Before 12-TET came into being, another ratio system was used - "Just Intonation" where simple ratios were used that sounded well enough. But it had a problem of the “Pythagorean / Diatonic Comma” – When we tune an instrument in “Just intonation” the octave gets divided into 12 notes approximately rather than perfectly.
If it were perfect, then any note will sound the same and will be an exact harmonic of its counterpart in an octave above or below: E.g. C4 will sound exactly like C2, C3, C5, C6, etc, these being perfect harmonics of C4.
But the ratios in just intonation do not aggregate up to harmonics for every note in every octave. Or, to put it simply, two notes that are one or more complete octaves apart do not sound the same. This means is that a note will not sound the same in different octaves; the C4 note of a justly tempered instrument will sound slightly different that it’s C2, C3, C5, etc. This, of course, has implications for the quality of the sound produced.
The 12-TET system sounds a lot like the just intonation system in terms of number of notes and their sounds, but it is a lot more mathematically sound. The frequencies of these notes are calculated by adjusting for the “comma”. Thus a near note here is a perfect harmonic of itself in different octaves.
Liked 'Pythagorean Comma or Diatonic Comma in Music Theory' enough to share / save?