In Ethernet-based time-triggered networks, like TTEthernet, a global communication scheme, for which the schedule synthesis is known to be an NP-complete problem, establishes contention-free windows for the exchange of messages with guaranteed low latency and minimal jitter. However, in order to achieve end-to-end determinism at the application level, software tasks running on the end-system nodes need to obey a similar execution scheme with tight dependencies towards the network domain. In this paper we address the simultaneous co-synthesis of network as well as application schedules for preemptive time-triggered tasks communicating in a switched multi-speed time-triggered network. We use Satisfiability Modulo Theories (SMT) to formulate the scheduling constraints and solve the resulting problem using a state-of-the-art SMT solver. Furthermore, we introduce a novel incremental scheduling approach, based on the demand bound test for asynchronous constrained-deadline periodic tasks, which significantly improves scalability for the average case without sacrificing schedulability. We demonstrate the performance of our approach using synthetic network topologies and system configurations. * The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n o 610640 (DREAMS).