© The Author(s), Published by EDP Sciences, 2026

1 Introduction

The making of musical instruments relies on two different paradigms. On one hand, it is based on craft apprenticeship and empirical knowledge transmission. On the other hand, it is associated with industrial contexts that produce similar instruments in much larger quantities. Although musical instruments have long been studied, and associated with makers’ practices [1], providing guidelines [2], many questions remain regarding the detailed understanding of their behaviour and the links between material properties, geometry of the instruments, sound produced, and appreciation by musicians. Such questions are particularly relevant in a domain undergoing rapid technological evolution, facing ecological constraints, and where beliefs and habits prevail. The study of musical instruments encompasses various fields and diverse agents, where acoustics and mechanics are particularly emphasised. These approaches historically relied mainly on analytical models and experimental studies. Although these approaches have proven useful in multiple domains, they seem limited in producing reliable knowledge, especially where uncertainties prevail. These uncertainties arise from both aleatory and epistemic factors. According to the uncertainty terminology used by the American Society of Mechanical Engineers (ASME) [3], aleatory uncertainty refers to irreducible variability (e.g., wood-to-wood scatter, climate fluctuations), whereas epistemic uncertainty refers to lack of knowledge or modelling idealisations (e.g., imperfect boundary conditions or omitted couplings) that can, in principle, be reduced through additional information or refined models.

Chordophones (notably those in the quartet, guitar, and keyboard families) are highly sensitive to material and organological factors as well as to their sound-production mechanisms. Considering dynamical behaviour, response variability can be attributed to:

– Structural variability:

• Assembly (glue, interfaces) [4].

• Geometry of each part [5].

• Making tolerances [6].

– Varnish mechanical properties [7–9]

– Material properties:

• Inter-/intra-species and inter-individual wood variability [10, 11].

• Frequency, temperature, and relative humidity dependence of wood properties [5, 12, 13].

• Anatomical singularities [14] or wood treatments for instrument making [15].

Under high variability, traditional approaches encounter two main issues. First, the number of samples required to obtain robust and statistically significant results is often too large; consequently, many published studies rely on small sample sizes. Secondly, analytical approaches postulate assumptions and simplifications that give tendencies but prevent studying fine instrument makers’ choices during the making process.

1.1 Finite element method and virtual prototyping

The Finite Element Method (FEM) is a numerical approach based on complex geometries obtained through computer-aided design and associated constitutive models. These approaches can be used as a virtual prototyping framework, like in [16, 17] for the study and conception of stringed instruments. In addition to the usual applications of FEM/virtual prototyping approaches (model updating, direct analysis, input identification, optimisation), this paper proposes, by inverting the statistical test approaches, an evaluation of the minimum number of instrument pairs required for a study to assess the impact of a modification (design or material). This paper aims at providing conceptual guidelines to implement FEM in the design of experimental campaigns, where physics-based modelling may minimise bias and avoid experiments that would not produce robust results.

1.2 Experimental paradigms: in vitro measurements vs. in vivo perception

Two distinct cases can be identified and analysed in parallel with musical acoustics, considering the study of an instrument. A first point of view would be “in vitro”, consisting of measurements with sensors, whose results may differ between experimental means, climatic conditions, or boundary conditions of the tested instrument. The second point of view would be “in vivo”, where the instrument is tested by a player, generally a musician, taking into account the playing characteristics of musicians. In this case, the excitation of the string on the instrument, which can be measured, is highly variable, as well as the feedback of the musician and its appreciation. Therefore, coherence for a tester and between testers has to be evaluated and can lead to dead ends. Moreover, the judgement is based on a vocabulary that is the object of studies, but for which terms used, however suggesting some consensus on their meaning, are not necessarily associated with quantified outputs. Therefore, considering the feeling of the musician, who is tested to give descriptor keys on functional instruments, too many parameters interfere: subjectivity, touch, ergonomics, leading to idiosyncratic descriptions. One can also mention the paronomastic effect when qualifying the sound of instruments, as the vocabulary used by musicians is itself subjective and often influenced by aesthetics (e.g., associating darker materials with a “darker” sound), creating potential semantic biases. Therefore, it is difficult to draw robust conclusions from studies with small sample sizes when the underlying mechanisms are either too subtle to be directly observed or quantified, or are governed by multiple interacting factors. Perceptual studies on functional instruments may therefore be flawed except in the case of large cohorts and sampling, as this casuistic approach is precluded by the variability of materials, structures, and musician’s biases and subjectivity. Therefore, increasing the number of measured features, refining the models, improving device precision, or increasing computational power does not automatically guarantee more reliable knowledge.

