Pitch and frequency

An oscillator running at 220 Hz and one running at 440 Hz sound like the same note, one higher than the other. That interval is an octave, and it is a doubling of frequency at every pitch: 55 to 110 Hz is an octave, and so is 2 to 4 kHz. Pitch is therefore heard on a logarithmic scale of frequency: each octave covers twice as many hertz as the one below it, and each sounds like a step of the same size.

The tool below sets a frequency anywhere from 27.5 Hz to 4.2 kHz, the range of a piano, and plots it on a scale marked in octaves. It also names the nearest note in twelve-tone equal temperament, the tuning used by most Western instruments and by synthesisers by default, and how far the frequency is from that note in cents. Every frequency on the slider is a pitch you can play, and the note name is given as a reference.

Ratios, not differences

Two frequencies 100 Hz apart can be almost any interval. At 110 Hz, adding 100 Hz gives 210 Hz, which is a little under an octave above. At 1760 Hz the same 100 Hz is less than a semitone. What the ear hears as the size of an interval is the ratio between the two frequencies. An octave is 2:1 at any pitch, and the fifth in the tool above is 3:2, the ratio between the second and third harmonics.

This is why a synthesiser's pitch controls are exponential. Adding a fixed amount to the control raises the frequency by a fixed ratio, so the same turn of a knob, the same change in control voltage or the same change in MIDI note number gives the same interval at any pitch. The chapter on one volt per octave describes the standard that most analogue and modular synthesisers use for it.

Notes, semitones and cents

Twelve-tone equal temperament divides the octave into twelve equal ratios. Each step, a semitone, multiplies the frequency by the twelfth root of two, about 1.0595, and twelve of them make exactly 2. The reference is A above middle C, written A4, at 440 Hz, and every other note follows from it. Intervals smaller than a semitone are measured in cents, with 100 cents to the semitone and 1200 to the octave.

Equal temperament is one way of dividing the octave among many. Just intonation uses whole-number ratios like 3:2 and 5:4 taken from the harmonic series, and those two intervals differ from their equal-tempered versions by 2 and 14 cents. Indonesian gamelan uses two tuning systems, slendro and pelog, with five and seven notes to the octave at intervals different from the equal-tempered ones, and Arabic maqam and Turkish makam music use intervals between the equal-tempered semitones. A synthesiser oscillator can produce any of these, because its frequency is continuous. The harmonic series guide compares the harmonic-series intervals with their equal-tempered neighbours, and where pitch comes from shows quantizers, which move a control signal to the nearest note of a chosen scale.