Three-string inharmonic networks

This paper studies the resonant frequencies of three-string networks by examining the roots of the relevant spectral equation. A collection of scaling laws are established which relate the frequencies to structured changes in the lengths, densities, and tensions of the strings. Asymptotic properties of the system are derived, and several situations where transcritical bifurcations occur are detailed. Numerical optimization is used to solve the inverse problem (where a desired set of frequencies is specified and the parameters of the system are adjusted to best realize the specification). The intrinsic dissonance of the overtones provides an approximate way to measure the inherent inharmonicity of the sound.


Introduction
Significant effort in musical instrument design has focused on instruments in which the sounding element vibrates periodically where the overtones lie in a harmonic series above a single fundamental frequency.While many vibrating systems (such as those built from single uniform strings and those based on air columns) are inherently harmonic, others (such as bells and vibrating bars) are inharmonic, and there is a sizeable literature showing how to modify the shapes and densities (and perhaps other parameters) so as to make them more harmonic (Fletcher and Rossing 1991).For example, Bork (1995) adjusts the contour of xylophone keys to make them more harmonic, while Fletcher (1993) engineers the pentangle (a bent vibrating bar) so as to achieve (approximately) harmonic overtones.
But it is also possible to begin with inharmonic sounds and to design instruments (and tunings for those instruments) that retain some of the desirable properties of familiar instruments despite the inharmonicity.For example, McLachlan (2004) re-shapes bells so as to achieve a "major-third" sound instead of the intrinsic "minor-third" overtone of Western bells.Examples in Sethares (2004) focus on sounds that can be generated via audio synthesis technologies: guitars with inharmonic strings retuned for 10-tone equal temperament or flute-like instruments with the spectrum of an ideal bar.More recently, Sethares and Hobby (2018) consider the design and specification of non-uniform strings, which can be used in the design of inharmonic stringed instruments like the hyperpiano (Hobby and Sethares 2016).The upper left shows a single "key" consisting of a circular frame (shaded) and three strings.The strings may be of different lengths and densities, and the tensions are adjustable.A mass in the centre is used both to secure the strings and to adjust the overtones of the resulting system.The two larger figures show two possible geometries that combine multiple three-strings keys to form musical instruments.
Recently, Brahmi and Gauthier (2019) have investigated networks of strings and have shown how the physical parameters such as lengths, densities, and tensions relate to the spectrum of the resulting vibrating element.In building their tritare (a guitar-like instrument with six three-string elements), Gaudet et al. (2005) note that the sound is essentially inharmonic and that the "unique frequency spectrum results in a very unique timbre for the tritare."Gaudet and Gauthier (2010) concretize this design.
This paper builds on the work of Gaudet, Gauthier and colleagues and their relevant results are summarized in Section 2 in the Spectral Equation (1), which expresses the frequencies of the overtones as roots of an equation parameterized by the lengths, densities, and tensions of the strings.Consider the artist's conception in the upper left corner of Figure 1 which shows a circular frame supporting three strings that join at a mass.The lengths and densities of the strings may be different and the sound will, in general, be inharmonic.The tensions may be changed using the small tuning pegs; this causes variations in the angles at the junction and in the timbre of the sound.It is intuitively clear that the overall pitch of the element should be lower if the radius of the circular support were larger (thus requiring longer strings).We find scaling laws that show how the lengths of the strings must vary in order to keep the spectrum fixed (other than being transposed).A collection of such scaling laws, including those with density and tension, are introduced in Section 3.
Section 4 examines the asymptotic behaviour of the three-string element, describing what happens when the length of one of the strings increases.We also consider asymptotics of the central mass, as it grows large (compared to the mass densities of the strings) and as it decreases to zero.The mathematical objects of study are the roots of the spectral equation and how they vary when certain parameters are changed.This leads to a kind of bifurcation diagram (or root locus) showing how the roots may legally vary.Interestingly, the parameterizations where the number of roots change are almost universally transcritical bifurcations.
Section 5 considers the relationships that exist between the tensions on the strings and the resulting angles between them.This is used as a constraint in the numerical optimization routine of Section 6 which designs three-string keys from a specification on a desired spectrum.One example specifies a stretched set of octaves, and the numerical optimization is able to locate feasible string lengths, densities, and masses to realize the specification.As far as we know, this is the first acoustic realization of an instrument containing stretched octaves.Another example specifies the overtones of a major third bell, and the optimization is able to realize a three-string design with this overtone structure.Section 7 suggests that a measure of intrinsic dissonance can be used as a way to describe the timbre of the inharmonic sound.This is simple enough that it can be incorporated into the optimization routine to help guide the designs.

