Deep learning models tend to perform best at large widths and depths. The growing interest in scaling laws, especially in the language modeling domain, has motivated procedures for scaling up network widths and depths in order to preserve stability and computational efficiency of training dynamics. Existing depth-scaling approaches force a choice between retaining locally maximal \emph{feature learning} and preserving a form of \emph{feature diversity} in the residual stream arising from random differences between channels. We examine a \textit{joint scaling recipe} that preserves both. With residual width $N_\flat$, hidden width $N_\sharp$, and depth $L$, the key quantity is, by default, the limiting ratio $N_\flat/(N_\sharp L)\toα$, while correlated initialization (Half Copy) can partially relax the resulting shape constraints. Theory and residual-MLP experiments support convergence at fixed positive $α$ to an SDE limit with locally maximal feature learning. Hidden computations can then be redistributed between width and depth while preserving this limit, like turning melody into harmony and back. Language-model experiments nevertheless show performance differences under regrouping at fixed parameter count. These results distinguish approximating a limit (ODE or SDE) from scaling for better performance.

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