The Legend of Zelda: Ocarina of Time introduced Epona the horse to the series, and they were created for a specific purpose. Epona is a recurring Zelda character that appears in multiple games, serving as Link's steed, yet she was an optional character in her debut appearance. Ocarina of Time represented a huge jump forward in technology from the previous entries in the series and Epona was created to fulfill a specific role in the game.
Ocarina of Time introduced big open environments that took a long time to cross. The most notorious of these was Hyrule Field, which could be a pain to sprint across on foot, especially at nighttime, when Link could be attacked by Stalchildren that crawl their way out of the ground. If the player makes their way to the Lon Lon Ranch and completes the Epona quest (which involves activities in both the past and present), then they can use Epona's Song to summon their horse and ride her across Hyrule. Epona was a returning Zelda: Ocarina of Time character in Majora's Mask, and they have made sporadic appearances since.
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The 8-bit and 16-bit entries in The Legend of Zelda franchise didn't require a mount, as the hardware of the time could only handle small-scale areas. Ocarina of Time had a much grander scope than A Link to the Past and Link's Awakening, which is why Link needed a new mode of transportation. Ocarina of Time producer Shigeru Miyamoto discussed the reasoning behind Epona's inclusion in an interview printed in The Legend of Zelda: Ocarina of Time Encyclopedia. The interview was translated by Shmuplations and the full piece can be read on their website, along with other rare interviews given by Japanese
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