Spectrum of a three-string key
The fundamental object of study in this paper is a junction of three ideal strings fixed at one end and attached together via a centre mass M as displayed in Figure 2; we call this a three-string system or a key when it is thought of as an element of a larger musical instrument, as in the designs of Figure 1.The ith string in the key is parameterized to have length l i , mass density μ i and tension τ i .
The model used in this paper to determine the resonant frequencies of a three-string key is taken from Brahmi and Gauthier (2019) where it was derived by representing the transverse displacement of each string and the centre mass using the wave equation.Similar techniques have also been used to examine the modes of vibration of other systems such tubular bells in Oliver and Arsie (2019).The three-string system has two boundary conditions: the displacement of each string and the centre mass must be equal at the centre of the key, and the string displacements are zero at their fixed ends.Using multiplicative separation of variables to solve the wave equation along each string, the spatial solutions are derived, each with a single arbitrary constant.Substituting the spatial solutions into the separated wave equation for the centre mass generates a system of three homogeneous linear equations which together dictate the relationship between the three arbitrary constants.This linear system has solutions if and only if the determinant of its corresponding coefficient matrix is zero.The roots of the resulting spectral equation are the resonant frequencies of the key.
Definition 1 Spectral equation The spectral equation for a three-string system with a single centre mass M is where The roots of this equation {λ i } ∞ i=1 are the resonant frequencies of the system, as presented in Brahmi and Gauthier (2019).Since s(0) = 0, λ = 0 is the trivial solution for any combination of τ i , μ i and l i .Physically, this corresponds to the system at rest.Observe that (1) is fundamentally different from the corresponding equation for a single nonuniform string in Sethares and Hobby (2018) though both consist of products and sums of sinusoids.

Scaling laws
As suggested in Figure 1, keys of different sizes can be combined to design a three-string instrument.In the case of stringed and keyboard instruments, it is desirable for each vibrating element to have different fundamental frequencies while retaining the "same" spectrum.For example, if one key has partials at {f 1 , f 2 , . . ., f n }, another key would have partials at {αf 1 , αf 2 , . . ., αf n } for some transposition factor α.For instruments with uniform strings, this is easy since the resonant frequencies of each string (regardless of string length, tension or mass density) are always harmonic: where l, τ and μ are the length, tension and mass density of the string (Morse and Uno Ingard 1968).Selecting the fundamental f 1 of a single string is as simple as changing l, μ or τ .For example, doubling the length halves the frequencies.Doubling the mass density scales the frequencies by 1/ √ 2. Doubling the tension scales the frequencies by √ 2. This section studies analogous scaling laws for three-string keys, which are useful for the purposes of musical instrument design.Two types of scaling laws are considered: the scaling of parameters that leave the resonant frequencies unchanged, and the scaling of the parameters that leave the spectrum unchanged.