1.3 Evidence-based method

Evidence-based practice originated in critical domains such as medicine and aerospace; its aim is to reduce bias and secure statistically significant, reproducible conclusions. This approach is particularly relevant to musical acoustics, a field marked by variability, uncertainties, and subjectivity [18]. Rather than relying on anecdotal or single-case observations, it emphasises peer-reviewed evidence, explicit statistical criteria, and dedicated methodology.

1.4 Statistical framework

A central question is whether observed differences between two instruments (or designs) exceed expected variability. Statistical hypothesis testing addresses this by quantifying the probability of misleading results. Here, α denotes the chosen Type-I error rate (false-positive risk), and the p-value is the probability, under the null hypothesis, of observing a result at least as extreme as the measured one. A practical planning tool is Lehr’s rule of thumb for a two-sample, two-tailed t-test at α = 0.05, which yields an approximate power of 0.80 [19]:

$$\begin{array}{c}\hfill n=16\frac{{\sigma }^2}{{(\mathrm{\Delta }\mu )}^2},\end{array}$$(1)

and, equivalently,

$$\begin{array}{c}\hfill \mathrm{\Delta }\mu =4\frac{\sigma }{\sqrt{n}},\end{array}$$(2)

where n is the number of pairs (assuming equal population sizes), σ the standard deviation (assuming homoscedasticity), and Δμ the difference of means for a given quantity of interest. This study applies these relations to dynamical features of guitar soundboards. As often recalled, all models are wrong, but some are useful [20]; numerical models are therefore used in this framework to (i) estimate the measurable effects of design changes, (ii) highlight variability and uncertainty, (iii) rank influential parameters.

1.5 Research aims

In this work, FEM is treated as a practical planning tool: combining FEM-based estimates of expected shifts and variances of eigenfrequencies with Lehr’s rule to specify how many paired specimens are required to test a hypothesised design effect with controlled error rates. The application is performed on acoustic guitar soundboards, with particular focus on the way the braces are arranged and shaped beneath the soundboard, that is, the bracing pattern. Numerous experimental and numerical works have examined brace variations [21–23], reflecting the primacy of soundboard dynamics and the roles of geometry and wood [24]. Bracing is widely used to tune dynamical behaviour [25] and to mitigate wood variability. Therefore, guitar braces, placed underneath the soundboard, contribute to the structural integrity but also to the vibrational behaviour of the instrument. As debate remains regarding the impact of brace modifications on the static and dynamic behaviour of the guitar, as well as on the resulting sound, strong evidence is still lacking. Therefore, this study addresses this archetypal case. The approach estimates expected effect scales and variability to decide in advance whether a bracing effect is testable with a realistic number of instruments. This study focuses on low-order modes in the low modal overlap regime and using FEM primarily as an experimental-planning tool rather than as a full-band stochastic predictor. Moreover, a perceptual study is performed to evaluate, with blind tests in playing conditions, if the measured difference is still relevant in terms of player perception.

The remainder of this paper is organised as follows. Section 2 presents the materials and methods, including the guitars under study, the experimental vibroacoustic protocol, the finite element model used for experimental planning, the discrimination test, and the associated statistical framework. Section 3 reports the main results, first from the numerical simulations, then from the experimental measurements and blind perceptual tests. Section 4 discusses the implications of these results for evidence-based instrument making, with particular attention to the gap between structurally measurable differences and perceptually meaningful effects, as well as to the epistemic role of FEM in experimental design. Finally, Section 5 summarises the main conclusions and outlines perspectives for future work.

2 Material and methods

2.1 Object of study

One of the most common bracing systems in steel-string acoustic guitars is X-bracing, used in archetypal OM guitars from MARTIN, in which two large braces cross to form an “X” near or beneath the bridge. Braces and bridge are glued on the soundboard, a wooden plate (usually spruce, but other species or composites can be considered). Instead of leaving the braces with a relatively straight profile, scalloped bracing consists of carving scooped-out sections between the ends of the braces and their high points. This gives the brace a series of shallow parts, reducing the mass in specific zones while leaving the strength at the ends and cross points. The effect on the structure has been evaluated using analytical models [26], and the modification is here studied through finite element method, based on computer aided design. The scalloped and regular bracing configurations are shown in Figure 1.

Figure 1. Cross-braced steel-string acoustic guitar soundboard. Top: upper view. Middle: scalloped brace. Bottom: regular brace.

