Why are equal-tempered steps multiplicative?

A melody can move higher on a keyboard while keeping the same pattern of musical intervals. In ideal twelve-tone equal temperament, the secret is a repeated multiplication. Adjacent semitone steps use the same frequency ratio, even though the extra number of vibrations per second grows as the notes rise. Equal musical steps are not equal additions of hertz.

Frequency counts cycles per second. An ideal octave has a frequency ratio of two to one: a 220 Hz reference reaches 440 Hz one octave higher. Twelve-tone equal temperament divides that octave into twelve semitone steps. If each step multiplies frequency by the same number r, twelve steps multiply by r twelve times. The requirement is r raised to the twelfth power equals 2. Therefore r equals 2^(1/12), approximately 1.059463.

The calculation gives a whole family of frequencies. After n semitone steps, the frequency is f times 2^(n/12), where f is the starting frequency. Six steps from 220 Hz reach 220 times the square root of 2, approximately 311.127 Hz. Twelve steps reach exactly 440 Hz in the model. The printed decimal for an intermediate pitch is rounded; the symbolic ratio preserves the exact relationship.

The growing size of a hertz step follows from multiplication. One semitone above 220 Hz is about 233.082 Hz, an increase of 13.082 Hz. One semitone above 440 Hz is about 466.164 Hz, an increase of 26.164 Hz. Both rises have the same ratio. The second hertz increase is twice as large because its starting frequency is twice as large. A graph against equally spaced semitone numbers curves upward when its vertical axis measures hertz on a linear scale.

This structure explains transposition within the tuning system. Moving every note of a melody up by k semitones multiplies all its frequencies by the same factor, 2^(k/12). When the ratio of any two new frequencies is calculated, that common factor cancels. Their interval ratio is preserved. The absolute pitches rise, while the mathematical relationships between them stay intact.

Equal temperament also involves a trade-off. Seven semitones have a ratio of 2^(7/12), approximately 1.498307, close to but not exactly the simple 3:2 ratio of a just perfect fifth. From 220 Hz those alternatives give about 329.628 Hz and exactly 330 Hz. The slight difference allows the twelve-step pattern to repeat consistently across the keyboard, while giving up exact agreement with some simple whole-number interval ratios.

This is a tuning convention, not a ranking of musical traditions or a claim that every instrument follows an ideal formula. Musicians use other tuning systems, and real instruments can depart from exact values. The striking connection is that a practical way to move melodies between starting notes grows from the same idea as proportional scaling: keep the ratios, change the overall size. A musical pattern can travel while its internal relationships remain recognizable.