Nature, Published online: 03 March 2026; doi:10.1038/d41586-026-00678-7
In 12-TET, you divide up the octave into twelve equally-sized semitones (the interval between two adjacent piano keys or guitar frets). To go up a semitone from any note, you multiply its frequency by the 12th root of 2 (about 1.05946). To go down a semitone from any note, you divide its frequency by the 12th root of 2. If you go up by an octave (twelve semitones), you’re multiplying your frequency by the 12th root of 2 twelve times, which works out to 2. That’s a perfect octave, hooray! Unfortunately, you can’t exactly create the other harmonics-based intervals by adding up 12-TET semitones; you can only approximate them.
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