefit from this muscular redundancy. Furuya et al. [28] investigated the “downswing” movement – elbow extension – performed by novice and expert pianists before a struck touch to achieve speed at key impact. The authors observed through EMG measurements that the onset of the downswing is not immediately accompanied by triceps activation, the primary agonist for elbow extension. Therefore, before triceps activation, elbow extension must be driven by gravity via the relaxation of the antagonist muscles (mainly the biceps). The instantaneous torque during the downswing, which combines muscular and gravitational contributions, was estimated, and the impulse of the total torque was calculated for the intervals before and after triceps activation, distinguishing the gravitational from the muscular contributions. The findings reveal that, for the same sound intensity, these contributions are comparable in novices, but among experts, the muscular contribution is negligible. To our knowledge, no studies have examined the balance between gravitational and muscular contributions during the key press itself, particularly in pressed touch. However, the selectivity shown by expert pianists in utilizing muscular redundancy during the preparation of a keystroke suggests that an isolated study of gravitational contribution is not entirely implausible.

Considering gravity, the model still effectively captures the existing correlation between the inertia, M_I, engaged by the I-th playing unit and the resulting increase in maximum attainable force. This correlation is derived from the downward driving force applied at the fingertip, due to the weight torque of the engaged segments, referred to as F_I herein. Understanding F_I provides a dependable measure of the force achievable by engaging each playing unit. Moreover, an assessment of its impact prompts a reevaluation of the concept of weight playing. Due to the finger’s very low F_I, any excitation solely by finger rotation is omitted from the present model.

Utilizing the developed model, we were able to estimate values for both the parameters M_I and F_I of the I-th playing unit by considering rotations (1) around the wrist, (2) around the elbow, and (3) around the shoulder. The mass distribution of each segment was determined based on anthropometric measurements [29] for the first author, a 75 kg male, whose limb segments (hand, forearm, upper arm) are 19.5, 30.5 and 31.5 cm long, respectively. Additionally, the geometric configuration required for parameter estimation was extracted from the illustration in Figure 2, which depicts a keystroke by releasing the weight of the arm, commonly known as the “fall” movement [19]. Each I-th limb segment, rotating around its respective center of rotation O_I, is considered as inclined at an angle θ_I with respect to the horizontal, assumed constant during the acceleration phase of the key depression, due to the small angles of rotation. The results of the estimation are reported in Table 1.

Figure 2 Keystroke through “throw” [Wurf] or “fall” of the arm ([19], p. 97).

3.2 The acceleration phase of a pressed keystroke

A piano key functions as a lever, rotating to elevate a system of levers, totaling N, which includes the key itself. In the case of the Stodart, the number of levers can be reduced to N = 2 during the acceleration phase, as the lightly spring-supported escapement hopper (Fig. 1) can be regarded as part of the key lever body. The rotation of the key lever is normally limited to very short angles, typically of the order of 2–3°, until some obstacle (the keybed itself in both modern pianos and the Stodart) prevents it from advancing. The pianists induce this rotation by pressing their finger on the visible surface of the key lever, at a distance from the lever’s pivot that can extend up to the key front, at a distance L_k (20 cm on the Stodart, about 25 cm on the modern piano). Due to the small angles involved, the rotation appears at the key front as an almost vertical depression. The minimum force initiating down-motion, F_down, expresses the static feel for the action opposing its displacement. It is a parameter regulated by technicians and called “downweight.”

By slowly and gradually depressing the key through the application of a force initially slightly greater than F_down, it is possible to glimpse under the strings the rising head of the hammer. This latter, a light-weight (its mass m_h spans from 7.3 to 4.7 g on the Stodart, and from 11–4 g on modern instruments [30]) piece of wood, composed of a shank (of length L_h = 13 cm on both the Stodart and modern grand piano actions) and a head covered with several layers of leather or felt, can be seen as the last lever of the transmission chain. The piano mechanism is designed so that the increase in elevation of the hammer head is much greater than the corresponding difference in depression of the key, by a factor α of approximately five for modern pianos, and eight for the Stodart. When the key reaches a depth Y_k, of the order of 5 mm for the Stodart and 8 mm for the Kawai, the hammer, now at a distance δ_h from the strings, is seen to “escape” the key, falling spontaneously onto the check, attached to the key in more modern instruments (including the Stodart). The angle swept by the hammer during the mechanical coupling can be estimated as the arctangent of αY_k/L_h, namely 17° for both instruments, and hence be considered as small for the scope of this study.

