The kernel computes a signed timing displacement for each musical event:
\[ T_{\text{final}} = Q_{\text{PPQN}}\!\left(T_{\text{grid}} + \beta \cdot \bigl[\Delta_L + \Gamma(\tau) \cdot \bigl(\Delta_C(n) + \Omega(v)\bigr) + \Psi(b)\bigr]\right) \]where each basis function is defined as:
\[ \Delta_C = D_{\max}(n/N)^\alpha \qquad \Omega = \rho(v - v_0) \qquad \Psi = A\sin(2\pi b / P) \]| Term | Name | Definition | Layer |
|---|---|---|---|
| \( T_{\text{grid}} \) | Grid Time | Quantized step position from BPM and step index | Transport |
| \( \beta \) | BPM Scalar | \( 90 / BPM \); normalizes to reference tempo | Transport |
| \( \Delta_L \) | Linear Offset | Static per-channel timing bias (ms) | Scheduler |
| \( \Delta_C(n) \) | Curvature | \( D_{\max} \cdot (n/N)^\alpha \) | Groove Field |
| \( \Omega(v) \) | Phase Coupling | \( \rho \cdot (v - v_0) \) | Physics |
| \( \Psi(b) \) | Macro Drift | \( A \cdot \sin(2\pi \cdot b/P) \) | Phrase |
| \( \Gamma(\tau) \) | Harmonic Gravity | Continuous on \( [0,1] \); elastic components only | Harmonic |
| \( Q_{\text{PPQN}} \) | Hardware Quantize | Rounds to nearest hardware pulse (final transform) | Hardware |
First, harmonic gravity \( \Gamma(\tau) \) multiplies only the elastic components \( (\Delta_C + \Omega) \), never the linear offset \( \Delta_L \) or macro drift \( \Psi \). This ensures that harmonic context modulates the feel of displacement without altering the structural timing skeleton. Without this constraint, changing musical key would cause snare drag to shift.
Second, hardware quantization \( Q_{\text{PPQN}} \) is applied as the final transform, after all displacement computation. This preserves the timing resolution of the target device.
Third, the kernel is a pure function: it receives a precomputed context and returns a displacement value. It performs no mutation, holds no state, and makes no scheduling decisions.
| Component | Domain | Bound |
|---|---|---|
| \( \beta \) | \( BPM \in [BPM_{\min}, BPM_{\max}] \) | \( 0 < \beta \leq 90/BPM_{\min} \) |
| \( \Delta_L \) | Per-channel constant | \( |\Delta_L| \leq L_{\max} \) |
| \( \Delta_C(n) \) | \( n/N \in [0,1] \) | \( 0 \leq (n/N)^\alpha \leq 1 \), so \( |\Delta_C| \leq D_{\max} \) |
| \( \Omega(v) \) | \( v \in [0,1] \) | \( |v - v_0| \leq 1 \), so \( |\Omega| \leq |\rho| \) |
| \( \Psi(b) \) | \( b \in \{0,...,P{-}1\} \) | \( |\sin(\cdot)| \leq 1 \), so \( |\Psi| \leq A \) |
| \( \Gamma(\tau) \) | \( \tau \in [0,1] \) (compact) | Continuous; \( |\Gamma| \leq \Gamma_{\max} \) by EVT |
Proof. Define the interior sum \( S = \Delta_L + \Gamma(\tau)(\Delta_C + \Omega) + \Psi \). By the triangle inequality:
\[ |S| \leq |\Delta_L| + |\Gamma(\tau)| \cdot |\Delta_C + \Omega| + |\Psi| \]Applying the individual bounds:
\[ |S| \leq L_{\max} + \Gamma_{\max} \cdot (D_{\max} + |\rho|) + A \;:=\; C \]where \( C \) is a finite constant determined by the profile coefficients. Since \( \Delta = \beta \cdot S \) and \( |\beta| \leq 90/BPM_{\min} \):
\[ |\Delta| \leq \frac{90}{BPM_{\min}} \cdot C = \Delta_{\max} \]The domain \( D = [0,1] \times [0,1] \times \{0,...,P{-}1\} \times [BPM_{\min}, BPM_{\max}] \) is a product of closed bounded intervals and a finite set, hence compact. Each basis function is continuous on its respective factor. The displacement field \( \Delta : D \rightarrow \mathbb{R} \) is therefore a continuous function on a compact domain. By the Extreme Value Theorem, \( \Delta \) attains its global maximum and minimum on \( D \).