For the experimental part, ten guitars were built at the Institut Technologique Européen des Métiers de la Musique (ITEMM) and divided into two groups of five according to the brace type. Two batches of five guitars regular Ri and scalloped Si were compared at different stages of manufacture. These guitars are Martin OM type, with a design adapted by the ITEMM, developed by the guitar making teaching section. Guitars were made during the 2021–2022 school year. The sides and back are made of Indian rosewood (Dalbergia latifolia). The neck is made of cedro (Cedrela odorata), and the fretboard is made of Pau Ferro (Caesalpinia ferrea). The assembly of the wooden parts was carried out with Titebond® aliphatic glue and the varnish was polyurethane-based. The vibrating length of the strings is 645 mm. The strings were all new, with reference D’Addario EJ16 with 12–53 gauge (expressed in thousandths of inches). The top and braces are made of spruce, in this case, Picea abies provided by a specialised sawmill in France. Braces were shaped by hand according to a detailed plan (not CNC-identical), and no iterative tap-tuning to modal targets was performed instrument-by-instrument; this avoids introducing operator-dependent feedback loops while reflecting common practice where unavoidable tolerance scatter contributes to aleatory variability.

2.2 Experimental setup

During measurements, guitars were in playable condition. Strings were damped using soft foam to reduce string-related resonances; the instrument body rested on compliant foam supports (back plate down) to improve repeatability while remaining close to free boundary conditions, rather than introducing the constraints associated with contact with the musician’s torso. The response (in acceleration) of the guitar top to an impact has been measured by a hit from an impact hammer measuring the input force. An accelerometer model PCB-352C23 was glued with wax at 20 mm on the right side at the level of the bridge, and 290 mm from the top of the body. A PCB-086E80 impact hammer was used. Measurements were carried out with the Danid software [27]. Each measurement was repeated five times. Modal analysis was performed with ModAn software [28] to estimate eigenfrequencies and associated modal damping using POLYMAX method. In the following discussion, the A ₀ mode denotes the coupled Helmholtz air resonance of the cavity; The “true” Helmholtz frequency appears above the A ₀ as an anti-resonance peak. T ₁ denotes the first soundboard-dominated bending resonance (often strongly coupled with A ₀ in the assembled instrument). In this paper F ₁ is reported as A ₀ and F ₂ as T ₁, followed by higher soundboard modes F ₃–F ₅ identified from the bridge mobility [29]. The observables will therefore be the resonant frequencies and their associated damping. During the measurements, the relative humidity was measured with an accuracy of ±2%. The mass of the guitars was measured with a balance (precision ±1g).

2.3 Finite Element Method model

2.3.1 Material properties

The material properties were derived from density measured on real samples. Then, elastic parameters based on linear elastic orthotropic hypotheses have been calculated with the density, following empirical correlations for tonewood (specifically Picea abies as proposed in [16, 30]). The material parameters are given in Table 1. The three principal material directions are longitudinal (L), radial (R), and tangential (T). In the modal computations (COMSOL), spruce was modelled as linearly elastic and orthotropic, with anatomical axes aligned to the real parts (soundboard: X = L, Y = R, Z = T; braces: L  =  along the bar length, R  =  across the width).

2.3.2 Boundary conditions

Regarding boundary conditions, the top plate was modelled as fully fixed (clamped) along its outer contour and at the upper/lower neck/heel interfaces, reflecting a construction-stage constraint. In the real instrument, however, the top is embedded in the rib/back/binding assembly, and the body is the element actually resting on the foam during tests. Therefore, model boundary conditions are stiffer than experimental support conditions and primarily affect absolute frequency placement while preserving mode shapes. Because the present study focuses on relative differences between bracing patterns (scalloped vs. regular) at low order, this boundary condition idealisation is considered acceptable for planning purposes. The mesh consists of tetrahedral elements with a target size below λ/6 for the frequency range of interest. A first modal analysis was then performed, and the lowest soundboard-dominated mode (T ₁) was identified for each brace design.

2.3.3 Modelling choice

The present FEM model represents the soundboard and bracing only and does not include the enclosed air cavity nor structural–acoustic coupling. Low-frequency guitar acoustics is known to involve coupling between the Helmholtz air resonance H and the first top-plate resonance (T ₁), which can affect absolute frequency placement [29], leading to A ₀ and shifted T ₁, around the anti-resonance peak of the uncoupled H. In this paper, the FEM model is used for planning (expected shifts and required sample sizes) rather than for absolute frequency matching; adding cavity air would introduce additional uncertain parameters, complex post-processing [31] and thus an additional source of uncertainty.

