Three studies were designed to determine the relation between subcortical neural temporal coding and the perception of musical consonance. Consonance describes the pleasing perception of resolution and stability that occurs when musical notes with simple frequency ratios are combined. Recent work suggests that consonance is likely to be driven by the perception of ‘harmonicity’, i.e. the extent to which the frequency components of the combined spectrum of two or more notes share a common fundamental frequency and therefore resemble a single complex tone (McDermott et al, 2010, Curr Biol). The publication in Chapter 3 is a paper describing a method for measuring the harmonicity of neural phase locking represented by the frequency-following response (FFR). The FFR is a scalp-recorded auditory evoked potential, generated by neural phase locking and named from the characteristic peaks in the waveform with periods corresponding to the frequencies present in the fine structure and envelope of the stimulus. The studies in Chapters 4 and 5 demonstrate that this method predicts individual differences in the perception of consonance in young normal-hearing listeners, both with and without musical experience. The results of the study in Chapter 4 also demonstrate that phase locking to distortion products resulting from monaural cochlear interactions which enhance the harmonicity of the FFR may also increase the perceived pleasantness of consonant combinations of notes. The results of the study in Chapter 5 suggest that the FFR to two-note chords consisting of frequencies below 2500 Hz is likely to be generated in part by a basal region of the cochlea tuned to above this frequency range. The results of this study also demonstrate that the effects of high-frequency masking noise can be accounted for by a model of a saturating inner hair-cell receptor potential. Finally, the study in Chapter 6 demonstrates that age is related to a decline in the distinction between the representation of the harmonicity of consonant and dissonant dyads in the FFR, concurrent with a decline in the perceptual distinction between the pleasantness of consonant and dissonant dyads. Overall the results of the studies in this thesis provide evidence that consonance perception can be explained in part by subcortical neural temporal coding, and that age-related declines in temporal coding may underlie a decline in the perception of consonance.

Ce mémoire traite de l’harmonicitié d’instruments de musique à travers la géométrie spectrale. Nous y présentons, en premier lieu, les résultats connus concernant la corde de guitare, le tambour circulaire et puis le tabla ; le premier est harmonique, le deuxième ne l’est pas et puis le dernier s’en approche. Le cas de la timbale est ce qui constitue la majeure partie de notre travail. L’ingénieur-physicien Robert E. Davis en avait déjà étudié la quasi-harmonicité et nous faisons ici une relecture mathématique de sa démarche. En alliant les méthodes analytiques et numériques, nous montrons que la caisse de résonance de la timbale permet à la fois d’ajuster les fréquences de vibration de la forme ω_(i1) , avec 1 ≤ i ≤ 5, afin qu’elles s’approchent du rapport idéal 2 : 3 : 4 : 5 : 6, et elle permet aussi d’étouffer certains autres modes dissonants. Pour ce faire, nous élaborons un modèle simplifié de timbale cylindrique basé sur la physique et sur ce que propose Davis dans sa thèse. Ce modèle nous fournit un système d’équations divisé en trois parties : la vibration de la peau et la pression à l’intérieur et à l’extérieur de la timbale. Nous utilisons la méthode des fonctions de Green pour trouver les expressions des deux pressions. Nous nous servons de celles-ci ainsi que d’un développement en série de Fourier-Bessel modifiée pour résoudre les équations de la vibration de la peau. La résolution de ces équations se ramène finalement à celle d’un système matriciel infini dont nous faisons l’analyse numériquement. À l’aide de Mathématica et de ce système matriciel, nous trouvons les fréquences de vibration de la timbale, ce qui nous permet d’analyser l’harmonicité de l’instrument. Grâce à une mesure de dissonance, nous optimisons l’harmonicité de la timbale en fonction du rayon du cylindre, de sa hauteur et de la tension. / This thesis deals with the harmonicity of musical instruments through spectral geometry. First, we present the known results concerning the guitar string, the circular drum and the tabla ; the first is harmonic, the second is not, and the last is somewhere in between. The case of the timpani constitutes the major part of our work. The physicist-engineer Robert E. Davis had already studied its quasi-harmonicity and here we undergo a mathematical proofreading of his approach. By combining analytical and numerical methods, we show that the sound box of the timpani allows an adjustement of the vibration frequencies of the form ω_(i1) , with 1 ≤ i ≤ 5, so that they get close to the ideal 2 : 3 : 4 : 5 : 6 ratio, while it also stifles some other dissonant modes. To do so, we develop a simplified model of a cylindrical timpani based on physics and on what Davis suggests in his thesis. This model provides a system of equations divided into three parts : the vibration of the skin and the pressure inside and outside the timpani. We use the method of Green’s functions to find the expressions of the pressures. We use these together with a modified Fourier-Bessel series development to solve the equations of the vibration of the skin. In the end, the solving of these equations is reduced to an infinite matrix system that we analyze numerically. Using Mathematica and this matrix system, we find the vibrational frequencies of the timpani, which allows us to analyze the harmonicity of the instrument. Thanks to a measure of dissonance, we optimize the harmonicity of different timpani models with different cylinder radii, heights and tensions.