1. Introduction ARTICLE SECTIONS Jump To

Musical instruments create sound by vibrating at a frequency within the range of human hearing. Different types of instruments have different vibrating parts, and this distinction has long been used to categorize musical instruments. For example, in the Hornbostel–Sachs system of musical instrument classification, membranophones like drums and kazoos have vibrating membranes; chordophones like pianos and guitars have vibrating strings; aerophones like flutes and pipe organs have vibrating columns of air; and in idiophones like bells and triangles, the whole instrument vibrates. (1)

The precise frequencies of sound created by a musical instrument are largely determined by the physical properties of that instrument. For example, the pitch of a drum is a function of the size and tension of the drum head; the pitch of a guitar is determined by the length, mass, and tension of the strings; the pitch of a pipe organ is a function of the length of the pipes; and the pitch of a bell is a function of its size and shape. When a musician tunes a musical instrument, they are altering the instrument’s physical properties to obtain the desired pitch.

The relationship between a musical instrument’s physical properties and its sound frequencies raises an interesting prospect: could an instrument’s sound be used to infer information about the instrument’s physical properties? More specifically, could we add a sample to a musical instrument, measure the resulting change in the instrument’s frequency, and use this change to determine information about the sample and its properties?

f n of a plucked mbira tine is (1) n is the mode of vibration (n = 1, 2, 3...), λ n are eigenvalue solutions of the frequency equation of a cantilever beam (constants), and the remaining variables are known properties of the tine: E is Young’s modulus of the tine material (often steel), I is the second moment of inertia (a function of the cross-sectional shape of the tine), ρ is the density of the tine material, A is the cross-sectional area of the tine, and L is the length of the tine. In this work, we demonstrate that musical instruments can be used as practical sensors in important real-world applications. For this initial demonstration, we focused our efforts on the mbira, a 3000-year-old African musical instrument (2) also known as the karimba, kalimba, or thumb piano. (3) The mbira ( Figure 1 A) usually consists of metal tines of different sizes and lengths attached to a wooden sounding box; these tines are plucked to create musical notes. As an idiophone, the mbira’s sound is influenced by the physical properties of the metal tines: longer or larger tines create low notes, and shorter or smaller tines create high notes. In general, the frequencyof a plucked mbira tine iswhereis the mode of vibration (= 1, 2, 3...), λare eigenvalue solutions of the frequency equation of a cantilever beam (constants), and the remaining variables are known properties of the tine:is Young’s modulus of the tine material (often steel),is the second moment of inertia (a function of the cross-sectional shape of the tine), ρ is the density of the tine material,is the cross-sectional area of the tine, andis the length of the tine. (4) Equation 1 shows that at least five different physical properties of the mbira tine influence the pitch of an mbira note (and could therefore in principle be measured by analyzing the frequency of an mbira note).

Figure 1 Figure 1. (A) This conventional mbira musical instrument (left) has 10 metal tines of different lengths mounted on a wooden sounding box; plucking the tines creates musical notes. By replacing these tines with a length of stainless steel tubing bent into a U shape (center), we create a sensor capable of accurately measuring the density of any sample inside the tubing with a resolution of 0.012 g/mL. Mbira sensors can also be made using scrap lumber and hardware (right). (B) Waveform plot of a sound recording of plucking an mbira sensor, obtained using a smartphone’s voice recorder app and our analysis website (http://mbira.groverlab.org). The early part of the sound (C) exhibits inharmonic overtones, whereas the rest of the sound (inset) consists of a pure tone. By performing a Fourier transform on this portion of the sound, we determine the fundamental resonance frequency of the tubing, which is inversely proportional to the density of the sample inside the tubing. Using the mbira sensor and a smartphone to test river water in California (D) and bison milk in India (E).

n = 1), (2) r i is the inner radius of the tubing, r o is the outer radius of the tubing, ρ t is the density of the tubing material, and ρ s is the density of the sample inside the tubing. If we know the tubing’s dimensions, density, and Young’s modulus, we can use s . And even without any information about the tubing, we can approximate (3) a and b are calibration constants for a particular mbira sensor. By measuring an mbira sensor’s frequency f when filled with two or more samples of different known densities ρ s and plotting f vs ρ s , we can obtain a and b for the mbira sensor from the slope and Y-intercept of the plot. Then, by filling the mbira sensor with a sample of unknown density, measuring the sensor’s frequency, and solving In this study, we primarily focused on ρ, the density of the tine material, as our physical property of interest when using the mbira as a sensor. If we change the density of the tine material, the frequency of the mbira note will change by a predictable (and measurable) amount. In particular, if we use hollow tubing as a tine, then the frequency of the mbira note will be influenced by the density of the tubing material (which is constant) and the density of the sample inside the tubing (which can be anything we like). When the tubing is filled with a low-density sample (like air), the net density of the tubing is lower, which results in a higher frequency when the tubing is plucked. When the tubing is filled with a higher-density material (like water), the net density of the tubing is higher, which results in a lower frequency. In general, for a hollow-tubing-based mbira vibrating in its first resonance mode (= 1), eq 1 becomeswhereis the inner radius of the tubing,is the outer radius of the tubing, ρis the density of the tubing material, and ρis the density of the sample inside the tubing. If we know the tubing’s dimensions, density, and Young’s modulus, we can use eq 2 to predict the frequency of the mbira sensor when filled with a sample of a density ρ. And even without any information about the tubing, we can approximate eq 2 aswhereandare calibration constants for a particular mbira sensor. By measuring an mbira sensor’s frequencywhen filled with two or more samples of different known densities ρand plottingvs ρ, we can obtainandfor the mbira sensor from the slope and-intercept of the plot. Then, by filling the mbira sensor with a sample of unknown density, measuring the sensor’s frequency, and solving eq 3 , we can determine the density of the sample. Figure 1 A shows two mbira density sensors, one made from a commercial musical instrument (middle) and one made from scrap lumber and hardware from the author’s garage (right).