The fires were started in parked cars, the Los Angeles Times reported. In some cases, the fires spread to nearby buildings.
At the house on Love Street in West Hollywood where Morrison once lived a fire that started in a parked Mazda Miata spread to the front of the building, officials said. The blaze was extinguished in what officials called a "major save" because of the 90-year-old house's age and its location on an overgrown hillside.
Morrison, lead singer with the Doors, lived in the house with his partner, Pamela Courson, before moving to Paris. His song, "Love Street," was inspired by the house.
Investigators said the first fires were reported just after midnight in West Hollywood. Calls continued to come in through the night.
On Thursday, three fires were started before dawn in a five-block stretch of Sunset Boulevard. A suspect, Samuel Arrington, 22, of Los Angeles was arrested.
Investigators are unsure if the new string of fires were set by copycats or if they involve the same arsonist as those on Thursday.
2014: The Year in Music [PHOTOS]