Mathematics were also used in musical tuning and music theory. The 2nd-century-BCE Philosophers of Huainan, compiled by eight scholars under the patronage of King Liu An of Han, outlined the use of the twelve tones on a musical scale. Jing Fang, a mathematician and music theorist, expanded the scale further to create a scale of 60 tones. While doing so, he realized that 53 just fifths, or an interval of five staff positions, is approximate to 31 octaves, or the interval between a musical pitch and another with half or twice its frequency. By calculating the difference at 177147⁄176776, Jing reached the same value of 53 equal temperament discovered by the German mathematician Nicholas Mercator later in the 17th century.