Frequency invariant scaling
Theorem 3.1 (Length and Tension vs. Density Scaling) Suppose there exists a three-string system with tensions {τ i } 3 i=1 , densities {μ i } 3 i=1 and lengths ), the resonant frequencies of the three-string system remain unchanged.
Proof Observe that each term in f (λ) and g(λ) in (1) remains unchanged under the specified substitutions.
Theorem 3.2 (Tension, Density and Center Mass Scaling) Suppose there exists a three-string system with tensions {τ i } 3 i=1 , densities {μ i } 3 i=1 , centre mass M, and resonant frequencies , the resonant frequencies of the three-string system remain unchanged.
Proof Under the specified substitutions in (1), Since r > 0, the roots remain unchanged under this transformation.
Scaling laws like those of Theorems 3.1 and 3.2 may be useful in the design of an instrument with many keys, since they show to how to trade off the various physical parameters without changing the sound.

Spectrally invariant scaling
As is the case with single strings, there are several ways to transpose the frequencies of a threestring key by adjusting the tension, densities, lengths, and centre masses.
Theorem 3.3 (Length Scaling) Suppose there exists a three-string system with lengths {l i } 3 i=1 and resonant frequencies {λ i } ∞ i=1 .If all of the string lengths l i and the centre mass M are scaled by r, the resonant frequencies of the three-string system become {λ i /r} ∞ i=1 .
A proof of Theorem 3.3 is provided in Appendix 1. Proofs of Theorems 3.4-3.5 are analogous.
Theorem 3.4 (Density Scaling) Suppose there exists a three-string system with densities {μ i } 3 i=1 and resonant frequencies {λ i } ∞ i=1 .If all of the string densities and the centre mass are scaled by r, the resonant frequencies of the three-string system become {λ i / √ r} ∞ i=1 .
Theorem 3.5 (Tension Scaling) Suppose there exists a three-string system with tensions {τ i } 3 i=1 and resonant frequencies {λ i } ∞ i=1 .If all of the tensions and the centre mass are scaled by a factor of r, then the resonant frequencies of the three-string system become { √ rλ i } ∞ i=1 .
Together, Theorems 3.3-3.5 demonstrate that the scaling laws associated with three-string keys are not that different from the familiar laws governing single (uniform) string scaling.

Asymptotic behaviour of a three-string key
For harmonic strings, the resonant frequencies obey two important asymptotic properties.First, as seen in (2), in the limit that the string length or density goes to infinity, the resonant frequencies decay towards zero.Second, in the limit that the string tension increases without bound, the resonant frequencies go to infinity (presuming, of course, that the string does not break).This section presents the analogous asymptotic behaviours of the three-string keys and extends this to include an analysis of how the centre mass alters the resonant frequencies of the system.

Large mass
The centre mass M plays a unique role in the resonant frequencies of three-string keys in determining the extent to which the strings interact.As can be seen in ( 1), as M goes to infinity, g(λ) dominates the spectral equation while as M decreases to zero, f (λ) dominates.This section studies the spectrum of an arbitrary three-string key in the limit that the centre mass M → ∞; as the mass increases, the three strings essentially decouple into three independent harmonically vibrating elements.
Proof See Appendix 2.
Theorem 4.2 (Asymptotic Behavior for Large M ) The lowest resonant frequency of any threestring system with centre mass M must approach zero in the limit as M approaches infinity.
Proof See Appendix 3.
Theorem 4.3 (Superposition for Large M ) Suppose there exists a three-string system with a heavy mass M such that the massive term in s(λ) is much larger than the mass-less term.Then, the resonant frequencies of the three-string system are a superposition of the "resonant frequencies" of each constituent string in the system.
Proof See Appendix 4.

