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A solution to the mystery of the sub-harmonic series and to the combination tone via a linear mathematical model of the cochlea

Ugo Boscain, Xiangyu Ma, Dario Prandi, Giuseppina Turco

TL;DR

The study tackles how sub-harmonic phenomena and Tartini's third tone can arise within a linear cochlear framework. It develops a continuum of noninteracting damped strings with position-dependent parameters and an energy-based readout $E(x,t)$ to the cortex, showing that subharmonics and combination tones emerge from linear dynamics plus energy nonlinearity in $E$. By analyzing sinusoidal and genuine sounds, it demonstrates that subharmonics (notably odd ones) appear near resonant strings and that two-tone inputs yield combination tones described by $( {\mathbf k}_1-{\mathbf k}_0)/q$, connecting to historical Helmholtz and Lagrange theories via a unifying energy-based picture. The authors validate the robustness of these phenomena across parameter sets drawn from Nobili 2003 and a modified NVTM03 scenario, and provide practical parameter choices for numerical exploration, highlighting the method's relevance for psychoacoustics and auditory perception.

Abstract

In this paper, we study a simple linear model of the cochlea as a set of vibrating strings. We make hypothesis that the information sent to the auditory cortex is the energy stored in the strings and consider all oscillation modes of the strings. We show the emergence of the sub-harmonic series whose existence was hypothesized in the XVI century to explain the consonance of the minor chord. We additionally show how the nonlinearity of the energy can be used to study the emergence of the combination tone (Tartini's third sound) shedding new light on this long debated subject.

A solution to the mystery of the sub-harmonic series and to the combination tone via a linear mathematical model of the cochlea

TL;DR

The study tackles how sub-harmonic phenomena and Tartini's third tone can arise within a linear cochlear framework. It develops a continuum of noninteracting damped strings with position-dependent parameters and an energy-based readout to the cortex, showing that subharmonics and combination tones emerge from linear dynamics plus energy nonlinearity in . By analyzing sinusoidal and genuine sounds, it demonstrates that subharmonics (notably odd ones) appear near resonant strings and that two-tone inputs yield combination tones described by , connecting to historical Helmholtz and Lagrange theories via a unifying energy-based picture. The authors validate the robustness of these phenomena across parameter sets drawn from Nobili 2003 and a modified NVTM03 scenario, and provide practical parameter choices for numerical exploration, highlighting the method's relevance for psychoacoustics and auditory perception.

Abstract

In this paper, we study a simple linear model of the cochlea as a set of vibrating strings. We make hypothesis that the information sent to the auditory cortex is the energy stored in the strings and consider all oscillation modes of the strings. We show the emergence of the sub-harmonic series whose existence was hypothesized in the XVI century to explain the consonance of the minor chord. We additionally show how the nonlinearity of the energy can be used to study the emergence of the combination tone (Tartini's third sound) shedding new light on this long debated subject.

Paper Structure

This paper contains 21 sections, 1 theorem, 58 equations, 11 figures.

Table of Contents

  1. Introduction
  2. A model of the basilar membrane as a set of noninteracting vibrating strings
  3. The physics
  4. The Mathematics
  5. Parameterization in function of $\xi$
  6. Unforced case
  7. Asymptotic response to a sinusoidal signal
  8. Asymptotic response to a genuine sound
  9. A model for the information sent to the auditory cortex
  10. Perception of a sinusoidal signal and emergence of the sub-harmonic series
  11. Perception of a periodic signal
  12. More on the nonlinearity of the response: combination tone
  13. Comparison with Helmholz's and Lagrange's theory
  14. Choice of parameters for numerical simulations
  15. Parameters directly deduced from nobili2003
  16. ...and 6 more sections

Key Result

Proposition 1

Let $\mathbf k_1>\mathbf k_0>0.$ Assume that $\frac{\mathbf k_1+\mathbf k_0}{\mathbf k_1-\mathbf k_0}=\frac{p}{q}$ with $p>q$ integers and $\frac{p}{q}$ an irreducible fraction. Then, for every $n=1,3,5,\ldots$ and $\xi>0$, $E_n(\xi,t)$ (and as a consequence $E(\xi,t) =\sum_{n=1,3,5, \ldots}E_n(\xi,

Figures (11)

  • Figure 1: Two periods of a sawtooth signal with a frequency of 262Hz with its first 3 Fourier components (left) and with their sum (right).
  • Figure 2: The harmonic and sub-harmonic series corresponding to the C$_4$
  • Figure 3: The sound (top) and its spectrogram (the "sound image", bottom) .
  • Figure 4: The Cochlea.
  • Figure 6: The function ${\cal R}_1$ for an input that is a sinusoidal signal corresponding to the C$_4$ for the set of parameters of Section \ref{['s-parameters-original']} and of Section \ref{['s-parameters-modif+A03']}.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Proposition 1
  • Remark 9