We present PitchFlower, a flow-based neural audio codec with explicit pitch controllability. Our approach promotes pitch disentanglement through a simple perturbation: during training, F0 contours are flattened and randomly shifted at the input, while the true F0 is provided as conditioning to regenerate the original audio. A vector-quantization bottleneck prevents pitch recovery, and a flow-based decoder generates high quality audio. Experiments show that PitchFlower achieves accurate pitch control at the level of DSP baselines but at much higher audio quality, and performs on par with state-of-the-art neural approaches. Notably, despite using WORLD-transformed audio for training, our method filters out the vocoder's inherent artifacts, revealing a strong resilience of deep generative modeling to input degradation. This finding suggests that our framework provides a simple and extensible path that could be extended to other speech attributes.