When someone shows you who they sound like, believe them

This is ongoing work related to our consulting project with Arno Pianos. Most of the work details are available in the GitHub repository for our consulting class STAT409.

A frequent problem…

One of the interesting problems of the trade is to figure out the dimensions of an organ pipe. In particular, the length of the pipe is a key parameter that determines the fundamental frequency of the pipe (and subsequent harmonics).

Some fun facts:

• Open pipes have a “complete” harmonic series, i.e., all harmonics are present.
• Closed pipes have a “partial” harmonic series, i.e., only odd harmonics are present.
• Both cases behave as a harmonic oscillator, which is interesting: one assumes a section of the pipe —a cylinder of air— behaves as a harmonic oscillator, in a “spring-like” fashion.
• In the case of open pipes, the fundamental frequency $\omega_0$ is given by $\omega_0 = \frac{c}{2(\ell + \Delta\ell)},$ where $c$ is the speed of sound in air, and $\ell$ is the length of the pipe, and $\Delta\ell$ is a correction term that accounts for the end effects.
• In the case of closed pipes, the fundamental frequency $\omega_0$ is given by $\omega_0 = \frac{c}{4(\ell + \Delta\ell)},$ where $c$ is the speed of sound in air, and $\ell$ is the length of the pipe, and $\Delta\ell$ is a correction term that accounts for the end effects.

…has a frequent solution

The Ising Number is a dimensionless number that is used to characterize the intonation of the pipe. It corresponds to the following ratio:

$$\mathsf{I} = \frac{1}{\omega_0}\sqrt{\frac{2 P d}{\rho h^3}} = \frac{v}{\omega_0} \sqrt{\frac{d}{\rho h^3}}$$

where $\omega_0$ is the fundamental frequency of the pipe, $P$ is the pressure of the air in the pipe, $d$ is the diameter of the pipe, $\rho$ is the density of the air in the pipe, $h$ is the cut-up height of the pipe, $d$ is the jet initial thickness, and $v$ is the velocity of the air in the pipe (which is related to the dynamic pressure of the air in the pipe).

A simple correction idea

We have tried some different approaches with the data we have. We have built a simple neural network model to predict the Ising Number using a specific softmax-softplus layer. My colleagues are working on GAMs and other models (for modeling the partials and the Ising Number).

Nevertheless a simple idea we haven’t fully explored is a (physics-informed) correction formula for the Ising Number, based on the flow rate conservation equation. Basicaly, it pertains a continuity equation where $A_1 v_1 = A_2 v_2$, where $A_1$ and $A_2$ are the cross-sectional areas of the pipe at two different points, and $v_1$ and $v_2$ are the velocities of the air at those points.

As the air flows through a system and goes through the toe hole, a modified Ising Number can be defined as

$$\hat{\mathsf{I}} = \frac{A_{\mathrm{sys}}}{\omega_0 A_{\mathrm{toe}}}\sqrt{\frac{2 P d}{\rho h^3}}$$

where $A_{\mathrm{sys}}$ is the cross-sectional area of the system where the velocity is measured, and $A_{\mathrm{toe}}$ is the cross-sectional area of the toe hole.