Design of a fretboard using the stiff string equation

TL;DR

This paper aims to improve guitar fretboard design by analyzing the effects of string stiffness on vibration, potentially leading to better intonation through a redesigned fret distribution.

Contribution

It introduces a method to incorporate bending stiffness into string vibration models for more accurate fretboard layout design.

Findings

Differences between ideal and stiff string vibration solutions identified

Proposed fret distribution adjustments to enhance guitar intonation

Analytical approach using separation of variables and perturbation analysis

Abstract

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 perfect, and guitars have right tuning just in an \emph{average} sense. There are commercially available fretboards that differ from the traditional design. (One example is the \cite{patent} by the Company True Temperament AB, where each fretboard is made using CNC processes.) As a final application of this work we would like to redesign the fretboard layout considering the effects of bending stiffness. The main goal of this project is to analyze the differences between the differences in the solution for vibrations of the ideal string and a stiff string. These differences should lead to changes in the fret distribution for a guitar, and, hopefully improve the overall intonation of the instrument. We will start analyzing the ideal string equation and after a good understanding of this analytical solution we will proceed with the, more complex, stiff equation. Topics like separation of variables, Fourier transforms, and Perturbation analysis might prove useful during the course of this project