Large string length; large mass density
This subsection studies how the resonant frequencies of a three-string key change in the limit when a single string length or density approaches infinity.The effect of an asymptotically large tension is not studied since a design where τ 1 τ 2 , τ 3 since it is physically impossible to have one tension increase without also increasing the others.
Theorem 4.4 (Approximate Harmonic Spectra with One Long String) Consider a three-string system with equal string densities μ 1 = μ 2 = μ 3 ≡ μ, equal string tensions τ 1 = τ 2 = τ 3 ≡ τ , centre mass M, and string lengths l 1 , l 2 and l 3 , where l 2 = l 3 ≡ l.For any integer n > 1, there exists a large string l 1 lnπ , such that the first n frequencies of the corresponding three-string system approach a harmonic spectrum where λ i ≈ 2π i √ τ/μ/l 1 for any integer 1 ≤ i ≤ n.
Proof See Appendix 5.
Theorem 4.5 (Approximate Harmonic Spectra with One Large Density) Consider a three -string system with equal string lengths l 1 = l 2 = l 3 ≡ l, equal string tensions τ 1 = τ 2 = τ 3 ≡ τ , a centre mass M, and string densities μ 1 , μ 2 and μ 3 , where μ 2 = μ 3 ≡ μ.For any integer n > 1, there exists a string density μ 1 √ μnπ such that the first n frequencies of the corresponding three-string system approach a harmonic spectrum where Proof Analogous to the proof of Theorem 4.4 in Appendix 5.

Angle parameterization
This section introduces the physical variables (other than lengths and densities) used to design three-string systems and the range of values they may assume.Section 5.1 discusses the angle convention used to parameterize a three-string key (without loss of generality).Section 5.2 studies the relationship between string tensions and the angles that govern three-string systems with the purpose of determining the values that each angle can take in order to maintain a physically realizable key.

Orientation
Consider an arbitrary three-string design.By rotating the frame of reference, any design can be viewed as a right-side-up "Y" such that the angles on either side of the "Y" lie between 90 and 180 degrees. 1 Let θ 1 and θ 2 be the angles on the left-and right-hand-side of the "Y", respectively, and let θ 3 be the angle of the branch as seen in the right-hand side of Figure 2. Any physically attainable design can be rotated into this form.Let γ ≡ 180 − θ 1 and θ ≡ 180 − θ 2 .With this substitution, γ and θ can be altered to generate any other design so long as 0 < θ, γ < 90.In designing these instruments, we will be altering γ and θ in place of the string tensions τ 1 , τ 2 and τ 3 .

Three-string junctions
When designing three-string systems, the string lengths l i and densities μ i can be varied independently while retaining designs that are physically plausible.The three-string angles (γ and θ ), however, must be chosen with care since these angles are proxy variables for the string tensions, and there are combinations of string tensions that violate Newton's second law and are thus physically unrealizable.This section studies what angle combinations (θ, γ ) constitute physically realizable systems.
By applying Newton's second law in the horizontal and vertical directions where γ and θ are angles, and τ 1 , τ 2 and τ 3 are tensions as indicated in the right-side diagram of Figure 2 (Thornton and Marion 2004).
There are also many combinations of tensions (τ 1 , τ 2 , τ 3 ) that correspond to physically realizable systems (solutions to (4)) but that may be undesirable for the purposes of building a musical instrument.For instance, it is possible to design a three-string system where (τ 1 , τ 2 , τ 3 ) = (100 N, 20 N, 100 N) (the subsequent angles are θ ≈ 85 • and γ ≈ 12 • ); however, the tension disparities between the strings would lead to complications in the assembly of the instrument.
We study systems with tensions such that 1/2 < τ i /τ j < 2 for any i, j ∈ {1, 2, 3}, which is a common ratio preference for stringed instruments (Guettler 2013).By rewriting (4) in terms of the ratios between tensions and imposing the physical preferences from ( 5), the resulting constraints on tensions are Inequalities ( 6) and ( 7) place a limit on the combinations of angles (θ, γ ) used to design threestring keys.The shaded region in Figure 3 displays the portion of (θ, γ )-space that satisfies these conditions.As indicated within the shaded region of Figure 3, conditions ( 6) and ( 7) are met for all 30 • ≤ γ , θ ≤ 75 • .

