  The thermodynamic free energy F(\beta) is calculated for a gas consisting of
the transverse oscillations of a piecewise uniform bosonic string. The string
consists of 2N parts of equal length, of alternating type I and type II
material, and is relativistic in the sense that the velocity of sound
everywhere equals the velocity of light. The present paper is a continuation of
two earlier papers, one dealing with the Casimir energy of a 2N--piece string
[I. Brevik and R. Sollie (1997)], and another dealing with the thermodynamic
properties of a string divided into two (unequal) parts [I. Brevik, A. A.
Bytsenko and H. B. Nielsen (1998)]. Making use of the Meinardus theorem we
calculate the asymptotics of the level state density, and show that the
critical temperatures in the individual parts are equal, for arbitrary
spacetime dimension D. If D=26, we find \beta= (2/N)\sqrt{2\pi /T_{II}}, T_{II}
being the tension in part II. Thermodynamic interactions of parts related to
high genus g is also considered.
