Joe Blochberger, PE

Digital waveguide synthesis and the Karplus–Strong algorithm assume ideal boundary conditions and uniform exponential decay across every harmonic. A real electric guitar places a permanent magnet within millimeters of a ferromagnetic string, so the string carries a position-dependent force on top of the tension-based restoring force.

An unwound G string was recorded through a direct-inject path at clearances from 5.00 mm down to 1.50 mm. Onset detection by amplitude envelope time-aligns the takes, short-time Fourier analysis resolves decay per harmonic, and a calibrated Karplus–Strong reference isolates the magnetic contribution by subtraction. Bootstrap resampling over 2000 iterations supplies nonparametric 95 percent confidence intervals.

Damping proves strongly harmonic-selective. The fundamental changes by a factor of 1.3 to 1.6 across the sweep, while the fourth harmonic changes by a factor of 3.8, and the equivalent T60 of the fourth harmonic falls from 5.0 s to 1.3 s, an audible loss of high-harmonic sustain. Magnetic and mechanical contributions become equal for the fourth harmonic near 3.0 to 3.5 mm, and the magnetic term reaches 2.8 times the mechanical term at 2.5 mm. One-way analysis of variance resolves significant clearance effects for three of the four harmonics. The fitted power-law exponent clusters near 1.0 and rises to 1.48 at 2.5 mm, well above the frequency-independent eddy-current prediction and below the jerk-dominated limit of 2. A localized reversal at 2.0 mm is retained in every table and excluded from trend fitting.

Principal component analysis, independent component analysis, singular spectrum analysis, and time-delay dynamic mode decomposition are applied to a two-source instantaneous mixture built from a NOAA Ship Ronald H. Brown recording and a fin whale recording. Kurtosis separates the two sources by two orders of magnitude, and the leading principal component explains 80 to 82 percent of the variance in every analysis domain, so variance concentration is close to domain-invariant for a rank-2 mixture.

Under matched instantaneous mixing, independent component analysis leads every metric with a source-averaged scale-invariant signal-to-distortion ratio of 38.25 dB over 30 random mixing matrices, followed by time-delay dynamic mode decomposition at 24.95 dB. Principal component analysis saturates near 1.6 dB and singular spectrum analysis stays negative.

A robustness study of 1680 runs across seven signal-to-noise levels and both mixing models changes the picture. Under convolutive mixing the independent-component ceiling falls from 38.2 dB to 7.1 dB even without noise, and the time-delay ceiling falls to 1.5 dB because a minimal delay embedding cannot span echoes of 5 to 50 ms. The high instantaneous figures are therefore matched-model ceilings rather than expected field performance. Embedding depth trades separation accuracy against modal interpretability: a depth of 2 separates best, while a depth of 12 exposes a complex-conjugate eigenvalue pair near the unit circle at 19.0 Hz.

The segregation model of Schelling links micro-level preference to macro-level structure, yet the spatiotemporal record has been summarized mainly through scalar indices measured near the absorbing state. A hybrid pipeline couples sparsity-promoting dynamic mode decomposition, which supplies an energy ranking through L1-regularized amplitudes solved by the alternating-direction method of multipliers, with multi-resolution dynamic mode decomposition, which separates rapid initial relaxation from slow patch coarsening.

Two complementary order parameters emerge. The modal-energy fraction decreases monotonically with tolerance, while the neighbor-based spatial correlation is non-monotonic, deepening in segregation just below the frustration threshold and collapsing beyond it. Divergence between the two resolves three regimes that no single index can distinguish: convergent segregation, a critical band of slow but strong segregation, and frustrated mixing. Both parameters place the transition at the discrete Moore-neighborhood tolerance of 0.75 to 0.76.

Boundary geometry decides what survives the transition. Regular periodic compartmentalization retains modal structure deep in the frustrated regime, because each enclosed compartment sustains a locally segregated pocket. The open torus, a digitized 1941 Baltimore slum-clearance boundary, and scattered random obstacles all collapse to a structureless field. One limitation is stated plainly in the manuscript: the modal hierarchy characterizes the transition without anticipating it.

Decomposition methods matured in fluid mechanics and structural dynamics, where the state is a field sampled on a grid. An agent-based model produces the same kind of high-dimensional spatiotemporal record, so the tooling transfers whenever the simulation layer is reproducible and the state can be written as a snapshot matrix.

None of the four methods is new. Continuous wavelet analysis of an order parameter, radially averaged spatial power spectra, normalized cross-correlation, and phase portraits are all standard, and each carries an established history on systems of exactly this class. The contribution is treatment rather than method: decompositions are time-resolved and shown against the generating simulation instead of reported at a single instant, one pipeline spans lattice, continuous-space, network, and profile models, and spectra are computed on indicator fields rather than on categorical state codes.

The analysis frame is selected by maximum lag-1 spatial autocorrelation rather than defaulting to the final state, which is frequently degenerate: a burnt-out forest or a fully diffused field carries no spatial structure left to transform.

Ten NSynth instrument families supply the label set. Mel spectrograms computed with a 2048-sample Hann window, 50 percent overlap, a 4096-point transform, and 512 bands serve as image input to a two-dimensional convolutional network, while a wavelet scattering ensemble works on the same audio through a complementary representation. The two classifiers are combined at the decision stage rather than at the feature stage.

Fusion outperforms every single-model approach attempted on the project. Accuracy reaches 95 percent against 90 percent for the Python two-dimensional network, 88 percent for the one-dimensional network, and 91 percent for principal component analysis with a support vector machine. The gap widens on the metrics that matter for an imbalanced label set: an F1 score of 75 percent against 58, 40, and 57 percent, and a false-positive rate of 3 percent.

Ten-fold cross-validation returns 95 percent accuracy in every fold, with precision between 73 and 75 percent, so the result is stable rather than a fortunate split. Audio recorded by team members outside the dataset classifies far less well, which reflects the controlled and idealized character of NSynth rather than a defect in the fusion rule.