2.4 Recognition test in playing conditions

To further investigate the perceptual impact of brace modifications, blind tests were conducted with musicians. Following the guidelines provided in [18], hypothesis testing in this study was structured to achieve statistically significant results with minimal sample sizes using Lehr’s rule of thumb. Participants were presented with two sets of guitars, one with scalloped braces (I) and the other with regular braces (II). The participants did not see the instruments during the test itself, as they wore blindfolds. The guitars were handed to them by the experimenter, who remained in the room. The aim was to prevent visual identification of the instruments. Musicians played guitars of a batch (six in total per participant, an aleatory mix between R_i and S_i guitars) and were tasked with correctly identifying the brace configuration used in each trial. Knowing that for each test, the tester has varying chances (depending on the number of instruments I and II) of correctly identifying each instrument, in this protocol, the instruments already tested are not replaced in the batch. The perceptual experiment was intentionally situated in the regime of ordinary, situated decision-making, rather than in idealised psycho-acoustic conditions. Ten musicians participated in the test. In a non-negligible number of cases, when players choose an instrument, they do so in environments characterised by limited time, small rooms, ordinary reverberation, rapid comparison, and strong contextual cues. Perceptual judgements formed in this setting are not “noisy approximations" of laboratory measurements, but the very judgements that structure musical practice. Replicating this everyday context therefore allows the experiment to operate within the same conditions that govern instrument selection. In this sense, the design prioritises capturing how differences become perceptually meaningful in real situations over maximal signal detection under optimised laboratory conditions. The question is not whether brace differences are detectable in principle, but whether they are detectable in the very circumstances where they are supposed to matter: when a musician picks up two guitars and decides whether one feels or sounds different enough to justify a choice.

The sampling-without-replacement protocol leads to the hypergeometric probability:

$$\begin{array}{c}\hfill P(X=k)=\frac{\left(\begin{array}{c}K\\ k\end{array}\right)\left(\begin{array}{c}N-K\\ n-k\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)},\end{array}$$(3)

with total items N (I (scalloped) +II (regular)) of which K belong to group II, and n draws. To avoid overstating success at low trial counts, the per-listener pass criterion was set to 6/6 correct (P ≈ 0.016 under p ₀ = 0.5), consistent with α = 0.05. For individual listener performance, statistical significance was assessed using an exact model with chance probability p ₀ = 0.5. With n = 6 binary trials, only a perfect score (6/6 correct) reaches the conventional α = 0.05 significance threshold (P = 1/64 ≈ 0.016), whereas 5/6 correct remains non-significant (P = 7/64 ≈ 0.11). The hypergeometric distribution, as in [32], is reported only to describe the finite, without-replacement structure of the stimulus sampling, not as the primary test of perceptual success. Accordingly, for an individual listener tested on six binary trials, the maximum number of errors compatible with significance at the 5% level is zero. Participants were not given specific strategies or constraints beyond the fact that they could not see the guitars. They were free to play as they wished (mostly chords, arpeggios, melodies) before answering.

3 Results

3.1 Numerical results

The paired modes (F₂/T₁, F₃, F₄, F₅) for each brace pattern are given in Figure 2. The modes have similar shapes and exhibit, with nominal values of parameters of the FEM-based models, deterministically calculated frequencies of T₁-pagination equal to 184 and 204 Hz, for scalloped and regular braces, respectively. The difference in mean, therefore, is equal to 20 Hz. The standard deviation obtained through stochastic computations for such structures (parametric variability, normal distribution injected into the FEM model and propagated to modal frequencies), from [33], is equal to 9 Hz for the T₁ mode. Although this value does not take into account the complete variability of a real system with different boundary conditions, geometrical tolerances and vibro-acoustic coupling, it was used as an estimator in the application of Lehr’s approach, when experimental results are missing before launching an experimental campaign. Therefore results indicate that at least three pairs are needed to achieve statistical significance. In the present paper, this value is used as an order-of-magnitude prior for planning. Because experimental standard deviations include manufacturing scatter and uncontrolled factors (supports, humidity, assembly details), while numerical standard deviation depends on which inputs are randomised and with which distributions, the comparison should be interpreted as indicative rather than as a strict validation.

Figure 2. First modes of the soundboard F 2, F 3 and F 4, left: scalloped brace 184 Hz, right: regular brace 204 Hz.

