Related papers: The wave equation for stiff strings and piano tuni…
Piano tones vary according to how pianist touches the keys. Many possible factors contribute to the relations between piano touch and tone. Focusing on the stiffness of string, we establish a model for vibration of a real piano string and…
Guitar fretboards are designed based on the equation of the ideal string. That is, it neglects several factors as nonlinearities and bending stiffness of the strings. Due to this fact, intonation of guitars along the whole neck is not…
In this paper we present mathematical and physical models to be used in the analysis of the problem of intonation of musical instruments such as guitars, mandolins and the like, i.e., we study how to improve the tuning on these instruments.…
The amplitude equation for an unstable electrostatic wave is analyzed using an expansion in the mode amplitude $A(t)$. In the limit of weak instability, i.e. $\gamma\to 0^+$ where $\gamma$ is the linear growth rate, the nonlinear…
The human sense of hearing perceives a combination of sounds 'in tune' if the corresponding harmonic spectra are correlated, meaning that the neuronal excitation pattern in the inner ear exhibits some kind of order. Based on this…
We investigate a dynamically adapting tuning scheme for microtonal tuning of musical instruments, allowing the performer to play music in just intonation in any key. Unlike other methods, which are based on a procedural analysis of the…
In this paper I want to suggest a new solution to the problem of musical tuning. On one hand, I see it as a generalization of Just Intonation (JI) to inharmonic timbers, on another, as a unification of spectral interference and harmonicity…
The mathematics of musical intervals and scales has been extensively studied. Vastly simplified, our ears seem to prefer intervals whose frequency ratios have small numerator and denominator, such as 2:1 (octave), 3:2 (perfect fifth), 4:3…
We give a new analysis of a tuning problem in music theory, pertaining specifically to the approximation of harmonics on a two-dimensional keyboard. We formulate the question as a linear programming problem on families of constraints and…
We study eigenvibrations for inhomogeneous string consisting of two parts with strongly contrasting stiffness and mass density. In this work we treat a critical case for the high frequency approximations, namely the case when the order of…
The end conditions of piano strings can be approximated by the input admittance at the bridge. Proper measurements of this value are therefore required. A method of validation of admittance measurements on simple structures is proposed in…
We investigate the instabilities of low winding number electroweak strings using standard numerical techniques of linear algebra. For strings of unit winding we are able to confirm and extend existing calculations of the unstable region in…
Several cases of the explanation of the phenomenon of standing waves in strings, there are few experimental measurement tools when demonstrating this phenomenon in a classroom, it is for this reason that we have implemented different forms…
In this paper we present new results on the preservation of polynomial stability of damped wave equations under addition of perturbing terms. We in particular introduce sufficient conditions for the stability of perturbed two-dimensional…
We study the stability/instability of standing waves for the one dimensional nonlinear Schr\"odinger equation with double power nonlinearities: \begin{align*} &i\partial_t u +\partial_x^2 u -|u|^{p-1}u +|u|^{q-1}u=0, \quad (t,x)\in…
We investigate the use of Physics-Informed Neural Networks (PINNs) for solving the wave equation. Whilst PINNs have been successfully applied across many physical systems, the wave equation presents unique challenges due to the multi-scale,…
In piano playing, different ways of touch lead to different tones. The effects of changing forces on keys are presented by changing interaction between hammer and string. The thesis focuses on several important variables in hammer-string…
We present a statistical-modelling method for piano reduction, i.e. converting an ensemble score into piano scores, that can control performance difficulty. While previous studies have focused on describing the condition for playable piano…
We study a class of time dependent solutions of the vacuum Einstein equations which are plane waves with weak null singularities. This singularity is weak in the sense that though the tidal forces diverge at the singularity, the rate of…
We consider a full set of harmonics for the Stokes wave in deep water in the absence of viscosity, and examine the role that higher harmonics play in modifying the classical Benjamin-Feir instability. Using a representation of the wave…