The piano action mechanism can be seen as a dynamic system, with the external force applied at the key front as its driving input and the hammer kinematics as its output. Thorin et al. [5] have shown that such a system displays a strong integrating behavior, as any irregularities in the force at the key, that may mainly arise as an effect of the interaction forces between its components during the acceleration phase, are smoothed out in the resulting displacement. As a result, when one aims to assess the effect on hammer velocity of the application of a force at the key front, the compliance of its components at the contacts can be disregarded, assuming that they are rigidly constrained and neglecting slips. Maintaining the assumption of small angles, the angular coordinates of the components of the mechanism can be considered coupled through simple geometric relationships, as shown in Figure 1. As a result, a 1-DOF model can be introduced for describing the hammer kinematics before escapement, especially when inertia dominates the system [5].

Stick-slip transitions at the contacts and pivots significantly affect motion when force profiles including values lower than the threshold F_down are applied. However, since F_down experimentally represents the minimum force required to just slightly depress the key a few millimeters, restricting our experiments to scenarios where a sound is effectively produced via the application of a step force profile to the actuator eliminates this issue. These hypotheses are not applicable in cases where the initial impact velocity of the actuator is non-zero, as loss of contact following the collision inevitably leads to zero applied force. Moreover, in this case the dynamics show a pronounced decoupling between hammer and key kinematics (as shown in Fig. 1 in [24]), highlighting the significant influence of contact compression forces. We will also assume that the intricate let-off phase, which immediately precedes escapement, is sufficiently brief compared to the acceleration phase. This allows us to neglect its impact on the hammer kinematics. Another simplification, enhancing the clarity of expressions, involves treating the hammer as a point-wise mass, m_h, connected to its rotation axle through a mass-less, rigid beam.

The 1-DOF piano action exhibits an equivalent inertia at the key front, expressed as the ratio of its 1-DOF moment of inertia, J, and the squared key lever arm, L_k:

$$m\equiv \frac{J}{L_k^2}=m^{\text{'}}+{\alpha }^2{m}_h,$$(1)

where m′ gathers the contributions from components other than hammer, reducing to the key lever body in the case of the English action, and α is the displacement ratio of hammer head and key front, as discussed above.

The complete model of the pressed keystroke (e.g., under zero-velocity initial condition) is depicted schematically in Figure 3. The mechanical system under investigation comprises both the rigid 1-DOF piano action mechanism and an ideal mass M in contact with it at the key front. M reproduces the amount of inertia that would be engaged by a pianist or by an exciter during the acceleration of the hammer. The forces acting on the coupled mechanical system are the downward driving force F, applied to the actuating mass M and due to an external source, and the reaction force of the action. This latter can be regarded as a torque $\vec{C}$ applied at the key lever center of rotation or equivalently as a force applied at the key front, f = C/L_k. This force gathers several contributions, whose nature is diverse, including the weight torques of the components as transmitted to the key front. f will be assumed here as a function of time and displacement, but not velocity. The average value of f(t) during the acceleration phase, 〈f〉, can hence be considered as independent of dynamics. When considering strains at the pivots, dry friction is effectively constant in the case of a strictly increasing displacement and is then included in such an assumption. On the other hand, viscous frictions are neglected and one should expect the model to be invalidated for sufficiently high dynamics.

Figure 3 Diagram representing the displacements of the key front and the hammer head under the effect of excitation by an actuator of mass M and subject to a driving force F, in our 1-DOF model of the acceleration phase of the mechanism.