Designing three-string keys
Using the physical model ( 1), the resonant frequencies of a given three-string key are defined to be the ordered set where l ≡ (l 1 , l 2 , l 3 ) are the lengths, μ ≡ (μ 1 , μ 2 , μ 3 ) are the densities, M is the centre mass, and the angles θ and γ are constrained as in Figure 3 to be surrogates for physically plausible tensions.can be calculated by solving for the roots of s(λ) with the aid of a numerical root finding method (i.e.Newton-Raphson).
The inverse problem, finding a design with resonant frequencies equal to some desired spectrum, is not as straightforward.This section presents a solution to the inverse problem by the implementation of a numerical optimization algorithm: simultaneous perturbation stochastic approximation (SPSA).SPSA provides a method for optimizing the partials of a key by altering its various parameters: Let the desired spectrum of a given three-string key be defined to be the ordered set of frequencies * .In designing a three-string key, the objective is to find the three-string parameters such that is minimized under some appropriate norm p.This optimization can be performed with a gradient descent method is the value of the key parameters at timestep i and α is the algorithm stepsize.
Although it is possible to calculate the gradient of J at each timestep, it is costly.One way to avoid this calculation is by using the Simultaneous Perturbation Stochastic Approximation (SPSA) method which substitutes the explicit derivative calculation for a randomized numerical approximation (Spall 1997).With SPSA the optimization algorithm becomes where (i) is a random perturbation vector re-evaluated at each timestep i of length nine (the number of parameters of the system).Note that in the limit that c → 0, (12) recovers the derivative as in (10).

Stretched spectra
The partials of a piano string are known to exhibit a stretching, such that the partials are a bit larger than the harmonic partials expected by the classical model of a fixed-fixed ideal string (Young 1952).This stretching inspired (Slaymaker 1970) to simulate the sound of chords comprised of frequencies with stretched partials using additive synthesis.These partials are located at frequencies where f is a fundamental frequency.For stretching factors S greater than 2, the pseudo-octave and partials are stretched while for S < 2, they are compressed.For S = 2, (13) reduces to a harmonic spectrum.Setting the desired spectrum of ( 9) equal to the first five partials of ( 13), an SPSA optimization is able to design three-string keys with stretched partials for stretching factors S = 2, 2.01, 2.02, . . ., 2.1.The resulting designs are listed in an online supplement.

A major third bell
Significant effort has been spent designing bells with major-third partials (McLachlan 2004;Peairs 2016).The first five relative frequencies of an ideal major-third spectrum are 1, 2, 2.5, 3, 4 (Fletcher and Rossing 1991).SPSA is able to design a three-string key with these partials with high accuracy as displayed in Table 1.
Table 1.Design parameters for a three-string key with major-third partials.

Desired
Achieved % error Notes: The resonant frequencies are not included in the above table; however, any fundamental can be picked while maintaining this spectrum by using the scaling laws presented in Section 3.

Intrinsic dissonance of three-string systems
When listening to a harmonic sound, all the overtones/partials merge into one perceptual object.Inharmonic sounds may be heard as one object, or they may fission into two (or more) objects of perception (Dewitt and Crowder 1987).A commonplace example occurs when listening to a chord on a piano or guitar.When listening analytically, the multiple harmonic series are heard as separate notes, each with a harmonic set of overtones.When listening holistically, all the overtones of all the notes may merge into one perceptual entity.One reason the three-string key is intriguing is because it can smoothly connect these two perceptual extremes.As shown in Theorem 4.3, when the mass is large, the sound of the element approaches that of a chord with three separate vibrating elements.When the mass is modest, the sound tends to fuse into a single inharmonic perceptual object.Huron (1991) studies the relationship between auditory fusion and consonance in Western music.