The Figure 3 shows the synthesised bridge mobility based on mass-normalised eigenvectors at the measurement point. A unit out-of-plane force (1 N) is assumed for comparison/normalisation. Because modal masses and damping ratios are not fully identified for all modes in the synthesis here, the absolute amplitude should be interpreted cautiously; the comparison is primarily on peak locations, relative differences between brace designs, and general curve shape.

Figure 3. Deterministic simulation of the mobility at the bridge for scalloped and regular, non scalloped braces.

The Figure 4 shows the mobility up to 600 Hz (including the four modes that are studied), but taking into account post hoc uncertainties, taken from experimental results: 3.8% of coefficient of variation on each frequency in average and ±1 dB. The first mode, T₁ remains clearly distinct even when these uncertainties are taken into account. The comparison shows a shift in the peak with regular brace, which is also observed in the experimental results, described below.

Figure 4. Deterministic simulation of the mobility at the bridge for scalloped (orange) and regular, non scalloped (blue) braces with added inferred variability using experimental values (±1 dB amplitude, ±3.8% frequency), showing clear separation of first mode despite added uncertainty.

3.2 Inter-measurements, intra and inter-guitar variabilities

One guitar (R₅) was measured multiple times in succession, with the accelerometer removed and the support conditions reset between repetitions. The bridge mobility with its mean  ±  standard deviation for the four repetitions is given in Figure 5.

Figure 5. Bridge mobility, mean  ±  standard deviation for the repetitions on the guitar R5, regular brace.

The Table 2 gives the intra-specimen/inter-measurements variability. In addition, the precisions of the device and post-processing have been added. The inter-measures intra-guitar eigenfrequency variability ranges between 0.1 and 0.8 Hz, leading to coefficient of variation between 0.1 and 0.3%. The variability of the modal damping ranges between 0.1 and 0.2%, leading to coefficient of variation of modal damping for a single guitar and multiple experiments close to 6%. These preliminary results highlight the difficulty of measuring modal damping with good reproducibility, which is up to 60 times worse than the reproducibility of eigenfrequency evaluation. A total variability was calculated for each frequency and associated damping. The same process is applied between guitars of a same batch (S) or (R). The total standard deviation is calculated, based on the superposition of the different sources of variability. The resulting values of σ_totalR or σ_totalS therefore represent the typical variability that should be expected when measuring only one guitar of a given type of brace. Therefore, using Lehr rule of thumb, such variability can be used to assess the number of samples required for a material/design change that would lead to a change of Δμ in the observed quantity.

3.3 Inter guitars comparison of bridge mobilities, eigenfrequencies and damping

The bridge mobilities for each brace type and guitars, are given in Figures 6 and 7 for scalloped (S) and regular (R) brace respectively, in the [60–6000] Hz frequency band. The guitars with scalloped brace show a relatively small dispersion, as expected for supposedly similar guitars, as shown in [34] where coefficients of variation of 2.9% and 3.1% were estimated for A₀ and T₁ frequencies, respectively. Here, the same mode frequencies show a coefficient of variation of 2.7% and 2.2% (Table 3). For the guitars with soundboards made with regular braces, the curves show a wider dispersion. Guitars R₁ and R₃ are notably different up to 200 Hz, and a wide scatter is observed above 700 Hz. Above 1000 Hz, R₃ and R₅ show a high difference compared with R₁, R₂ and R₄ that are more similar to each other. During making, there was no notable phenomena that could explain this. The thicknesses, wood species, assembly method were similar. Therefore no notable explanations can be proposed to explain the difference in reproducibility between S_i and R_i guitars. The Figure 5 highlighted that even for the same guitar, between experiments, anti resonance peaks and global amplitude between resonance can vary, as exhibited with R5₃, the third experiment on the guitar R₅. Repeatability issues can be pointed out, as well as some wood anatomical features, like bear claw spruce [14] that can lead to strong variations of wood properties, for a given density, and may have been omitted. Nevertheless, the coefficients of variation on the identified frequencies of the modes range from 2.2 to 5.3 % (2.2 to 6% for Si). Therefore, on the observables considered for the determination of the number of pairs, the impact of the dispersion of the admittances is smaller than expected. It has to be pointed out that, for the tests with musicians, if they are more sensitive to global admittances changes rather than several eigenfrequencies and the associated amplitude, this would lead to a population R_i where one guitar of the batch would not necessarily be representative of the batch and a listener might be more sensitive to specific instruments rather than brace type.

