Pythagorean tuning is a method of tuning the 12-note chromatic musical scale based on a series of perfect fifth intervals. The method is named for the 6th centrury BC mathematician and philosopher Pythagoras. Pythagoras and his students, notably Nicomachus, defined harmonics as being mathematical relationships between the vibrational frequencies of various notes. A perfect fifth is defined as a frequency ratio of 3:2, whereas an octave is 2:1. Consequently, a problem of the Pythagorean tuning method is that no series of perfect fifths can be made equal to an octave. This resulted in higher octave notes being out of tune with respect to their lower octave counterparts. This problem was corrected by Bach with his well-tempered clavier, known today as equal temperament, by slightly flattening each fifth to make up for the discrepancy - in effect, distributing the problem throughout the entire scale so that is it not noticeable.