The model proposed facilitates the derivation of concise expressions elucidating the influence of construction and regulation parameters on the transformation of work of the driving force F into kinetic energy of the hammer. Considering first the 1-DOF piano action in isolation, one may write the kinetic energy of the hammer when it strikes the strings, K_h, as a function of the average acceleration of the key front during the acceleration phase, 〈A〉:

$$K_h={\alpha }^2{m}_h{Y}_k\langle A\rangle -m_{h}g{\delta }_h,$$(2)

where α is the displacement ratio of hammer head and key front, m_h the hammer mass, Y_k the key depth at escapement, g the acceleration of gravity, and δ_h the let-off distance. It can be seen that the acceleration of the key is transformed into hammer kinetic energy through the product of α², m_h and Y_k, three parameters that play an eminent role in dependable conversion of mechanical work at the key front into hammer’s kinetic energy when striking the strings.

Considering the system as a whole, the hammer’s kinetic energy can also be expressed as a function of the average excitation force 〈F〉 applied through the exciter inertia M:

$$K_h=\frac{{\alpha }^2{m}_h{Y}_k}{M+m\text{'}+{\alpha }^2{m}_h}\left(\langle F\rangle -\langle f\rangle \right)-m_{h}g{\delta }_h,$$(3)

where the total inertia of the system is the sum of those of the exciter and of the key, according to equation (1). It appears that greater amounts of inertia M reduce the hammer kinetic energy, as some of the mechanical work done by F is dispersed in order to set M in motion.

4 Measurement protocol and signal analysis

The subsequent section delineates experiments conducted on the Stodart pianoforte to investigate the impact of driving force and actuator inertia on tone production. The “mechanical arm” is a device designed for repeatable excitation of piano keys (Fig. 4). The mechanical arm can be set to engage with a piano key through mechanical contact with this latter at its “finger” (P in the figure, at a horizontal distance X_a from the center of rotation O). In essence, the weight-driven force at its finger and its moment of inertia can be independently controlled, as they both depend on the positions of two masses, m₁ and m₂, suspended along the two sides of the mechanical arm, at horizontal distances x₁ and x₂ from O. While the torque due to the lever weight is negligible by design, the force created by the suspended weights at the finger reads as:

$$F(m_1,m_2,x_1,x_2)=\frac{m_1{x}_1-m_2{x}_2}{X_a}g,$$(4)

where g is the acceleration of gravity. In the limit of small rotations, this force remains constant during the key deepening, and the mechanical arm exhibits an inertia at its finger given by:

$$M(m_1,m_2,x_1,x_2)=\frac{J_a}{X_a^2}+\sum _{i=1}^2\mathrm{}{\left(\frac{x_i}{X_a}\right)}^2,$$(5)

where J_a is the moment of inertia of the lever with respect to O.

Figure 4 Illustration of the mechanical arm positioned to engage with the piano key at contact point P. The arm is initially held in place by a current generator (G) that powers the solenoid of an electromagnet (E). The magnetic force generated by the current keeps the arm attached to a fixed support. Movement begins when the current supply is switched off.

Utilizing the mechanical arm, measurements were conducted on three keys of the Stodart and Kawai pianos, representing distinct registers of the instrument: D₂ (two octaves below middle C), D₄, and F₆. For each predetermined value of inertia, corresponding to the values M_I simulating attacks from the “wrist,” “elbow,” or “shoulder” (see Tab. 1), excitations were administered by applying force ranging from the minimum requisite to elicit sound to approximately the weight of the respective playing unit, F_I. The driving force was applied with the mechanical arm’s finger placed at 2 cm from the key front. Collectively, approximately 80 data points were recorded per key.

