  In this first of two papers, we present a new method for searching for
oscillatory features in the primordial power spectrum. A wide variety of models
predict these features in one of two different flavors: logarithmically spaced
oscillations and linearly spaced oscillations. The proposed method treats the
oscillations as perturbations on top of the scale-invariant power spectrum,
allowing us to vary all cosmological parameters. This perturbative approach
reduces the computational requirements for the search as the transfer functions
and their derivatives can be precomputed. We show that the most significant
degeneracy in the analysis is between the distance to last scattering and the
overall amplitude at low frequencies. For models with logarithmic oscillations,
this degeneracy leads to an uncertainty in the phase. For linear spaced
oscillations, it affects the frequency of the oscillations. In this first of
two papers, we test our code on simulated Planck-like data, and show we are
able to recover fiducial input oscillations with an amplitude of a few times
order 10^{-2}. We apply the code to WMAP9-year data and confirm the existence
of two intriguing resonant frequencies for log spaced oscillations. For linear
spaced oscillations we find a single resonance peak. We use numerical
simulations to assess the significance of these features and conclude that the
data do not provide compelling evidence for the existence of oscillatory
features in the primordial spectrum.
