Common Origin

The engine. State is a density matrix over 32 positions on a cycle crossed with a two state coin, so 64 by 64 complex. Each step applies U = S·C, where C is the coin [[cosθ, sinθe^{iφ}],[sinθe^{-iφ}, −cosθ]] and S shifts position conditioned on the coin. A dephasing channel then damps every off diagonal element of ρ by (1 − p). Trace holds at 1.000000000000 across hundreds of steps. At p = 1 the output matches the analytic classical random walk to 1e−16, using the same code path. That is the control group, and it is not an approximation of one.

Anchored coupling, and why free running fails. The obvious build is to let both voices run loose on a shared random stream and drift apart. Measured, it does not work. They decorrelate within two notes and the interval between them goes flat: 12, 15, 12, 13, 11, 9, 13, 11 percent across every spacing from unison to a seventh. A flat distribution is random harmony, which is mud rather than comparison. Periodic resyncing does not rescue it either, since the divergence arc lasts about one note. Anchored mode restarts both engines from the same position every note, so what you hear is the difference between two processes stepping from an identical origin, measured fresh each time. Bounded, repeatable, and the quantity actually worth comparing.

Ballistic against diffusive. From a common origin the quantum voice leaps in proportion to the number of steps while the classical one manages only its square root. Measured spread was σ = 5.5, 10.9, 16.6, 22.4, 28.6, 35.4 at t = 10 to 60, against 3.16, 4.47, 5.48, 6.32, 7.07, 7.75. That second column is √t to three digits. Musically it is a melody with a shadow that can never keep up.

Steps per note is a harmony control, and it is periodic. Because the cycle is only 32 notes wide, the ballistic walker runs out of room, wraps, and comes back. Separation peaks at 26 steps with a ratio near 3×, falls to a revival around 48 to 52 where the quantum voice is actually tighter than the classical one, then spreads again toward 70, contracts near 95, and continues with a quasi period around 45. The spread map under the slider is that curve, computed exactly rather than sampled, since in anchored mode the distribution is identical every note. The amber bands mark the inversions, where turning up the quantum knob makes the harmony narrower.

The phase knob. Moving φ from 0 to π/2 with everything else fixed, same seed, changes 32% of the notes while leaving every summary statistic identical. It rerolls the melody without changing its character. No classical parameter does that.

Reported honestly. At one step per note the two voices are indistinguishable, because collapse destroys the interference before it can form. If you never collapse, they are also indistinguishable, because sampling repeatedly from a near stationary distribution is noise either way. The effect lives in the middle band only. Pitch autocorrelation is near zero at every lag for both, so no motifs emerge on their own: this generates texture, not structure. The fact that only even intervals occur is parity, not quantum mechanics. And in anchored mode the classical voice is a shadow of the quantum one rather than an independent part, which is what makes the comparison legible but does make this a demonstration instrument more than a two part composition. Free mode is there if you want to hear the unstructured version.

Try this. Steps at 26, decoherence at 0, then drag decoherence slowly to 1.00. The stereo image closes to a point in the middle of your head. Then set decoherence back to 0 and sweep steps from 26 to 50 and listen to the harmony narrow as the walker wraps the cycle.