The fine structure of the temperature power spectrum of the cosmic microwave background (CMB) radiation is investigated in the presently accessible multipole range up to l ∼ 104. The temperature fluctuations are reproduced by an isotropic Gaussian random field on the unit sphere, whose Green function is defined by a Hermitian matrix kernel inferred from the data sets by way of spectral fits. The reconstruction of the temperature autocorrelation function from the measured multipole moments Cl is a classical inverse problem, which does not require specification of cosmic evolution equations for the photon density. The scale-invariant correlation function admits a multipole expansion in zonal spherical harmonics. The multipole coefficients are obtained as averages over Hermitian spectral matrices determining the angular power spectrum of the spherical random field. The low-l multipole regime of the CMB temperature fluctuations is composed of overlapping Gaussian peaks, followed by an intermediate oscillatory regime manifested by a modulated exponentially decaying Cl slope. The high-l regime above l ∼ 4000 comprises a power-law ascent with exponential cut-off. The fine structure of the Gaussian, oscillatory and high-l regimes is reproduced by zooming into the respective l intervals on linear and logarithmic scales.