This paper introduces a new family of constraints in OT, linear faithfulness constraints, through an analysis of the Xiamen Tone Circle. The first highlight of the analysis is its exhaustivity. The Xiamen Tone Circle is a phenomenon in which the tone borne by a word in its citation form goes to a different tone (called sandhi tone) when the word is realized in a non prominent prosodic context, the mapping from citation tones to sandhi tones forming a circle. Previous research on this circular chain shift (Barrie: 2006) have made clear that a great number of candidates (3,125 under the analytical assumptions of the author) have to be evaluated for an analysis of the circle to be considered complete, but no analysis had addressed this issue so far. In this paper, a successful OT analysis of the circle is presented together with a computational procedure that demonstrates its exhaustivity. The second higlight of the analysis is the introduction of linear faithfulness constraints. Linear faithfulness constraints penalize an output that differs from its input for some feature, only if the value of this feature in the output is higher than the value of this feature in the input on a given scale of feature values. It is argued that although linear faithfulness constraints combine aspects of faithfulness and markedness constraints, their use is more restricted and thus more desirable than markedness and faithfulness constraint conjunction.