Figure 6. Bridge mobilities of the scalloped (S) brace guitars, for frequency range [60–6000] Hz.

Figure 7. Bridge mobilities of the regular (R) brace guitars, for frequency range [60–6000] Hz.

The mobilities of both guitar groups are given in Figure 8 with mean  ±  standard deviation. This comparison between scalloped and regular bracing, including variability, shows significant discrepancies. Up to 600 Hz, regular brace shows for the A₀ (near 100 Hz) and T₁ (between 180 and 220 Hz) a shift to higher frequencies, with lower amplitude for A₀ (−6 dB), similar amplitude for T₁, and higher mobility between 350 and 500 Hz. In higher frequencies, the scalloped-braces guitars show much higher amplitude between 800 and 900 Hz (up to 14 dB), whereas R brace soundboards exhibit a clearly higher mobility from 1800 to 6000 Hz where the discrepancy increases with increasing frequency.

Figure 8. Bridge mobilities, mean  ±  standard deviation, comparison of the scalloped (S) and regular (R) brace guitars, for three different frequency range range [60–6000] Hz.

The results for the individual guitars are given in Table 3, reporting the total mass of the guitars, frequencies, and damping ratios of the first five identified modes. The table also gives mean μ, standard deviation σ and coefficient of variation CoV, and frequency of the finite element model simulations.

The A₀ frequencies are equal to 102.2  ±  2.7 Hz and 109.8  ±  2.4 Hz for scalloped and regular braces, respectively. This implies that the difference in brace patterns also affects the A₀ acoustic mode of the guitar. The frequencies of the T₁ mode are 187.0  ±  4.0 Hz and 216.5  ±  5.1 Hz, for scalloped and regular braces, respectively. The discrepancy in mean frequencies is 29.5 Hz, implying that only one pair is required according to Lehr’s rule, as shown in Table 4. By contrast, the modal damping of T₁ is much harder to differentiate, since 16 pairs would be required on the basis of the present results (Tab. 4). Finally, the mass of each type of brace would be theoretically differentiable, but with at least 178 pairs. The mean frequencies obtained for mode F₃ are 358  ±  14 Hz for the scalloped braces and 396  ±  21 Hz for the regular braces. The difference of 38 Hz is significant, but the larger standard deviation requires at least n ≥ 4 pairs to reach statistical significance. The numerical results (386 Hz vs. 429 Hz) confirm this trend, with a slightly lower requirement of n ≥ 3. This intermediate mode is therefore sensitive to the brace design, but requires a larger sample size than mode F₂. For the dipole mode, the measured frequencies are 376.1  ±  16 Hz (scalloped) compared to 430  ±  15.3 Hz (regular). The difference of almost 54 Hz is sufficient to ensure a robust differentiation with only n ≥ 2 pairs experimentally, is lower than with the numerical values (406 Hz vs. 440 Hz, n ≥ 4). This mode thus appears particularly discriminant between the two brace designs. The measured frequencies of 549.9  ±  33 Hz (scalloped) and 595.5  ±  28.7 Hz (regular) show a shift of 46 Hz, but the higher inter-specimen variability implies a minimum of n ≥ 8 pairs to reach significance. The numerical results (548 Hz vs. 598 Hz, n ≥ 6) are consistent and confirm that this mode requires a large number of samples to overcome natural dispersion. Unlike the lower-frequency modes, F₅ is therefore less exploitable for small sample sizes.

3.4 Blind test results

The results of the test are given in Table 5. Of the ten participants, only one achieved a sufficient score (6/6), while the number of errors ranged from 0 to 5 across listeners. However, this isolated result should be interpreted with caution. Under the null hypothesis of random guessing with success probability p₀ = 0.5, the probability that a given participant obtains a perfect score over six trials is ≈0.016. When ten participants are tested independently, the probability that at least one of them obtains a perfect score purely by chance is

$$P(\text{at least one}6/6)=1-{(1-\frac{1}{64})}^{10}=1-{\left(\frac{63}{64}\right)}^{10}\approx 0.15.$$

Thus, observing one perfect score among ten listeners is not, by itself, exceptional evidence of reliable perceptual discrimination at the population level.