Dissonance of a spectrum
We posit that when listening holistically to a three-string key, the overall inharmonicity is related to the intrinsic sensory dissonance: harmonic sounds will have low dissonance, while inharmonic clusters will tend to have relatively high dissonance.This can be concisely modelled based on the work of Plomp and Levelt (1965) as parameterized in Sethares (1993Sethares ( , 2004) ) where the sensory dissonance of two pure tones is where x is the frequency difference.
Calculating the dissonance of a more complex sound can be done by summing over all pairs of component frequencies within the spectrum as where n is the number of partials of the sound and i is the ith element of in (8).Note that the i = j terms do not contribute to this sum since d(0) = 0.

Root bifurcation and dissonance
The spectral Equation (1) has a rich root bifurcation structure.This section shows conditions under which the resonant frequencies of the system bifurcate into two neighbouring frequencies.The closeness of the neighbouring frequencies can be quantified by the dissonance incurred, and this follows the same general contour as the function d(x) of ( 14): near zero for x ≈ 0, rising to a peak at some intermediate value of x, returning to zero as x increases.Thus the points of maximum dissonance correspond (approximately) to sounds with maximum inharmonicity while the bifurcation points themselves correspond to points of (locally) minimum sensory dissonance.Thus (15) can be incorporated into the design procedure (as in Section 6) to control the amount of inharmonicity present in the sound.Four different types of transcritical bifurcations are easily described in the roots of the spectral equation.
Theorem 7.1 (Single String Length Bifurcation) Consider a three-string system where μ 1 = μ 2 = μ 3 ≡ μ, τ 1 = τ 2 = τ 3 ≡ τ but l 1 = l 2 ≡ l and l 3 = ql for some positive real number q.As a consequence of the q and λ derivatives of the spectral equation being equal to zero, this system exhibits transcritical root bifurcation points.A subset of these bifurcations take place at frequencies λ = (nπ √ τ/μ)/l for all n ∈ Z ≥1 and q ∈ Q >0 on the condition that qn ∈ Z ≥1 .
Proof See Appendix 6.A graph illustrating the spectral dissonance for a possible range of string lengths q:1:1 for q ∈ (1, 5) (all else being equal) is displayed in Figure 4. Instead of changing q to study all of these possible string length ratios an alternative parameterization with parameter δ is used with string lengths 1 + δ : 1 − δ/2 : 1 − δ/2.The benefit of this parameterization is that it (mostly) preserves the fundamental frequency for the designs studied.This allows for the values of the spectral dissonance to be comparable between the different designs.Notice that in Figure 4, the local minima for dissonance are all bifurcation points according to Theorem 7.1.The following three results (Corollaries 7.2 and 7.3, and Theorem 7.4) are all proven analogously to Theorem 7.1 of Appendix 6.
Corollary 7.2 (Single String Density Bifurcation) Consider a three-string system where l 1 = l 2 = l 3 ≡ l, τ 1 = τ 2 = τ 3 ≡ τ but μ 1 = μ 2 ≡ μ and μ 3 = qμ for some positive real number q.As a consequence of the q and λ derivatives of the spectral equation being equal to zero, this system exhibits transcritical root bifurcation points.A subset of these bifurcations take place at frequencies λ = (nπ √ τ/μ)/l for all n ∈ Z ≥1 and q ∈ Q >0 on the condition that n √ q ∈ Z ≥1 .
Corollary 7.3 (Single String Tension Bifurcation) Consider a three-string system where l 1 = l 2 = l 3 ≡ l, μ 1 = μ 2 = μ 3 ≡ μ but τ 1 = τ 2 ≡ τ and τ 3 = qτ for some positive real number q.As a consequence of the q and λ derivatives of the spectral equation being equal to zero, this system exhibits transcritical root bifurcation points.A subset of these bifurcations take place at frequencies λ = (nπ √ τ/μ)/l for all n ∈ Z ≥1 and q ∈ Q >0 on the condition that n/ √ q ∈ Z ≥1 .
Theorem 7.4 (Multiple String Length Bifurcation) Consider a three-string system with equal string tensions τ i ≡ τ and equal densities μ i ≡ μ, but string lengths such that l 1 /l 3 ≡ k ∈ Q >0 and l 2 /l 3 ≡ q ∈ Q >0 .As a consequence of the k, q and λ derivatives of the spectral equation being equal to zero, this system exhibits transcritical root bifurcation points.A subset of these bifurcations take place at frequencies λ = nπ √ τ/μ/l 3 for all n ∈ Z ≥1 when kn ∈ Z ≥1 and qn ∈ Z ≥1 .For Theorem 7.4, a graph can be drawn similar to that presented in Figure 4.In order to introduce the second varying parameter, however, a dissonance surface must be sketched.Figure 4 can be thought of as a slice of that dissonance surface (where, in the language of Theorem 7.4, q is the varying parameter and k = 1).

