Thermoacoustic instabilities originate from the interaction of unsteady heat release rate associated with flames and the acoustic modes of a combustor. A feedback loop not involving the natural acoustic modes has been observed in single-flame configurations with anechoic terminations: an acoustic wave emitted by the flame travels upstream, and the associated velocity fluctuation again excites the flame. This feedback cycle gives rise to thermoacoustic modes intrinsic to the flame. An analytical model for an annular thermoacoustic system is formulated, and the existence of intrinsic modes of various azimuthal orders is demonstrated. The spectrum of an annular combustor is computed with a three-dimensional thermoacoustic Helmholtz solver. The configuration resembles those commonly found in gas turbines. In addition to the observations in previous studies, numerous intrinsic modes are found, with frequencies close to the lowest acoustic modes. All of the intrinsic modes can be grouped into clusters, at frequencies corresponding to multiples of the inverse flame response time delay. It is demonstrated that the newly observed intrinsic modes belong to the same mechanism that has recently been studied in single-sector/flame configurations. An analysis of the evanescent character of cut-off azimuthal modes explains the pattern in the spectrum. The underlying physical mechanism is generically present in any annular combustion chamber and a possible source of instability. (C) 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.