  Spins and oscillators are foundational to much of physics and applied
sciences. For quantum information, a spin 1/2 exemplifies the most basic unit,
a qubit. High angular momentum spins and harmonic oscillators provide
multi-level manifolds (e.g., qudits) which have the potential for
hardware-efficient protected encodings of quantum information and simulation of
many-body quantum systems. In this work, we demonstrate a new quantum control
protocol that conceptually merges these disparate hardware platforms. Namely,
we show how to modify a harmonic oscillator on-demand to implement a continuous
range of generators associated to resonant driving of a harmonic qudit, and
then specifically design a harmonic multi-level spin degree of freedom. The
synthetic spin is verified by demonstration of spin coherent (SU(2)) rotations
and comparison to other manifolds like simply-truncated oscillators. Our scheme
allows universal control of the qudit, and, for the first time, we use linear,
harmonic operations to accomplish four logical gates on a harmonic qudit
encoding. Our results show how motion on a closed Hilbert space can be useful
for quantum information processing and opens the door to superconducting
circuit simulations of higher angular momentum quantum magnetism.
