visualize

If you want the audience—and yourself—to feel the shape of Echelon’s inner life as you play, you have to turn an invisible vector (x_t^*\in\mathbb{R}^D) into a picture that moves with the same inevitability as the music. The challenge is not just to plot numbers; it’s to build a visual grammar where geometry equals meaning, where distance means “more different,” curvature means “changing intention,” thickness means “tension,” and motion means “you.” The trick is to choose a projection that preserves the two invariants your engine already enforces: contraction in the fast loop and phase alignment in the scheduler. When you respect those invariants visually, the image becomes a second instrument—honest, rhythmic, and interpretable at a glance. Start by deciding what the axes should mean. You do not want an arbitrary 2D scatter that jitters because the optimization changed its mind. You want a stable map that you can revisit next week and still recognize. There are two good ways to get that stability. One is to learn, offline, a parametric projector (f_\phi:\mathbb{R}^D\to\mathbb{R}^2) on your rehearsal data—a tiny two-layer network trained to preserve local neighborhoods the way UMAP does—then freeze its weights and run it in real time. The other is to compute a fixed linear basis with PCA on your rehearsals, take the top two or three components, and project live latents onto that plane. The first gives you more curvature and semantic separation; the second gives you maximum predictability with almost no compute. Either way, once you freeze (f_\phi) or the PCA matrix, the map stops drifting; a region that meant “hand-driven shimmer” last month still means it today. Now tie the map to rhythm. Phase is already a circular variable in Echelon, a number (\psi_t) between 0 and (2\pi) that tells you where inside the beat you are. If you draw your manifold as a flat scatter, phase is homeless. If you draw it on a ring, phase feels like home. A simple and powerful visual is to embed (\psi_t) as angle and latent energy (|x_t^*|) as radius: a polar plot where each frame lands on a circle and the point breathes in and out with effort. That picture alone—the wheel spinning, the comet tail fading behind the head—tells you whether your body is in time. If you want the full manifold and the phase wheel at once, you can lift the 2D projector into a cylinder: use (f_\phi(x)) for the floor plan and draw a thin halo around the current point whose hue rotates with (\psi_t). The halo locks the picture to the beat; the floor plan tells you what kind of motion you’re in. Contraction belongs in the picture as thickness. The fast loop’s residual (r_t=|x_{t}^{(k+1)}-x_{t}^{(k)}|) is a measure of how hard the solver is working. Draw the live point with an outer glow whose radi