We embed the scheme of Natural (Axionic) Inflation within a string theory
framework that satisfies the so-called "Weak Gravity Conjecture". There,
the pure cosine potential of natural inflation is replaced by modular
functions. This leads to "wiggles" in the inflationary potential that
modify the predictions with respect to CMB observations. This scheme of
"Modulated Natural Inflation" is compatible with all existing CMB data.
It predicts a lower limit of the scalar-to-tensor-ratio r>0.002 as well
as a running of the spectral scalar index. Future CMB experiments could
directly probe the modulations through their improved sensitivity.