 The remainder of the work is taken up with a discussion of the many parts of music according to the order which he had himself prescribed.While it is often held among modern scholars that Aristoxenus rejected the opinion of the Pythagoreans that arithmetic rules were the ultimate judge of intervals and that in every system there must be found a mathematical coincidence before such a system can be said to be harmonic, Aristoxenus made extensive use of arithmetic terminology, notably to define varieties of semitones and dieses in his descriptions of the various genera.In his second book he asserted that "by the hearing we judge of the magnitude of an interval, and by the understanding we consider its many powers."