Generator

Three octaves, and none of them is 2:1

How far above an exact doubling the upper note of an octave is set, against frequency. The listener's octave is measured with pure tones, which have no partials to beat against each other, so nothing about a stiff string can account for it. The piano's stretch is a different quantity with a different cause, and the two are drawn together only so that the difference is visible.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 5 of its 5 placements carry sound, built from the same numbers as the drawing, offering these buttons: An exact 2:1, Stretched 17 cents, The same two, with partials.

Called by 4 essays

the blast radius of changing it

The octave that is not two to one

The octave is the one interval nobody argues about: two to one, exact, in every tradition that has one. Asked to set an octave by ear, listeners set it wide — and they do it with pure tones, which have no partials to beat against each other. Whatever is stretching the octave, it is not the stiffness of a piano string.

perception · Pitch-acuity

Two ways to detune a pair, and they disagree by an octave. A pair of instruments tuned deliberately apart, drawn across 4 octaves. Holding the detuning at 10 cents gives a beat rate that rises from 0.64 to 10.20 beats a second — a factor of 16, one doubling per octave, because a fixed ratio is a growing number of hertz. Holding it at 3 beats a second instead gives a flat rate and an interval that shrinks from 46.6 cents at the bottom to 2.9 at the top. A tuner has to choose, the choice is audible across the range, and a table of cents can only write down the first of the two.

A tuning is not a table of cents

Two instruments of a pair are tuned deliberately apart, and the tuner has to choose between a fixed number of cents and a fixed number of beats — which differ by a factor of sixteen across four octaves and cannot both be held. A list of pitches records the first, cannot record the second, and cannot record at all that there are two instruments.

tuning · Beyond twelve

It was never the strings that stopped the climb

Every figure so far holds the instrument still and moves the standard. History did the opposite — instruments were rebuilt to suit the pitch, string by string. Hold the tension instead and each string's gauge is forced: the diameter goes as one over the frequency and the stress as its square. Gut breaks at A = 525 hertz and steel at 604, and the climb stopped at 466. The ceiling was somewhere else entirely.

tuning · Pitch standard

The setting a listener reports is not the interval they preferred. Five published preferences for an interval size, and the value a listener would have to play in a key context for that preference to be what they hear. The key pulls a heard interval back toward the scale, so a listener adjusting until it sounds right has to overshoot — and the reported setting, which is what they played, exaggerates the preference. The corrections run from 1.6 cents to 12.8 on notes of 0.25 seconds. The largest is nearly a syntonic comma, on a preference of thirteen cents, which is to say that the correction is bigger than the effect it is a correction to.

The setting is not the preference

Every published number for an interval listeners prefer — the pure third a quartet is said to find, the raised leading note, the harmonic seventh — is a value somebody adjusted until it sounded right. A listener adjusting inside a key is adjusting through the posterior computed just before, so the value they stopped at is not the one they preferred: it is the one whose heard size equals it. On quarter-second notes the correction for a pure major third is 12.8 cents, which is nearly a syntonic comma and is larger than the 13.7-cent preference it corrects.

intervals · Pitch-acuity

All figures · What can be heard