We measured the key downward motion via a Fiberoptic Displacement Sensor (Philtec-D171), positioned at a height of approximately 2 mm from the surface of the key at rest and at a distance of about 10 cm from the key pivot. The measured signal, lasting 4 s, was hence scaled to obtain the displacement of the key front. The hammer acceleration was captured using a light-weight (0.7 g) accelerometer (PCB Piezotronics, model 352B10) placed on it, at a distance of approximately 10 mm from the pivot. The kinematic signals were acquired at f_s = 51.2 kHz. The tone produced was sampled in the near field through the average signal coming from a pair of omnidirectional microphones (DPA 4006A). The rods supporting them were placed about 35 cm above the passage of the A₄ strings on the soundboard bridge, corresponding approximately to a 1 ms propagation delay from the soundboard to the microphones. The microphones were spaced 35 cm apart and aligned. The microphone signal acquisition was performed separately and at a sampling frequency of 64 kHz, resampled at f_s and synchronized with the kinematic measurements.

The hammer acceleration signal was subsequently integrated to determine its maximum velocity. An example of the resulting signal is provided at the top of Figure 5. The maximum hammer velocity (MHV) serves as an estimate for the let-off moment, signifying the end of the acceleration phase. A parabolic regression has been realized on the key front displacement signal (center of Fig. 5), windowed between 10% and 90% with respect to the displacement corresponding to the estimated let-off moment. The regression provides an estimate of the average acceleration of the key front, 〈A〉.² From the acoustic pressure signal (bottom of Fig. 5), the perceived tone volume S was estimated as the maximum of its time-varying loudness according to [31]. In practice, the algorithm developed by the company Genesis was used [32].

Figure 5 Estimated signals for a key depression with F = 3.36 N and M = 0.443 kg on the Stodart’s key playing D2. From top to bottom: hammer head and key front velocities; key front displacement; acoustic pressure of the produced sound. Vertical lines in the figure indicate the onset time of hammer free rotation (MHV), estimated as the local maximum of hammer velocity, and a representative time of the hammer-string impact phase (HS), estimated as the maximum of the raw hammer acceleration signal (not displayed). The parabolic regression on the displacement signal in the interval (TOn, TOff), corresponding to displacements (DOn, DOff), is superimposed on the key displacement signal.

5 Results and discussion

In Figure 6, we illustrate the relationship between the average acceleration of the key front and the initial weight force, applied through the mechanical arm at 2 cm from the key front. As anticipated by our model, the resulting acceleration is significantly influenced by the inertia involved by the actuator. By varying such inertia across three distinct values, the data segregate into three distinct groups, each delineating its own trend. A preliminary linear analysis reveals that, for each key, these trends converge towards a singular force value, F₀, indicative of the average opposing force at the key front during depression, 〈f〉 (see Eq. (3)). Upon deriving the F₀ coefficient through a weighted average of the estimations obtained for the three different values of excitation inertia, a secondary linear regression ensures alignment with this value, resulting in three lines emanating from (F = F₀, 〈A〉 = 0).

Figure 6 Average acceleration of the key front, 〈A〉, as a function of the initial weight force F applied through the mechanical arm approximately 2 cm from the key front. Excitations are obtained for 0 < F < FI at an excitation inertia MI, where FI and MI respectively reproduce the effective weight and inertia of the engaged human limb playing unit (Tab. 1). The engaged playing unit (hand only, hand plus forearm, whole arm) is indicated by the increasing size of markers for the experimental points. Experimental data is approximated by three linear regressions, depending on the engaged inertia MI, passing through the point (F0, 0), with F0 being an estimation of the minimum force to accelerate the key (the standard deviation of the estimates as uncertainty on the least significant digit is reported in round brackets). The values AI = 〈A〉(FI, AI) are reported on the ordinate.

The estimated slope of the isoinertial curves, $(\partial \langle A\rangle /\partial F{)}_{M_I}$ (Tab. 2), serves as an indicator of the efficiency of energy transmission from the exciter to the string. Consistent with the model’s predictions (as it is apparent combining Eqs. (2) and (3)), this slope diminishes with increasing the total inertia, with lower notes generally exhibiting lower slopes for identical exciter inertia. The disparate behaviors observed across the three keys of the two pianos primarily stem from variations in inertia, influenced by hammer mass (Eq. (1)), highlighting the substantial contribution of the key to the overall inertia set in motion. The conducted linear regressions facilitate the estimation of the accelerations A_I corresponding to a given key excitation force, determined by the effective weight of the engaged segment F_I with an inertia M_I.