4 Discussion

The results have shown that an FEM model, though simplified, was able to predict the behaviour of a complex structure in a sufficiently realistic way to evaluate the differences in mean between two different making choices during the construction of a musical instrument. By incorporating measured variability into the computed features, the approach makes it possible to predict the number of specimens per batch when comparing musical instruments, with features that can be easily simulated (namely eigenfrequencies, eigenmodes). Bridge mobility does not directly measure radiated sound; extending the protocol with a microphone FRF (sound pressure per unit force) would be valuable and is feasible in principle, as discussed in studies on mobility and radiated response [29]. Additional microphone measurements could not be collected here because the instruments were returned to the makers. The blind test results challenge the conventional assumptions about the perceptual discernibility of brace modifications. Despite the measurable differences in vibrational behaviour, with the present low-trial design (n = 6), no evidence of reliable discrimination at α = 0.05 (only 1/10 listeners reached 6/6) was found. The present data support that significantly assessed low-order modal differences between two batches do not automatically translate into reliable perceptual differences under the proposed conditions. The participants’ difficulty in discriminating between the two brace designs raises questions about the perceptual threshold for differences in the dynamical behaviour of chordophone soundboards and highlights the importance of evidence-based approaches. While modifications to the internal structure of the guitar may lead to quantifiable changes in vibrational behaviour, these changes might not necessarily translate to perceptible differences in sound perception for experienced musicians. Beyond statistical considerations, the perceptual protocol was deliberately constructed to consider the conditions under which instruments are usually chosen. In most real situations, such as shops or workshops, players evaluate a guitar by performing a few short trials in immediate succession, in an acoustically ordinary room, and without long-term familiarity with the instrument. This context is characterised by limited exposure time, rapid alternation between instruments, and a strong reliance on first impressions. Longer and more highly controlled psychoacoustic sessions, or evaluations based on extended playing time, would be more powerful and would probe another equally relevant context, namely the progressive familiarisation of a player with an instrument and the possible evolution of preference after repeated use. This second context is important for the makers or manufacturers, since it may lead to different longer-term design decisions if an instrument that is initially appreciated becomes less satisfying after continued use. They address a different perceptual question from the one considered in the present study. The current choice favours realism of tests conditions over maximal statistical power and aligns with the objective: assessing, helped by FEM, whether brace differences are prominent enough to be detectable and, ultimately, whether they matter in practice. Beyond individual scores, it is also informative to consider the aggregated responses across all participants. A total of ten listeners each performed six trials, resulting in N = 60 binary classification responses. Among these responses, 39 were correct and 21 incorrect. Under the null hypothesis of random guessing with probability p₀ = 0.5, the number of correct responses follows a binomial distribution. The probability of observing k or more correct responses out of N trials is:

$$\begin{array}{c}\hfill P(X\ge k)=\sum _{i=k}^N\left(\begin{array}{c}N\\ i\end{array}\right)p_0^i{(1-p_0)}^{N-i}.\end{array}$$(4)

For the present data (k = 39, N = 60), this yields

$$P\approx 0.015.$$

This aggregated result indicates a slight overall deviation from chance-level performance. However, several factors call for cautious interpretation. First, the responses may not be strictly independent, as trials were performed by the same participants within short experimental sessions. Second, the effect size remains modest, with an accuracy of 39/60 ≈ 65%. Taken together, these elements suggest that while a weak aggregate trend may exist, the results do not provide strong evidence of robust perceptual discrimination of brace configuration under the present experimental conditions. The results of the blind test showed no correlation between the number of years of musical practice and the ability to successfully discriminate the brace configurations. Whether playing experience improves the perceptual discrimination of subtle structural modifications would require a dedicated study on a larger population. From a practical standpoint, results indicate that brace modifications, while measurable in terms of vibrational response, may not justify the additional design or production costs if their perceptual effect on players has not been robustly established. As both empirical studies and theoretical frameworks emphasise, instrument variability, environmental conditions, and subjective player responses all contribute to the challenges in achieving robust measurements in musical acoustics. The evidence-based approach suggests that multiple measurements across varied samples increase the robustness of findings by accounting for:

• Inter- and intra-sample variability, particularly relevant in wood-based instruments whose acoustic properties can vary with climate and humidity, including short term changes, detailed in [35].

• Statistical analysis using paired testing, ensuring that the observed differences between braces are due to the modification rather than to random chance [33].

By planning experiments with explicit variance models and power targets, the evidence-based workflow clarifies the burden of proof and mitigates diffusion of underpowered results. In this sense, FEM and power analysis jointly function as epistemic instruments that help construct testable, reproducible propositions.

