Why is horn in woodwind quintet?

You may ask, since the French Horn is a brass instrument, why is it a member of the Wind / Woodwind Quintet?

There's no definitive answer to that question, but the likely reason would be due to the Horn's tonal quality and it closeness to that of the woodwind's, particularly the Bassoon. And the Horn blends well with Woodwind instruments.

The Horn can also unleash its 'brassy' quality to enhance the dynamics and vary the colours of the Woodwind quintet.

On a whole, the additional of Horn really enhances the middle to lower voice of the quintet, provides greater colour and dynamics, as well as provides a full and richer harmony to the Wind quintet.

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https://www.youtube.com/watch?v=uj79mBUPzfM

For further reading, check out these resources,

http://woodwindchambermusicrepertoire.weebly.com/a-woodwind-blog/why-is-the-french-horn-in-the-woodwind-quintet

http://theidiomaticorchestra.net/12-timbre-sound-combinations-and-imitation/