The Pitch Circle

A pure tone is a sine wave: a point going round a circle at a steady rate, with its height sent to a speaker. Four hundred and forty times a second gives the A an orchestra tunes to, and the rate of turning is the tone’s frequency. At twice the rate the ear does not hear an unrelated note; it hears the same note an octave higher. This is the fact that turns the line of frequencies into a circle.

how fast the arrow turns440.0 HzA4

f — turns per second: 440.0 Hz
laps above A4 — log₂(f / 440): 0.000
the fold — laps mod 1: 0 → mark 0 of 12 → A4

The slider walks laps, so equal drags are equal intervals. It parks on the marks. Sound is a sine at exactly the frequency shown, and nothing else. Every number is computed from how fast the arrow turns.

The arrows

slowed 440×each arrow turns f / 440 times a second on screen; the ear is sent its height

Frequency is a number on a line: 55 Hz, 110, 220, 440, and so on without end. Because the ear treats a doubling as a return to the same note, the quantity that matters for pitch is how many doublings a frequency sits above some reference, with the whole doublings discarded:

\theta = \log_2\!\left(\frac{f}{440}\right) \bmod 1 .

A fraction of a turn is a point on a circle. The spiral in the instrument is the line of frequencies drawn with one lap per doubling; the ring is what remains when the laps are discarded. The twelve note names are twelve equally spaced marks on the ring, a factor of $2^{1/12}$ apart in frequency. The twelve marks are a convention; the circle itself follows from the octave alone.

Why an octave sounds like a return

A sine wave at $2f$ completes two cycles in every cycle of one at $f$ , and that is all the two waves have to do with each other; the sense of return is in what the ear has always heard. A plucked string, a sung vowel or a struck bell is not one sine wave but a set of them at $f, 2f, 3f, 4f, \ldots$ — the harmonics — so a component at $2f$ is present inside every natural note at $f$ . The ear treats a tone at $2f$ as belonging with the tone at $f$ , and likewise at $4f$ and $8f$ , closely enough that a melody sung an octave up is heard as the same melody. The circle comes from hearing, not from anything in the sound wave itself.

Just ratios against the twelve marks

Turn on a second arrow and choose 3:2. Two tones whose frequencies are in the ratio 3:2 form the interval called a fifth, the one the ear picks out most readily after the octave. On the ring it falls near the seventh mark but not on it: about two cents past, a cent being a hundredth of a mark. Choose 5:4, the major third, and it falls fourteen cents short of the fourth mark.

No power of $2^{1/12}$ short of a whole octave is a ratio of whole numbers, so the marks cannot land on the just ratios. Equal temperament spaces the marks evenly so that every fifth is out by the same small amount rather than some being exact and one being unusable: twelve fifths stacked, $\left(\tfrac{3}{2}\right)^{12}$ , overshoot seven octaves by about a quarter of a mark, and that error is shared out across the twelve. A piano is tuned to the marks. An unaccompanied choir tends to drift toward the ratios.

Beats

Choose a hair apart. The second tone is one cycle per second faster than the first. The ear does not hear two notes; it hears one note that swells and fades once a second. In the lower strip the two waves drift in and out of step, and their sum is

\sin 2\pi f_1 t + \sin 2\pi f_2 t = 2\cos\!\big(\pi (f_1 - f_2)\, t\big)\,\sin\!\big(\pi (f_1 + f_2)\, t\big),

a fast oscillation whose amplitude varies slowly. The slow variation is the beat, at the difference of the two frequencies. Piano tuners listen for it and turn the peg until it stops.

The Shepard scale

Choose climb forever. Eight tones an octave apart rise together; each fades out as it reaches the top of the range while another fades in at the bottom. After one lap the set of tones is exactly what it was at the start — the check confirms this — and yet the ear hears a climb that never arrives. The ear registers the position on the ring, and that position keeps going round.

What reaches the speaker is not the wave but a list of numbers: the wave’s height, written down some tens of thousands of times a second. Two different frequencies can produce the same list. The next page is about that list and the second circle it contains.