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Equal temperament divides the octave into 12 equal semitones; just intonation uses pure frequency ratios — the two differ by cents

Concept L3Craft A · Music theory & musicianshipB · Sound design & synthesis

Equal temperament (ET) divides the octave into 12 mathematically equal semitones of 100 cents each. This allows music to be played in any key without retuning, but every interval except the octave is slightly impure — the major third in ET is 400 cents vs. a pure 386 cents (difference of 14 cents). Just intonation (JI) uses whole-number frequency ratios (pure 5th = 3:2 = 702 cents; pure major 3rd = 5:4 = 386 cents), sounding cleaner in a single key but creating problems when modulating. Pythagorean intonation tunes all notes in pure 5ths (3:2). The Pythagorean comma (approximately 23.5 cents) is the accumulation of error when 12 pure 5ths do not close the octave exactly.

Examples

Cubase micro-tuning: adjust each key by +/- cents. Balinese gamelan uses a non-equal 7-tone tuning unique to each instrument set.

Assessment

Explain in one sentence why equal temperament is a compromise. What is the cent difference between a pure major third and an equal-tempered major third?

Prerequisites — understand these first

Don't confuse with

Sources

“The equally tempered tuning that is forcedupon us by instrument manufacturers is not actually true to pure intervals. Compare, for example, a pure major third with an equally tempered major third. The tempered third is equivalent to 400 cents, while the pure major third is 384 cents”
corpus · michael-hewitt-music-theory-for-computer-musicians · chunk 43

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