Just intonation was designed to exploit the resonances inherent in the overtone series. Its strength lies in the fact
that many of its intervals have low-numbered ratios such as 2/1, 3/2, 4/3, and 5/4, so the higher partials for notes sounding together
in these frequency ratios will match up, creating harmony that sounds more pleasing to our ears. The major chords built on the
first, fourth, and fifth scale degrees are each in the nice ratio 4:5:6. Problems with just intonation include that there
are two different whole steps (9:8 and 10:9; off by the syntonic comma 81:80), two half steps do not equal either whole step, the circle of fifths does not
close up, B♯ ≠ C (enharmonic equivalence fails), it is hard to change keys, and melodic drift.
Equal temperament was created to solve many of the problems arising with just intonation. By making a consistent half step throughout
the scale and lower/higher octaves, it became feasible to change keys and play more chromatic melodies and harmonies. Now, two half steps
equals one whole step and the circle of fifths closes up properly (enharmonic equivalence is restored). Weaknesses include that the
just 3:2 perfect fifth is lost and that some intervals (e.g., the major third and sixth) are far from their just values, sometimes leading
to difficulties with tuning and harmonizing. Other than the octave (2:1) or unison (1:1), all frequency ratios in equal temperament use irrational numbers.