# Modeling Consonance and its Relationships with Temperament, Harmony, and Electronic Amplification

## TL;DR

This paper presents a computational model of musical consonance based on Helmholtz's theory, applied to various scales, chords, and electronic amplification, revealing insights into scale classification and effects of nonlinearities.

## Contribution

It introduces a simple yet effective mathematical model of consonance, representing temperaments as graphs, and analyzes the impact of electronic nonlinearities on musical perception.

## Key findings

- Model aligns well with observed properties of scales and chords
- Graph representation reveals two main groups of temperaments
- Nonlinear electronic amplification significantly alters consonance patterns

## Abstract

After briefly revising the concepts of consonance/dissonance, a respective mathematic-computational model is described, based on Helmholtz's consonance theory and also considering the partials intensity. It is then applied to characterize five scale temperaments, as well as some minor and major triads and electronic amplification. In spite of the simplicity of the described model, a surprising agreement is often observed between the obtained consonances/dissonances and the typically observed properties of scales and chords. The representation of temperaments as graphs where links correspond to consonance (or dissonance) is presented and used to compare distinct temperaments, allowing the identification of two main groups of scales. The interesting issue of nonlinearities in electronic music amplification is also addressed while considering quadratic distortions, and it is shown that such nonlinearities can have drastic effect in changing the original patterns of consonance and dissonance.