Six silent defects in a generative music system

The doubling map, x → 2x mod 1, is the canonical chaotic map — and in binary it is a left bit-shift. Each iteration discards one bit of the seed and reveals the next. A float64 carries 52 mantissa bits. After ~52 iterations the orbit reached exactly zero and stayed there.

The specific case was worse than the general one. The seed encoding placed 440 Hz at exactly 0.5, which is one bit-shift from zero. Every A-natural in the range — including the application's own default reference pitch — collapsed to a drone within one to three notes.

Sustained chaotic wandering for the length of a phrase.

A transform labelled “chaos” that emitted the same note repeatedly.

The function returned a valid orbit of the correct length and type. Its contract was honored. Only the content was degenerate, and no test asserted on content.

The tent map's height parameter defaulted to 2.0 — and the full tent map at height 2 is exactly conjugate to the doubling map. Same bit-shift, same mantissa exhaustion, frozen at step ~52. At height 1.9 it never froze. At 1.99 it never froze. Only the shipped default failed.

Behaved as the chaotic map it is at essentially any other height.

Nothing, unless they moved the slider — at which point the transform started working, with no explanation of why.

Testing a parameterized function at an arbitrary parameter value passes. The defect lives at a single point in parameter space, and that point happened to be the default.

The equations and the fourth-order Runge–Kutta integrator were correct. The defaults ran the system for 0.64 Lorenz time units. The attractor's e-folding time is ≈1.10 time units — so a default run was shorter than a single e-folding time.

Measured on the shipped code: two seeds one cent apart ended the run closer together than they started. Maximum divergence across all sixteen output tones was 0.25 Hz, well below audibility. Zero wing switches — the orbit stayed on one lobe for the entire run. The parameter controlling this was capped, and the cap was below the useful range. At maximum, the transform reached 3.2 time units; meaningful behavior needs roughly 16.

Audible butterfly-wing switching and sensitive dependence.

A single smooth glide, indistinguishable from an ordinary envelope.

The system was implemented correctly and integrated correctly. The defect was not in the code but in the window onto it — roughly 25× too short. No test can detect that without knowing the physical timescale of the phenomenon being modeled.

The pitch-to-plane encoding placed a note at a radius proportional to its octave, unbounded. The quadratic escape radius is 2.0. A sufficiently high note landed outside it, the iteration returned an empty orbit, and the note contributed nothing. At the default exploration setting, notes three octaves above the anchor vanished. At the setting a curious user would naturally reach for — “show me more of the plane” — everything above the anchor octave disappeared.

Mapped every note in the piece's range to a point inside the window.

Missing high notes, worse the further they explored, with nothing in the interface explaining it.

An empty orbit is a legal return value. The failure path and the success path were the same code path.

The Weierstrass function is nowhere-differentiable when ab ≥ 1. The four presets used ab = 0.60, 1.50, 2.40, and 4.80. The first is an ordinary smooth differentiable function — a pleasant gentle wobble, but not a fractal, offered from a menu of fractal geometry.

Delivered the property named on the label.

A working transform producing a musically fine result, mislabeled as something it mathematically was not.

Nothing in a test suite computes the box dimension of a preset. The check requires knowing the theorem that defines the category.

Each successive note was divided by Feigenbaum's δ = 4.669. On the default dimension, an eight-note phrase of half-second notes ran 0.500, 0.107, 0.023, 0.005, 0.001 seconds and onward — five of eight notes under 10 ms, inaudible clicks. The transform was unusable past three or four notes.

Beneath the practical problem sat a conceptual one. δ is a ratio in parameter space: it describes how the gaps between successive bifurcation points shrink as a control parameter increases. It is not a ratio of states, durations, or frequencies. Used as a geometric scaling factor on note lengths, the constant was decorative — 4.5 would have done identical work.

Either not invoked Feigenbaum, or used δ for what δ governs.

A transform that collapsed into silence after four notes.

A test asserts that durations shrink by the expected factor. They did. Correct arithmetic on a quantity the constant does not describe still passes.