5 Conclusion

The Finite Element Method (FEM) is a powerful tool for simulating and analysing the behaviour of musical instruments. Once such models are validated and correctly account for material parameters, their variability, and the relevant boundary conditions, they can reproduce realistic behaviour. By using FEM, it is possible to anticipate the number of instruments required per group to assess reliably whether a design modification produces a statistically significant effect. Furthermore, the variation in acoustical features can be quantified using standard deviation calculations, and rules of thumb such as the one proposed by Lehr can be used to guide experimental design. Associated with the criteria of evidence-based practice used in other domains, this can increase the robustness of models. The results show that an FEM-based model of a guitar soundboard, with associated variability, can be used to predict the minimum number of guitar soundboards per batch (here 3 instruments) to robustly evaluate the impact of scalloped or regular braces on the first modes of the soundboard. This approach can therefore provide a better estimate of the cost of an experimental campaign, or show that a phenomenon will not be assessed unless a sufficient number of pairs are gathered, depending on the variability and the initial impact of the modification. For this calculation, it is necessary to estimate the expected mean difference and the associated standard deviation of the quantities of interest in each batch to obtain the number of pairs. Nevertheless, establishing that the impact of scalloped or regular braces is significantly measured using modal analysis or dynamical tests does not guarantee that such a modification would be audible or perceived by musicians, because sound production involves multiple interacting phenomena, of which modal behaviour is only one component. Through a combination of dynamical analysis and blind tests, it has been demonstrated that while brace modifications yield measurable vibrational changes, these changes may not necessarily be discernible to skilled musicians based solely on auditory cues. This study highlights the importance of considering both objective measurements and perceptual assessments when evaluating the impact of instrument modifications. For guitar makers, this means that resources and time might be more effectively directed toward factors with reliable perceptual impact (such as setup, playability, or material consistency) rather than pursuing subtle variations whose audible consequences are marginal. High material variability, climatic conditions, uncertain boundary conditions at bonded joints, and manufacturing tolerances all constitute sources of bias in the objective characterisation of the instruments. Compounding this problem, any characterisation of an instrument by anything other than an objective device is also subject to numerous biases, and such studies need to be performed under severe constraints that only partly reduce biases. Nevertheless, such studies remain one possible way of addressing this issue. It is essential to keep in mind that the burden of proof must always be made explicit. Experiments designed to refute insufficiently supported claims may fail to deliver decisive results, yet they may still be scientifically fruitful. For instance, blinded studies have shown that experienced players and listeners do not reliably identify old Italian violins over new instruments under controlled conditions [36–38].

In sum, evidence-based practice provides a structured framework integrating virtual prototyping, experiments, and perception tests to support makers’ in modifying designs without wasting resources or being misled by ordinary variability. FEM, under the proposed paradigm, may help decide what is measurable and how many samples are needed before committing resources to experiments. From an epistemological perspective, the present case shows how modelling and measurement co-produce experimental evidence by structuring power and error control. This suggests a transferable template for robustness in instrument making practice and, more generally, in applied vibroacoustics where variability and subjective descriptors are the rule rather than the exception.

Funding

This work received no specific grant from public, commercial, or not-for-profit sectors.

Conflict of interests

The authors declare no competing interests.

Data availability

Data are available on request from the authors.

Informed consent

Informed consent was obtained from all participants.

References

All Tables

All Figures

	Figure 1. Cross-braced steel-string acoustic guitar soundboard. Top: upper view. Middle: scalloped brace. Bottom: regular brace.
In the text

	Figure 2. First modes of the soundboard F 2, F 3 and F 4, left: scalloped brace 184 Hz, right: regular brace 204 Hz.
In the text

	Figure 3. Deterministic simulation of the mobility at the bridge for scalloped and regular, non scalloped braces.
In the text

	Figure 4. Deterministic simulation of the mobility at the bridge for scalloped (orange) and regular, non scalloped (blue) braces with added inferred variability using experimental values (±1 dB amplitude, ±3.8% frequency), showing clear separation of first mode despite added uncertainty.
In the text

	Figure 5. Bridge mobility, mean  ±  standard deviation for the repetitions on the guitar R5, regular brace.
In the text

	Figure 6. Bridge mobilities of the scalloped (S) brace guitars, for frequency range [60–6000] Hz.
In the text

	Figure 7. Bridge mobilities of the regular (R) brace guitars, for frequency range [60–6000] Hz.
In the text

	Figure 8. Bridge mobilities, mean  ±  standard deviation, comparison of the scalloped (S) and regular (R) brace guitars, for three different frequency range range [60–6000] Hz.
In the text