Seasonal adjustment methods process and publish millions of time series across the world each month, and judgment of the adequacy relies heavily upon frequency domain diagnostics. In particular, peaks in the spectral density estimates of seasonally adjusted data are indicative of an inadequate adjustment. Spectral peaks are currently assessed in the X-12-ARIMA program via the visual significance approach, but this method lacks a rigorous statistical foundation. This paper provides such a foundation by providing measures of uncertainty for spectral peak measures, allowing for formal hypothesis testing. To apply the test we develop fixed-bandwidth fraction asymptotics for taper-based spectral density estimates.