Acoustics & Music Theory
why it sounds like that
Generator

The simple ratios, and 12 equal steps

One octave laid out in cents. Above the line, the frequency ratios of small whole numbers, where they actually fall; below it, 12 equal steps of 100.00 cents each. The two sets almost never coincide, and how nearly they do is the case for or against the division.
The simple ratios, and 12 equal steps. One octave laid out in cents. Above the line, the frequency ratios of small whole numbers, where they actually fall; below it, 12 equal steps of 100.00 cents each. The two sets almost never coincide, and how nearly they do is the case for or against the division.

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. 16 of its 16 placements carry sound, built from the same numbers as the drawing, offering these buttons: 27/25: just, then 13-step, 3/2: just, then 12-step, 5/4: just, then 12-step, 6/5: just, then 12-step, 7/5: just, then 13-step, 9/7: just, then 13-step, fifth: just, then 12-step, fifth: just, then 41-step and 17 more.

Called by 9 essays

the blast radius of changing it

Every equal division from 5 to 60, and how wrong it is. For each number of equal steps in the octave, how far its best fifth and its best major third fall from the pure ratios, in cents. The divisions people have actually used are the ones with small errors in both, and no other criterion was applied to pick them out.

Nineteen, thirty-one and fifty-three

Divide the octave into a different number of equal parts and different problems disappear. Which ones vanish is not a matter of taste — it can be computed from the prime factors of the intervals, and each division makes a different decision.

tuning · The comma Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A440. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded.

The pitch nobody agreed on, for four hundred years

A440 is a committee decision from 1955. Before it, the same written note was played anywhere between about 392 and 465 hertz depending on the town, the building and the decade — a spread of very nearly a minor third, and every piece of theory here is untouched by it.

tuning · Pitch standard Roughness across an octave. Sensory dissonance between two complex tones as the upper one is swept through 19.02 semitones, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit on 7/5, on 11/7, on 5/3, on 9/5, on 11/5, on 7/3, on 13/5, found by scanning the curve rather than by marking them. The wells are where this spectrum's partials coincide, and for a spectrum with no even partials they are not where an octave-based scale would put them.

A scale without an octave, and the spectrum that asks for it

Every scale so far repeats at the 2:1. That looks like a law and it is a consequence — the octave fuses because every partial of the upper note lands on one of the lower. Take a spectrum with no even partials and the 2:1 stops doing that, while the 3:1 starts.

scales · Beyond twelve The smallest audible difference, and what has to clear it. The difference limen for frequency, converted from Wier, Jesteadt and Green's 1977 fit into cents, against the intervals and commas the rest of these essays argue about. Anything drawn below the curve is a quantity nobody can hear as a change of pitch; anything well above it is a quantity a listener can be asked about. The limen is for pure tones, successive, with trained listeners — the most favourable case there is, and therefore the right one to test a claim against.

How small a difference is audible

Every essay here about tuning has assumed a listener who can hear the difference between two systems. The assumption has a number: about five cents in the middle of the range. It clears the two commas fourfold and it does not clear the schisma at all, which sorts the whole subject of tuning into the part that is about music and the part that is about arithmetic.

perception · Pitch-acuity 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.

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 A string struck at one 7th of its length. The amplitude of each partial of an ideal string excited at 0.1429 of its length. The mode shape is a sine, so a partial with a node at the excitation point cannot be set moving at all: partials 7, 14 are silent here. The envelope over the rest is one over n, a struck string's.

Where the hammer lands

Strike a string at exactly one over n and the nth partial is silent, because the hammer has landed on that mode's node and cannot move it. A piano's hammers strike between a seventh and a ninth of the way along, which puts the seventh partial — the most dissonant member of the series — at or near a null. That is a design decision made in wood, and it takes one line of trigonometry.

instruments · Excitation point Nine commas, and the line under which none of them matters. Each named comma at its true size in cents, against the difference limen of 3.87 cents at 500 Hz — the smallest change of frequency a listener can detect, computed from the same formula the perception essays use. One comma falls below it, and it is the only gap in the subject that no tuning system has to do anything about.

A comma under the threshold

Eight pure fifths taken downwards arrive at a major third 1.95 cents flat of a perfect 5:4. That gap has a name and a history, and it is the only one in the subject nobody has ever had to hide — it is under the smallest difference a listener can detect, which makes a chain of untempered fifths a source of almost-just thirds and makes one eighteenth-century tuning nearly free.

tuning · The comma Where the series stops being a chord and becomes a scale. The interval between each partial and the next, in semitones, against partial number. It is an octave, then a fifth, then a fourth, a major third, a minor third — and it crosses 2 semitones between partials 9 and 10. Each shaded band is one octave of pitch and holds twice as many partials as the band below it: 2 between 1 and 2, 3 between 2 and 4, 5 between 4 and 8, 9 between 8 and 16. Nothing about the series changes with height; what changes is the density, and a set of notes becomes usable as a melody when its neighbours are a step apart.

The register where the series becomes a scale

A melody needs steps, and a natural trumpet has only the harmonic series. The gap between one partial and the next falls as the series rises — an octave, a fifth, a fourth, a third — and does not reach a whole tone until the eighth. That single arithmetic fact decides what two centuries of brass writing could be: fanfares below, and melody only in a band one octave wide with nine notes in it, three of them badly out of tune and every one of them lipped into place by a player who had no valve to press.

instruments · Melody Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A415. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded.

The note that moved by a minor third

A has been anywhere between 392 and 465 hertz in the surviving record, which is 296 cents — a minor third short of six. The usual conclusion is that absolute pitch level is a convention and nothing musical depends on it. That is true of everything written on paper and false of everything that happens in a throat or a body: a singer's register break sits at a fixed frequency, so across the historical range it lands three semitones further down the written page. Transposing a piece is not a uniform operation, because the performer does not transpose.

tuning · Pitch standard

All figures · What can be heard