Figures 7a and 7b illustrate the kinematic chain that links key acceleration to hammer velocity at the end of the acceleration phase, enriching the experimental validation of the proposed model. Figure 7a shows the relationship between the increment in the key’s squared velocity (measured between 10% and 90% of its displacement during the acceleration phase) and the average acceleration during the same time window. Both variables are interpolated from the parabolic regression of the measured displacement, demonstrating that the increase in the key’s kinetic energy (and, consequently, the hammer’s kinetic energy) is proportional to the key’s average acceleration. This result is expected: in a uniformly accelerated motion, as assumed in our model when applying a step excitation force profile, such proportionality scales with twice the displacement occurred. Notably, a consistent slope can be observed for intra-instrument data, with significantly different inter-instrument values, reflecting differences in key displacement during the acceleration phase. Indeed, the two pianos show markedly different key depths at escapement, Y_k, which we estimated as the average key position at the moment of MHV across each set of three keys (see top of Tab. 3). Figure 7b explores the relationship between MHV and key velocity at the same instant, highlighting their proportionality through a constant we identify as the displacement ratio, α (Eq. (2)). This ratio is confirmed to be inherent to the instrument, estimated as the slope of a linear regression over each intra-piano dataset (see bottom of Tab. 3). The proposed model underscores not only that Y_k and α are critical regulation parameters, specific to the type and period of manufacture, but that the product α²Y_k also plays a crucial role in the physical control of tone production.

Figure 7 Kinematic key and hammer data for all the studied keys, obtained from all considered excitations using the mechanical arm, under controlled conditions of inertia and excitation force (M = MI and 0 < F < FI, see Tab. 1). (a) Increment in the key squared velocity, $V_{\mathrm{On}}^2-V_{\mathrm{Off}}^2$, between 10% and 90% of its displacement during the acceleration phase, as a function of the average acceleration 〈A〉 in the same time window. (b) Maximum Hammer Velocity (MHV), as a function of key velocity at MHV time. The dashed lines indicate the average slope of the data for each piano, estimated by robust linear regression and constrained to pass through the origin.

In Figure 8, we present the estimated loudness S for each excitation plotted against the average key acceleration 〈A〉. Each key demonstrates unique behavior influenced by hammer size, string and soundboard properties, and independent of the excitation conditions having produced such an acceleration, in accordance with our model (see Eq. (2)). Generally, in our measurements lower notes appear to exhibit louder sounds at equivalent key accelerations, indicating greater effort due to heavier hammer masses in lower keys. However, previous research has shown that the pattern of variation of peak sound levels for all tones as a function of pitch is unsystematic, and largely specific to the location of the microphone [33]. The conclusion can be drawn [34] that the peak loudness estimate of an audio signal coming from one pair of microphones is neither conclusive about the intrinsic extent of tone intensity, with respect to the tone dynamics of a single key, nor allows to compare the dynamics of tones produced through two distinct keys.³

Figure 8 Estimated loudness for the acoustic pressure signals, S, as a function of the average key acceleration 〈A〉. The data are obtained from all considered excitations using the mechanical arm (see Tab. 1). The maximums of S, Smax, produced on each key by the first author (who is also a pianist) with a pressed touch are shown. S is reported in sones on the left-side y-axis and as percentage over Smax on the right-side y-axis. The data have been approximated by a robust non-linear regression with a power law of the form $S=b_1+b_2{\langle A\rangle }^{b_3}$, where 〈A〉 is expressed in cm/s2. The coefficients of the estimation are shown in figure. The standard deviation of the estimates is indicated in parentheses as uncertainty on the least significant digit.

