  We describe a class of observational strategies for probing the anisotropies
in the cosmic microwave background (CMB) where the instrument scans on rings
which can be combined into an n-torus, the {\em ring torus}. This class has the
remarkable property that it allows exact maximum likelihood power spectrum
estimation in of order $N^2$ operations (if the size of the data set is $N$)
under circumstances which would previously have made this analysis intractable:
correlated receiver noise, arbitrary asymmetric beam shapes and far side lobes,
non-uniform distribution of integration time on the sky and partial sky
coverage. This ease of computation gives us an important theoretical tool for
understanding the impact of instrumental effects on CMB observables and hence
for the design and analysis of the CMB observations of the future. There are
members of this class which closely approximate the MAP and Planck satellite
missions. We present a numerical example where we apply our ring torus methods
to a simulated data set from a CMB mission covering a 20 degree patch on the
sky to compute the maximum likelihood estimate of the power spectrum $C_\ell$
with unprecedented efficiency.
