Equal temperament is one of the most common temperament in modern tuning systems. It uses a series of mathematical formulas to divide an octave into twelve equal intervals, enabling it to omit obvious dissonances when transposing.
Modern keyboard instruments (such as piano, organ) are generally used the equal temperament.
Just intonation is a tuning system that is based on the proportions of natural numbers. It is characterized by large frequency differences between notes.
Just intonation is different from the twelve-tone equal temperament which frequently used in the modern keyboard instruments (e.g.: piano). Just intonation allows for larger frequency differences between notes while twelve-tone equal temperament limits the frequency ratios between notes. These differences can be discerned by the human ear.
Just intonation has been used since ancient Greece, where many musicians and philosophers believed that the essence of music lies in its adherence to mathematical principles found in nature. This tuning system is still used in the modern time by many musicians and composers.
Xenharmonic music is a modern harmony or tuning method.
In traditional Western music, an octave is made up of 12 semitones while xenharmonic music uses intervals smaller than the traditional tuning, resulting in an octave with 13 or more pitches.