eli5 what just intonation is
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To understand just intonation in music, we need to start with a background on the shortcomings of equal temperament.
Equal temperament is a tuning system that works by simply spreading all pitches in an octave logarithmically through that octave. In 12-tone equal temperament, for example, all 12 pitches are spread evenly between one note and the note an octave above it. Pitches are divided on a logarithmic scale because then the frequency ratio between C and C# is equal to the frequency ratio between C# and D, D and Eb, etc. which means that all keys are equivalent. This is great for flexibility as it means that all keys can be used interchangeably.
The problem with equal temperament is that some intervals are out of tune. By this, I mean that those intervals are pretty far off from the simple frequency ratio they are trying to approximate. As an example, an equally tempered major third is 4 semitones, or 400 cents, wide, but a perfectly tuned major third has a frequency ratio of 5:4 and a size of 386 cents (3.86 semitones). Some frequency ratios also cannot be well approximated at all. The 7th harmonic (a 7:4 relationship) is 969 cents wide, which is 31 cents off from the minor seventh, and the 11th and 13th harmonics (11:8, 13:8) essentially fall right in between two notes a semitone apart and do not have a good approximation.
Just intonation solves these problems by ignoring any sort of pre-existing temperament system and just tuning the notes of every chord precisely to certain frequency ratios. While the sound is very consonant and pleasing, it is less flexible as you can only play in one or two keys (unless you write and play the music using software).
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