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Harmonic seeding of free electron lasers has attracted significant attention as a method for producing transform-limited pulses in the soft x-ray region. Harmonic multiplication schemes extend seeding to shorter wavelengths, but also amplify the spectral phase errors of the initial seed laser, and may degrade the pulse quality and impede production of transform-limited pulses. In this paper we consider the effect of seed laser phase errors in high gain harmonic generation and echo-enabled harmonic generation. We use simulations to confirm analytical results for the case of linearly chirped seed lasers, and extend the results for arbitrary seed laser envelope and phase.