Just-noticeable difference — where it appears
Named by 18 essays across 4 fields — each of them below, with the objects they name alongside it.
The ear sorts into boxes, and the boxes are the theory
Slide one note slowly upward against another and the interval between them changes continuously. What a listener reports does not. It stays a minor third, stays a minor third, and then in the space of about twenty cents becomes a major third — and nothing in the sound corresponds to the moment of the change.
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.
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.
A note that is never at its pitch
Eleven essays in this field argue about differences of one to twenty-four cents. An ordinary operatic vibrato is a hundred and forty cents wide and completes six excursions a second, so every one of those distinctions fits inside a single sung note several times over — and the beat rate a tuner would null passes through zero eleven times a second.
The series has three tops
How far up the harmonic series can an ear go? The question has three answers and they are an order of magnitude apart. Consecutive partials stop being separately resolvable somewhere around the eighth, and where exactly depends on the fundamental. They stop being a semitone apart at the seventeenth, at every fundamental, because the ratio does not know what it is measured in. And they stop being distinguishable in pitch at all between the thirty-fourth and the hundred and fortieth. Every claim about where the series runs out is a claim about which of the three was meant.
How many boxes an octave holds
An identification model of pitch is usually handed twelve categories, and nothing ever asked how many an octave can hold. There are two answers and they are a factor of forty apart. Resolution allows between 151 and 356 — a listener can tell that many pitches apart in a direct comparison. Naming one of them without a comparison is a different faculty and it runs out at six or seven, which is where every mode in every tradition compared here sits. Turkish theory names fifty-three commas to the octave and a makam uses seven of them, and the gap between those two numbers is the whole of the argument.
How long a note has to be
Every difference limen quoted so far is for a tone that lasts as long as the listener needs, and no note in music does. A tone of duration T occupies a band about 1/2T wide whatever the ear does with it, so at 440 hertz the quoted five-cent limen is the right number only for notes longer than 486 milliseconds. A tenth of a second gives 19.6 cents, which does not clear the syntonic comma. Most of the tuning arguments in this collection are about a quantity that only exists in long notes, and the essays that made them said so about the listener and not about the note.
An interval is two errors
Two notes in succession are two pitch estimates, and what a listener judges is their difference — so a melodic interval is heard less finely than either of the notes in it. At a semiquaver the limen is nineteen cents, which lands inside a published range long quoted from the literature and never derived. Sounded together instead of one after the other, the same interval is judged eighteen times more finely.
The boundary that barely moves
Every identification figure here has fixed category centres, and the essay before this one ended by admitting that real boundaries are supposed to move with context. Three mechanisms could move one, and their predictions are an order of magnitude apart and in two different directions. Expectation on its own — a listener who thinks one interval twenty times more likely than the other — is worth three and a half cents.
How much an anchor would have to be worth
Two pitch errors added in quadrature assume an independence nobody measured — a listener inside a key hears a note as a scale degree, and a shared reference is exactly a correlated error. Turning the dial is not evidence. What is evidence is that the dial is not free: a shared error cancels out of a difference completely, so a listener's single-note limen and their interval limen give the two components with nothing left over, and halving the interval limen needs a correlation of exactly 0.75.
The notes in between
Every figure until now is about two notes with nothing between them, and a melody is notes with other notes between them. Two published accounts of what the intervening material does predict opposite signs — one says the key is restated and the shared reference is refreshed, the other says each note competes for the same memory and it decays. By eight notes they differ by four and a half cents, which is nearly half the limen the interval would have with no anchor at all.
The number every claim here has been quoting
Sigma is the internal noise a listener's pitch judgements carry, it is measured in a laboratory on isolated intervals, and music never presents an isolated interval. Every resolution claim here rests on the laboratory value. Sweep it and the headline finding moves: at eleven cents the ear has about six nameable categories per octave, and at six it has twelve — which is the chromatic scale, and turns 'the ear has fewer boxes than the notation' into 'it has exactly as many'.
The quantity a rival account says is not there
Two earlier essays measure how much two notes' errors are correlated through a shared anchor, and price what that correlation would be worth. There is a rival account in which a listener refers each note to a key and never forms the distance at all — under which the correlation is not small, it is a description of something that is not happening. The two accounts agree on almost everything and disagree on one manipulation, and the manipulation costs an afternoon.
The turn is half the angle
A stationary head cannot tell a sound in front from the same sound behind, and an earlier essay said so at length. The turn that breaks the confusion is 0.84 degrees — exactly half the angle a source would have to move for the same listener to notice it moving, and half for a reason. In a hall the same turn does something else: the source swings at 8.9 microseconds a degree and the room swings at 2.7, so a listener who moves is separating the soloist from the reverberation as well as the front from the back.
A smaller head in the same hall
Ten earlier essays draw one head. Every parameter belonging to the room has been varied by some figure and the one belonging to the listener never has, and it is the only one whose change the detection threshold does not follow: a six-year-old in the same seat receives the same fifty-four reflections at the same instants and reads them onto an axis with seventy distinguishable positions instead of eighty-seven. The speed of sound, swept over every temperature a hall is ever at, changes nothing at all — and the reason it cannot is the reason head size can.
The part of the error a key cannot touch
Four earlier essays turn one dial — the correlation between two notes' pitch errors — and apply it to the whole of a note's limen. Half of that limen is not the listener's: a note of finite length does not carry its frequency more finely than 1/2T, and no context can put information into a signal that is not there. So the correlation has a ceiling, it is 0.21 at a quarter-second note at A4 and 0.07 at A2, and the figure that prices a correlation of one half is drawn where one half is unavailable.
Where a wrong head gives itself away
Every claim so far maps a delay to a direction through one fixed geometry, and the listener acquires that map while the geometry grows under them by seventy per cent. So the map can be wrong — and the essay before this one said the error would be largest on the median plane, where the delay curve is steepest. It is exactly zero there. The steepness is in the error and in the threshold and cancels between them, which leaves a listener whose internal head is 1.3 millimetres out with one place to catch it: hard to the side, where nobody localises well.
The error that moves straight ahead
The essay before this one found that a listener whose internal head is the wrong size makes no error at all on the median plane, and has to look hard to the side to catch it. Every head drawn here has its ears at equal radii, which makes the delay curve odd and every error a factor — and a factor cannot move a zero. Real heads are not symmetric. A constant offset of twenty microseconds displaces a listener's straight ahead by two and a quarter degrees, and it displaces every other direction by the same number of just-noticeable steps, exactly.
Named alongside it
The objects these essays reach for when they reach for this one.
Difference limenCentsCategorical perceptionHead shadowInteraural time differenceIntervalLocalisationScale degreeIntonationLimenPitch discriminationPitch memory