Your ear hears musical steps as equal, but the frequencies multiply
Climb a scale one octave and the pitch doubles in frequency; climb another and it doubles again. Musical intervals are ratios, not fixed distances, so equal-sounding steps are really an exponential climb. That is why musicians measure them in cents, a logarithmic unit where an octave spans exactly 1200.
An interval is the pitch difference between two sounds. Played one after another, as in a melody, it is called melodic or horizontal; sounded together, as in a chord, it is harmonic or vertical. In Western music intervals usually come from the notes of a diatonic scale, and the steps between neighbouring notes are called scale steps, the smallest being the semitone. Anything smaller is a microtone, and the tiniest discrepancies, such as the slight gap between C sharp and D flat in some tunings, are called commas.
Physically, every interval is a ratio of frequencies. Two notes an octave apart stand at 2 to 1. In just intonation the main intervals use small whole-number ratios: 3 to 2 for a perfect fifth, 4 to 3 for a perfect fourth, 5 to 4 for a major third and 6 to 5 for a minor third. Such intervals are called pure.
Most instruments today are tuned instead to 12-tone equal temperament, in which all twelve semitones are the same size. The price is that most intervals no longer match neat ratios. An equal-tempered fifth, for instance, works out to about 1.498 to 1, very close to, but not exactly, three halves. The cent makes comparison easy: divide the octave into 1200 logarithmic slices and, in equal temperament, each semitone is precisely 100 cents. The formula multiplies 1200 by the base-two logarithm of the frequency ratio.
Names describe two things: quality, such as perfect, major, minor, augmented or diminished, and number, such as unison, third or fifth. A perfect unison, two identical notes, measures zero cents; a whole tone spans two semitones; a tritone spans six. The spelling matters because musicians historically tuned G to G sharp differently from G to A flat, even though the two can sound alike.
Source: Interval (music)