Six canonical groove styles from 1990s popular music production, each produced by the identical kernel function \( \Delta(x;\theta) \) with different parameter values.
| Style | \(\Delta_L\) | \(\alpha\) | \(A\) | \(\rho\) | PPQN | Character |
|---|---|---|---|---|---|---|
| Boom Bap | 10 | 1.0 | 0 | 0 | 96 | Static snare drag |
| Neo-Soul | 0 | 1.25 | 4 | 0.2 | 96 | Elastic pocket |
| Timbaland | 0 | 1.0 | 8 | 0 | 96 | Phrase breathing |
| MPC60 | 6 | 1.0 | 0 | 0 | 96 | Hardware emulation |
| G-Funk | 8 | 1.0 | 3 | 0 | 96 | Sub-bass lag |
| New Jack Swing | 2 | 1.0 | 0 | 0.3 | 96 | Velocity interaction |
Interpolation between profiles is trivial: linearly blend the coefficient vectors.
The kernel is implemented as a pure function suitable for execution in a Web Audio worklet. The core function accepts an event descriptor and a precomputed context object, and returns a signed displacement value in milliseconds. For identical inputs, it produces bit-identical output under IEEE-754 deterministic evaluation.
Context assembly occurs outside the kernel boundary: the context assembler constructs the coefficient context from the active groove profile and the current musical state. The scheduler applies PPQN quantization as the final transform before dispatching to the audio clock.
Bounded: For all profiles and all BPM values in the production range (60–180), the absolute displacement does not exceed the profile-specific \( \Delta_{\max} \). This is verified by exhaustive sweep over discretized \( D \).
BPM-Scaled: Displacement at 70 BPM is exactly twice the displacement at 140 BPM, for identical coefficients. Since \( \beta = 90/BPM \), the ratio \( \beta_{70}/\beta_{140} = 2 \). Mathematically exact.
Deterministic: Ten consecutive evaluations with identical context produce bit-identical output under IEEE-754 compliant evaluation. No randomness is permitted inside the kernel boundary.
Resettable: Macro drift is periodic via bar index modulo \( P \). Phrase reset clears accumulated drift to zero.
The Unified Displacement Kernel differs from prior work in three respects. First, it is additive rather than template-based, enabling continuous interpolation between styles and formal analysis of the displacement space. Second, it is deterministic rather than stochastic, essential for reproducible audio rendering and automated testing. Third, it is formally bounded by construction; the boundedness proof is not an empirical observation but a mathematical consequence of the kernel architecture.
The displacement field \( \Delta(x;\theta) \) is Lipschitz continuous in each coefficient. Since each basis function is a polynomial, sinusoidal, or linear term, the partial derivatives \( \partial\Delta/\partial\theta_i \) are bounded on \( D \). This implies that small perturbations in the coefficient vector produce proportionally small changes in displacement, guaranteeing smooth interpolation between groove profiles.
To quantify how responsive the displacement field is to input variations, we define two orthogonal sensitivity indices evaluated at the current playground state:
A limitation of the current work is that the coefficient profiles have not been validated through perceptual listening tests. Future work should evaluate perceptual indistinguishability from original production styles.
v1.0 — Initial arXiv submission.
v1.1 — Added Lipschitz continuity discussion.
v1.2 — Interactive companion page with coefficient playground.
v1.3 — Radar identity, dual sensitivity indices, causality enforcement, JSON export.