The most popular poetic form in the Heian period was the 31-syllable waka. Its phrasing consisted of 5 lines in the sequence of 5 syllables, then 7, then 5, (the body) followed by a cap of two lines of 7 each. The body of the waka is what eventually became the haiku.
An educated person's sensibility was on display every time he or she composed a waka. Not only was the content and sensitive use of images appreciated, but the handwriting was carefully scrutinized as well. Murasaki's hero, Prince Genji, was of course a paragon in this regard, and every lady in Murasaki's tale is judged on her skill in writing waka.
(Waka are also known as tanka.)