Tonal Frequencies, Consonance, Dissonance: A Math-Bio Intersection

TL;DR

This paper develops a mathematical model using differential equations to calculate musical note frequencies and explores the neurobiological basis of consonance and dissonance in music, explaining why certain notes sound harmonious.

Contribution

It introduces a novel differential equations approach for frequency calculation and provides a neurobiological explanation for musical consonance and dissonance.

Findings

Mathematical model for tonal frequencies using differential equations

Theoretical explanation of consonance and dissonance based on neurobiology

Analysis of harmonic richness and sound interference patterns

Abstract

To date, calculating the frequencies of musical notes requires one to know the frequency of some reference note. In this study, first-order ordinary differential equations are used to arrive at a mathematical model to determine tonal frequencies using their respective note indices. In the next part of the study, an analysis that is based on the fundamental musical frequencies is conducted to theoretically and neurobiologically explain the consonance and dissonance caused by the different musical notes in the chromatic scale which is based on the fact that systematic patterns of sound invoke pleasure. The reason behind the richness of harmony and the sonic interference and degree of consonance in musical chords are discussed. Since a human mind analyses everything relatively, anything other than the most consonant notes sounds dissonant. In conclusion, the study explains clearly why musical notes and in toto, music sounds the way it does.