The measurements can be contrasted with the greatest loudness, S_max, achieved by the first author, who is also a pianist, by pressing the key for maximum output, with the sole constraint of never lifting the finger from the key surface. The loudness ratio S/S_max, over a key-specific reference that approximates the maximum loudness achievable by a human stroke on that key, provides an estimate of the key’s dynamic range, enabling a comparison between the dynamics of two distinct keys. It is important to note that the control parameters of the mechanical arm are set to rely solely on gravity, whereas the loud human-produced reference theoretically involves muscular force. In our experiments, the loudness from a purely passive stroke never exceeds that of the loud human-produced reference. Additionally, as a rule of thumb keys of the same piano exhibit comparable loudness ratios, with the Kawai piano showing lower ratios compared to the Stodart. Particularly, the excitation provided by the mechanical arm effectively covers almost the entire Stodart’s dynamic range, but is only sufficient to achieve barely more than half of Kawai’s. If the mechano-acoustic transmission chain in the piano is linear in a first approximation, the relationship between acceleration and loudness can be expressed by a power law, in accordance with Stevens’ empirical law. We conducted nonlinear regression analysis to extrapolate loudness from given accelerations, yielding the results depicted in Figure 8.

The A_I values (see Fig. 6) can now be leveraged to estimate the corresponding loudness S_I through the regression analysis depicted in Figure 8. The analysis of the resulting loudness ratios S_I/S_max (Tab. 4) provides insight into the interplay between increasing equivalent inertia, M_I, and the corresponding applied effective weight, F_I. While there is a general trend of loudness ratio augmentation with increasing inertia, “elbow” and “shoulder” excitation types yield comparable loudness ratios (D₂ on Kawai, D₄ on Stodart) or even a decrease in ratio as inertia increases (D₄ on Kawai and F₆ on both pianos). These findings suggest that, contingent upon the key’s inertia, an escalation in exciter equivalent inertia disproportionate to effective weight can result in diminished tone loudness. The angular configuration of each playing unit is crucial (Fig. 2), shedding light on the pivotal role of the pianist’s posture in tone production.

6 Conclusion

Motivated by our claim that the study of the key coupled with the mechanical actuator might clarify the assessment of its dynamical properties and the extent of control by pianists, the investigation proposed in this study underscores the significance of considering the mass of the actuators in the mechanical force-driven excitation of piano keys. The conducted measurements provide a pathway towards the non-invasive characterization of touch-tone relationship properties of a piano, compatible with pianists’ control physical framework.

A comparison between the same three keys of the Stodart piano and of its modern counterpart the Kawai reveals significantly higher per key F₀ values (reported in Fig. 6) for the latter instrument. This appears to quantify the notion of “lightness” in touch associated with historical pianos. However, despite this difference, when considering the slopes of the isoinertial curves of the average key acceleration as a function of the driving weight force (Fig. 6), the two pianos appears to exhibit similar trends (Tab. 2). This suggests that while the historical instrument indeed requires lower minimum force for key acceleration, the inertia exhibited by the key behaves independently and may explain distinct touch qualities as perceived and exploited by pianists. The results of the conducted measurements show that differences in the actuator inertia M affect diversely and significantly the efficiency of the energy transduction on the three keys of the two pianos studied here, characterized by different hammer masses and hence having different inertia m (Eq. (1)). This means that the amount of inertia typically engaged by the pianist and exhibited by the key at its front are comparable.

The conducted measurements have the merit of describing the pianos under examination in a way that is relevant to the pianist, without necessitating the costly, invasive, and often delicate measurement of force at the key front, N. Nonetheless, the model proposed in this study allows to write N as follows:

$$N=\frac{f+\mu F}{1+\mu },$$

where μ ≡ m/M, F is the driving force and f is the force opposed by the key to its depression. The greater μ, the less N is sensitive to f. Such a condition shows the interdependence between disparate actuator-action inertial balances and how variations in the dynamic characteristics of distinct actions can be assessed through contact force. For example, among two keys of a same instrument, the higher one appears as more reactive to key depression, merely because of its lower inertia. Conversely, keys played by non-expert pianists, aiming at soft tones, might appear as less reactive than when played aiming at louder tones, given that there might be an unnoticed correlation in such pianists between the extent of the playing unit and the intended tone volume. This aspects should be more extensively studied in the future.

