A few years ago, a friend of mine, who is a very accomplished player of the violin & viola and who is an amateur builder of violins & violas, sent me a video on YouTube [LINK] of the violin maker Peter Westerlund showing how to create violins (and violas, by the same logic and with only slightly different overall dimensions) that consistently produce sounds that people well-versed in European classical music judge to be of high quality. Many such people have praised the consistent warm & rich aural tones of violins & violas made in Italy by Andrea Guarneri around 1650 & Antonio Stradivari around 1700, yet it has been difficult for any violin maker to replicate such warm & rich tones in the violins that they have made, in part because there is little scientific consensus about the specific acoustic spectral qualities that make those violins so highly valued. Further complicating matters, as explained in a news article by Science magazine in 2017 [LINK], is the fact that many blind & double-blind studies have left musicians & keen listeners preferring contemporary violins over those made by Stradivari or Guarneri. Even if the instruments by made by Stradivari or Guarneri aren't consistently rated the best, they are rated very highly far more consistently than those made by other violin makers (though this may reflect selection bias in such tests), so there may be merit to considering how to replicate those instruments' aural qualities.
Traditionally, violin makers have carefully studied the exact dimensions, shapes, and materials used in instruments made by Stradivari or Guarneri and done their best to replicate them. However, the resulting aural qualities have typically been judged to be unsatisfactory. This is in various parts due to the effects of aging on the material components of violins (which generally cannot be artificially accelerated), one-off differences in instances of even the same materials used, slight deviations in the craftsmanship, shapes, or dimensions, and other factors that have yet to be explained. This motivated Westerlund (who is probably not the first person to come up with this method) to develop a method to reliably create instruments with highly consistent sounds even if they don't replicate sounds from instruments made by Stradivari or Guarneri per se. The method is as follows. Each violin or viola has a top plate and a back plate, and these are initially constructed separately, joined only later on corresponding sides of the ribs. For each plate, after getting the basic shape & thickness correct, the violin maker should further adjust the shape by tapping or rubbing the plate in various locations, ideally using the same tapping or rubbing motion with the same intensity every time. The violin maker should iteratively shape (by cutting or scraping) at & around those locations of tapping or rubbing and then continue to tap or rub at all locations, until eventually, the tapping or rubbing motion produces approximately the same tone at every location. Once that has been achieved, shaping of the plate is finished.
My background in theoretical & computational nanophotonics from graduate school, even though I was already a few years removed from that even by a few years ago (having changed fields to transportation research after graduate school [LINK]) when my friend shared that video with me, made me interested in trying to mathematically & physically understand why Westerlund's process is successful in creating instruments that have strong internal tonal consistency. However, I wasn't able to come up with a satisfactory answer until much more recently. Follow the jump to see more of my thinking about this, leading to the resolution.
Initial mathematical formulation of and attempts to solve this problem
Initially, I formulated this problem mathematically as follows. An impulse, corresponding to a tap on the plate (or violin), at a location →x′ on the plate at a single frequency ω produces the acoustic response G(ω,→x,→x′) for a listener at location →x, where G is the acoustic Green's function. (The notion of the impulse being a quick tap is a little misleading when considering a single frequency, but I don't know how to describe it better.) Suppose that the listener's ear is at a fixed location →x and that the plate's position & orientation relative to the listener's ear are also fixed. Furthermore, suppose that the plate is close enough to the listener's ear that at all relevant frequencies, the finite travel time of the sound wave to the listener's ear due to the finite speed of sound can be ignored; this is a near-field approximation. For a fixed listener location →x, it is convenient to define H(ω,→x′)=G(ω,→x,→x′). This function of two variables H(ω,→x′) can be interpreted as a linear algebraic operator, specifically a continuous analogue of a matrix, in which the tap/rub locations →x′ label the columns and the frequencies ω label the rows, such that each "column" at a fixed →x′ represents the spectral response at the listener's ear as a function of ω. If any tap or rub at any location →x′ on a finished plate leads to H(ω,→x′) being some multiple (perhaps a different multiple at different tap/rub locations →x′) of the same spectral function y(ω), then H(ω,→x′) is of rank 1. Thus, the objective to get a consistent tone would seem to be to minimize the rank of the operator H(ω,→x′)=G(ω,→x,→x′) for fixed →x.
I was happy that I had formulated the problem in such a precise way. However, for a few more years until very recently, I had no idea why this rank minimization procedure should yield strongly internally consistent tones, and I struggled to figure it out. The only other intuition that I could have is that an actual tap or rub of a very short duration in time would excite almost all frequencies at nearly equal amplitudes, so having a consistent tone in response to a quick tap or rub would imply internal consistency in the response within each & every frequency.
Ultimately solving this problem
The solution is as follows. If H(ω,→x′) for fixed →x has a rank close to 1, then pressure waves generated by the vibration of the violin's strings that are incident upon the plates will produce spectral responses that have the same functional dependence on ω, apart from overall scalar multiplication that may depend on →x′, that do not otherwise have a complicated dependence on →x′. This means that the incident pressure waves will excite few new modal frequencies in the material volumes of the wood plates, so the overall response will thus be dominated by the modes at the desired frequencies from the string excitation & the resonance in the volume of air in the violin box (bounded by the plates & ribs).
More formally, H(ω,→x′) having a rank of 1 allows for writing H(ω,→x′)=g(ω)u(→x′) where g(ω) is the fundamental spectral response of the plate and u(→x′) simply gives the scalar multiplicative dependence on →x′. If the incident pressure wave can be described by a dimensionless amplitude A(ω,→x′), then the overall response in the listener's ear can be written as a(ω)g(ω) having defined a(ω)=∫VA(ω,→x′)u(→x′) d3x′ in which the integral is taken over the spatial volume V of the solid parts of the plate. Thus, there is essentially very little spatial dependence that can be picked out in the response at the listener's ear. Moreover, in the near-field approximation, a long string, when excited, will generate pressure waves whose amplitude varies only logarithmically with distance from the wire (according to Poisson's equation for an infinitely long wire); logarithmic dependence is almost like a lack of dependence (constant function), so if A(ω,→x′)≈B(ω) for some incident spectrum B(ω) describing the string, then a(ω)≈B(ω)∫Vu(→x′) d3x′ is effectively just a scalar multiple of B(ω), so the overall response is approximately a scalar multiple of B(ω)g(ω). This means that the overall response basically only depends on the tonal quality of the string in isolation when excited, which is calibrated to produce a fairly clean sound exciting only a few specific frequencies, and the fundamental spatially independent tonal quality of the plate; excitation at different parts of the plate won't unnecessarily add or subtract unexpected frequencies at the listener's ear. This is why violins and violas constructed in this manner have such strong internal tonal consistency.
Strictly speaking, tonal consistency does not necessarily imply that the tones are of high quality per se. The fundamental plate spectrum g(ω) could have strong responses at different frequencies that sound jarring when played together at similar amplitudes, so tonal consistency would lead to an instrument that most people would judge to have low quality. Moreover, subtle differences in the materials & initial shape conditions will lead to different combinations of final shapes & materials, so every plate will have a different g(ω), even if a given plate's Green's function H(ω,→x′) has a rank that is exactly or approximately 1.
