Lately I am thinking about music from a mathematical point of view. Let us consider $\mathbb{H} =\mathcal{L}^2([0,T])$ as the Hilbert space of functions that represents the time-intensity plot of a sound. Quite tautologically, playing together sounds amount to sum them. So, up to scale volumes, the chords one can produce with $n$ sounds $f_1, \ldots, f_n \in \mathbb{H}$ lives in the convex envelope of $f_1, \ldots, f_n$.
Now let us restrict to simple wave sounds $f_{\lambda}(t) := e^{i \lambda t} \in \mathcal{L}^2([0,T])$. If I consider $n$ different frequencies $\lambda_1, \ldots, \lambda_n$, the functions $f_{\lambda_i}$ will be linearly independent (and orthogonal actually), so that their convex envelope is just the simplex $\Delta^{n-1}$. This approach unfortunately does not shed light on the underlying geometry.
My second approach is more differential. Let us consider $f: (0,\Lambda) \to \mathbb{H}$ defined as $f \mapsto f_{\lambda}(\bullet)$ for some big $\Lambda$. This is a curve in $\mathbb{H}$, and I'd like to understand if there is some way to compute its "curvature" even though we have infinite dimensions. Trying to remember differential geometry from University and reading off Wikipedia, there should be infinitely many "curvatures" associated to a "Frenet orthonormal system", but I have not been able to generalize the latter.
On the other hand, the simple definition for a curvature in two dimensions generalize easily. We can compute the norm of the "second-derivative" along $\lambda$, obtaining $$ || \partial_{\lambda}^2 f_{\lambda} || = \frac{1}{T} \int_0^{T} ((it)^2 e^{i \lambda t}) (-it)^2 e^{-i \lambda t } dt = \frac{1}{T} \frac{T^5}{5} = \frac{T^4}{5} $$ Quite nicely, it does not depend on the frequency (but unfortunately it does depend on the length of the interval that is arbitrary). However, I am not really sure about how I should interpret a measure of curvature in $(\textrm{seconds})^4$.
Does any of the latter make sense? Is there a way to compute the curvature in infinite dimensions? One could then look for a curve in the space that has similar features and get a grasp of the geometry of $f_{\lambda}$!
As a last intuition, note that at each fixed time $t$ the function $f_{\bullet}(t)$ draws a circle at a speed proportional to $t$. It does make sense then that the curvature does not depend on the frequency since the circle has constant curvature. The method based on the inner product integrates all the contributes at fixed time.