The slope coefficients of the isoinertial curves derived from our experiments (Tab. 2) illustrate the influence of the key-actuator system’s inertia: as our model predicts, lower slope values are associated with greater total system inertia. The measurements we conducted did not allow for the isolation of the keys’ inertia, knowing the actuator’s nominal contribution. This limitation may stem from a lack of repeatability in the experimental protocol, potentially affecting the precision of our measurements, and will require further investigation. However, focusing on energy transmission efficiency is more crucial for understanding a pianist’s control over tone production than dissecting the individual contributions that make up its reciprocal.

The methodology presented in this paper can be extended to study key deepening types beyond “pressed”, although the proposed model is not suited to their specific mechanical conditions. For example, a temporary loss of contact between the finger and the key would require accounting for stick-slip transitions at the pivots, while mechanical decoupling between the hammer and the rest of the action would necessitate modeling the interaction at the hammer-jack contact point within a 2-DOF framework. Both of these phenomena have been observed in the so-called “struck” touch, where the characteristic key velocity profile theoretically underscores the significant yet often overlooked role of bodily mass. Finally, within the dynamic range observed during the experiments, the influence of dynamic friction torques at the pivots appeared to be negligible. However, the excitation conditions allowed us to explore a significant portion of the dynamic range of the keys on only one of the two pianos examined (Tab. 4). For the Kawai piano, achieving higher dynamics may require extending the model to include dynamic friction effects.

By integrating the pianists’ arm mass into our framework, this study initiates a more accurate physical representation of the control strategies over the keyboard, highlighting the fact that the pianists act on their own biomechanical frame during key depression more than on the key itself. This opens up new perspectives on understanding the type of keyboard control operated by pianists. Now, even in the realm of expert pianists, Ortmann associates the intended volume with the extent of the playing unit. This assumption seems to imply that skilled pianists transition from a force-driven manner for soft tones to a motion-driven manner for intense tones, suggesting a control strategy which is founded on a general principle of aligning the pianist’s bio-mechanical apparatus with the energy demands of performance. While sacrificing the efficiency in translating the mechanical work of the driving force into kinetic energy of the hammer, highlighted by the present study, such a strategy would offer the advantage of supplementing the efforts of bigger muscles with the effective weight of the playing unit. However, the decrease in energy transmission efficiency resulting from increased actuator inertia can also be seen as a reduced sensitivity of tone volume to the applied driving force. In essence, a greater inertia results in a smaller change in tone volume for a given increase in driving force. Optimizing the sensitivity of the mechanical system would suggest a control strategy contrary to the one initially proposed but that aligns more closely with its musical extent, as controlling nuances is a significant challenge for musicians.

A principle aimed at optimizing sensitivity would not necessarily contradict what pianists typically assert they do, as it may initially appear. Indeed, the notion itself of associating louder tones with increasing the playing unit has not received universal acknowledgment, with the suitability of playing primarily by the arms being viewed as highly circumstantial during the 19th century. Future research should more explicitly delve into the variances in control strategies, contingent upon the style of playing. In this regard, the findings of the present study advocate for a reevaluation of the concept of “weight” within piano technique [20]. The results suggest that addressing the term “weight” in a purely physical framework is misleading, as an increase in the effective weight of the playing unit does not necessarily imply louder tones, due to a potentially detrimental increase in the overall inertia of the system. Given the common association between “weight” and “relaxation” in pianistic pedagogy, it is fair to conclude that the topic cannot be addressed unless one considers the muscular activity and particularly its effect on the compliance exhibited by the pianist. To this aim, future research should endeavor a more refined model of the biomechanics of the pianist, in order to assess the extent of a finite stiffness and its effect on the behavior of the piano action.

