We consider the n(n-1)/2 uniquely defined positive intervals among the first n=106 prime numbers as a probe of the global nature of the sequence of primes. A statistically strong periodicity is identified in the counting function giving the total number of intervals of a certain size. The nature of the periodic signature implies that the sequences of intervals spanning fixed numbers of gaps repeat quasi-cyclically. From the distribution of intervals we extract also the characteristic period of the repetition, which increases with n in a step-wise manner between consecutive primorial numbers and coincides with the most commonly occurring interval. The relationship between the most common interval and the primorial numbers is noteworthy independently of the periodic behaviors.