Absolute pitch (AP), a superior ability of pitch letter naming in the absence of a reference note, has long been viewed as an indicator of human musical talent and thus as evidence for the adaptationist hypothesis of music evolution. Little is known, however, whether AP possessors are superior to non-AP possessors in music processing. The present study investigated whether the AP ability facilitates musical tension processing in perceptual and experienced tasks. Twenty-one AP possessors and 21 matched non-AP possessors were tested using novel melodies in C and non-C contexts. Results indicated that the two groups provided comparable ratings of perceived and felt tension for melodies in both contexts. While AP possessors demonstrated lower accuracy with longer reaction time than non-AP possessors in naming movable solfège syllables for pitch in the pretest, their tension rating profiles showed a similar tonal hierarchy as non-AP possessors in regard to the stability of the ending tones of the melodies in both major and minor keys. Correlation analyses suggested that musical tension ratings were not significantly related to performance in pitch letter, movable solfège syllable naming, pitch change detection threshold, or pitch direction discrimination threshold for either group. These findings suggest that pitch naming abilities (either pitch letter or movable solfège syllable naming) do not benefit processing of perceived or felt musical tension, providing evidence to support the hypothesis that AP ability is not associated with advantage in music processing.