For questions related to tuning systems that privilege justly tuned intervals as opposed to equal temperament, meantone temperament, etc. Questions will likely also require the tuning tag.
Just Intonation is a term that encompasses a wide variety of different tuning systems which aim to create the most perfectly consonant intervals possible. In accordance with the natural frequency ratios generated within the harmonic series, many just intonation tuning systems aim for simple mathematical integer frequency ratios between the notes in the tuning system; for example, JI systems often tune the interval of a perfect fifth with a frequency ratio of 3:2 in accordance with the exact 3:2 ratio between the 3rd and 2nd harmonic in the overtone series.
One common obstacle to any justly-intonated music is that it is mathematically impossible for both the 2:1 octave and the 3:2 perfect fifth to remain perfectly in tune with each other over a series of interval leaps. As proof, consider that 2^x = 3^y only when x and y both equal zero, meaning that only after zero octave leaps and zero perfect fifth leaps will the two notes be on the same frequency. In other words, 12 perfect fifths up from a note should equal an octave multiple of that note, since we will have cycled through every note in the 12-note system. (3/2)^12 = 129.75, meaning that after 12 perfect fifths we have a total ratio of about 130 times the starting note's frequency. Unfortunately, this supposed octave multiple seven octaves up from the starting note should then also be equal to (2)^7, which is equal to exactly 128. Since 129.75 is slightly greater than 128, the 3:2 perfect fifths don't meet up with the 2:1 octaves.
This occurs for many other JI intervals, and thus almost every just intonation tuning system will have many (mathematically) consonant intervals at the cost of a few extremely dissonant intervals, such as the famed "wolf fifth" that aligns the very last perfect fifth to the octave in one of the most well-known JI systems.
Just intonation stands in stark contrast to the relatively newer Equal Temperament, which "tempers" the mathematically pure intervals by setting a standard semitone interval as 2^(1/12):1 so that all twelve notes in the temperament are equidistant from their neighbors and that the octave is set at exactly 2:1. From a JI perspective, equally tempered intervals (aside from the octave) are ever so slightly out of tune, but there are no especially out-of-tune notes such as the "wolf fifth" since all intervals are multiples of the same common frequency ratio.