Acknowledgments

The authors extend their sincere gratitude to Xavier Boutillon for his invaluable contributions to the discussions, which have greatly enriched the insights of the present study. They also wish to thank the reviewers for their help in clarifying several points in the writing of the paper.

Conflicts of interest

The authors declare no conflict of interest.

Data availability statement

The data are available from the corresponding author on request.

References

All Tables

All Figures

	Figure 1 View of an English escapement action (Stodart) during the acceleration phase (damper not reported). The lever arms, indicated by dashed arrows, determine the mechanical coupling between the key lever body and the hammer lever, through the escapement hopper.
In the text

	Figure 2 Keystroke through “throw” [Wurf] or “fall” of the arm ([19], p. 97).
In the text

	Figure 3 Diagram representing the displacements of the key front and the hammer head under the effect of excitation by an actuator of mass M and subject to a driving force F, in our 1-DOF model of the acceleration phase of the mechanism.
In the text

	Figure 4 Illustration of the mechanical arm positioned to engage with the piano key at contact point P. The arm is initially held in place by a current generator (G) that powers the solenoid of an electromagnet (E). The magnetic force generated by the current keeps the arm attached to a fixed support. Movement begins when the current supply is switched off.
In the text

Figure 5 Estimated signals for a key depression with F = 3.36 N and M = 0.443 kg on the Stodart’s key playing D2. From top to bottom: hammer head and key front velocities; key front displacement; acoustic pressure of the produced sound. Vertical lines in the figure indicate the onset time of hammer free rotation (MHV), estimated as the local maximum of hammer velocity, and a representative time of the hammer-string impact phase (HS), estimated as the maximum of the raw hammer acceleration signal (not displayed). The parabolic regression on the displacement signal in the interval (TOn, TOff), corresponding to displacements (DOn, DOff), is superimposed on the key displacement signal.

In the text

Figure 6 Average acceleration of the key front, 〈A〉, as a function of the initial weight force F applied through the mechanical arm approximately 2 cm from the key front. Excitations are obtained for 0 < F < FI at an excitation inertia MI, where FI and MI respectively reproduce the effective weight and inertia of the engaged human limb playing unit (Tab. 1). The engaged playing unit (hand only, hand plus forearm, whole arm) is indicated by the increasing size of markers for the experimental points. Experimental data is approximated by three linear regressions, depending on the engaged inertia MI, passing through the point (F0, 0), with F0 being an estimation of the minimum force to accelerate the key (the standard deviation of the estimates as uncertainty on the least significant digit is reported in round brackets). The values AI = 〈A〉(FI, AI) are reported on the ordinate.

In the text

Figure 7 Kinematic key and hammer data for all the studied keys, obtained from all considered excitations using the mechanical arm, under controlled conditions of inertia and excitation force (M = MI and 0 < F < FI, see Tab. 1). (a) Increment in the key squared velocity, $V_{\mathrm{On}}^2-V_{\mathrm{Off}}^2$, between 10% and 90% of its displacement during the acceleration phase, as a function of the average acceleration 〈A〉 in the same time window. (b) Maximum Hammer Velocity (MHV), as a function of key velocity at MHV time. The dashed lines indicate the average slope of the data for each piano, estimated by robust linear regression and constrained to pass through the origin.

In the text

Figure 8 Estimated loudness for the acoustic pressure signals, S, as a function of the average key acceleration 〈A〉. The data are obtained from all considered excitations using the mechanical arm (see Tab. 1). The maximums of S, Smax, produced on each key by the first author (who is also a pianist) with a pressed touch are shown. S is reported in sones on the left-side y-axis and as percentage over Smax on the right-side y-axis. The data have been approximated by a robust non-linear regression with a power law of the form $S=b_1+b_2{\langle A\rangle }^{b_3}$, where 〈A〉 is expressed in cm/s2. The coefficients of the estimation are shown in figure. The standard deviation of the estimates is indicated in parentheses as uncertainty on the least significant digit.

In the text