Discussion
This paper has studied the properties of the frequencies of vibration of a three-string network.Similar analyses and results can be readily obtained for n-string configurations, as long as the elements are constrained to vibrate in a single direction (that is, for the network to lie in a plane).The spectral equation (and its roots) for truly three-dimensional vibrations are considerably more complicated.
Using numerical optimization, keys with stretched spectra and major-third partials are designed.While not discussed above, designing spectra with highly inharmonic partials, such as odd harmonics or ideal-bar spectra were not possible using SPSA.Such designs may require more complex networks such as: (1) keys comprised of non-uniform strings, (2) keys with more than three component strings, or (3) networks with many connected n-string keys.Increasing the number of design variables should provide a greater range of potential spectra.
One intriguing challenge is that of designing a complete musical instrument.What is the best mechanism to physically join the strings at the mass?What governs optimal choices for the "desired" spectra?One possibility is to use dissonance curves to help select compatible threestring keys in harp-like instruments such as those of Figure 1.Fretted instruments present an even more challenging design problem since the spectrum/timbre of the keys will change along with the pitch as any single string is fretted.
The excitation mechanism of a three-string key is also an important consideration.Typically, the excitation in a stringed instrument may be plucked, hammered, or bowed.The pluck (and the hammer) can be studied by looking at the pattern of nodes and anti-nodes for each frequency: if the excitation occurs near a node, very little energy will be transferred to the string while if it occurs near an anti-node, the frequency will be strongly excited.This mechanism is analogous to the situation in familiar single string instruments, though of course the nodes and anti-nodes are placed differently.When bowing, a mode-locking mechanism (Fletcher 1978) will likely occur in which one of the prominent modes dominates and oscillations occur at harmonic multiples of the frequency of that mode.Considering the radiative properties of a three-stringed instrument, analysis would proceed with all the issues normally associated with the study of the sound-box of a guitar or a piano (Fletcher and Rossing 1991).Clearly, each of these issues requires further study.So if the roots (frequencies) of s(λ) are defined to be {λ i } ∞ i=1 , the frequencies of s(λ)| (rl i ,rM ) must be stretched/compressed along the λ-axis as {λ i /r} ∞ i=1 .

Figure 1 .
Figure 1.Artist's conception of musical instruments based on three-strings.The upper left shows a single "key" consisting of a circular frame (shaded) and three strings.The strings may be of different lengths and densities, and the tensions are adjustable.A mass in the centre is used both to secure the strings and to adjust the overtones of the resulting system.The two larger figures show two possible geometries that combine multiple three-strings keys to form musical instruments.

Figure 2 .
Figure 2. Two diagrams of a three-string key with centre mass M, string lengths l i , tensions τ i , mass densities μ i for 1 ≤ i ≤ 3. The right diagram displays the θ and γ variables used for optimization.
Lemma 4.1 (Location of Lowest Nontrivial Resonant Frequency) For any three-string system, the lowest resonant frequency occurs within the interval k

Figure 4 .
Figure 4. Graph of spectral dissonance as a function of a varying string length.The string length combinations are parameterized by δ such that the lengths are 1 + δ : 1 − δ/2 : 1 − δ/2.

Figure A1 .
Figure A1.Roots of spectral equation for large M.