Where one interval stops being itself
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. 70 of its 70 placements carry sound, built from the same numbers as the drawing, offering these buttons: 100¢ — semitone, 12 equal steps, the count at 6 cents of noise, 18 equal steps, what a listener with 4 cents of noise names at 95%, 2 equal steps, the count at 30 cents of noise, 2 equal steps, the count at 35 cents of noise, 200¢ — major 2nd, 25 equal steps, the count at 6 cents of noise, 250¢ — the boundary itself and 64 more.
Called by 16 essays
the blast radius of changing 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.
The chord is still major, and that is why temperament works
A major third can be seventeen cents wrong and still be a major third. That tolerance is not a failure of hearing — it is the reason the whole subject of tuning is a discussion rather than a catastrophe. Every temperament ever proposed moves intervals around inside their categories, and the one thing none of them may do is push one across a boundary.
How late is a different note
A deviation of thirty milliseconds is expression and a deviation of two hundred is a wrong note, so there is an edge. The edges in time are arithmetic — the midpoints between the simple ratios — and the swing ratio crosses two of them as the tempo rises, at 171 and at 240 beats a minute, while the notation says triplet feel throughout.
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.
A scale built downward from a fourth
A tetrachord is a bounded span with two free notes inside it, which is a different generator from a chain of fifths and a different object from a subset of an equal division. At the maqam tradition's own quarter-tone resolution the construction admits thirty-six fillings of the fourth and 1,296 octaves built from them; the census here can see six of the first and 2.8 per cent of the second, and one of the octaves it cannot see is an ordinary mode of a living tradition.
The same distance, under two names
Four hundred cents is a major third or a diminished fourth, and on a keyboard nothing in the sound distinguishes them. An earlier essay was about the boundary between two categories; this is about two categories at one acoustic value, and the surprise is where the ambiguity comes from. In quarter-comma meantone a major third is 386 cents and a diminished fourth is 427 — two names, two pitches, forty-one cents apart. Equal temperament collapsed them, and what a listener now supplies from context used to be in the sound.
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.
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 listener the model was never run for
Five earlier essays rest on one number — how finely a listener resolves a pitch — and every one of them used a trained listener's eleven cents. The model's dependence on it is not gentle: the capacity goes as its reciprocal and the expectation shift as its square, so an untrained listener at thirty-five cents has two nameable categories per octave rather than six, and a foreign tuning system is not mis-transcribed by them but absorbed.
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'.
A boundary costs the same wherever it is put
Every capacity figure drawn until now cuts the octave into equal categories, and no scale in the world is equal. Putting the real step patterns through the same model returns exactly the same accuracy to four decimal places — because naming is lost at boundaries, an n-degree scale has n of them wherever they are, and each costs the mean absolute value of the noise. That turns the most-quoted result about hearing from a search into one line: 1504 times one minus the criterion, over sigma.
An interval is two posteriors subtracted
Treating a key as a prior over one note predicts that an interval's pull is not the single-note pull doubled, because the two degrees are not equally weighted. Half of that is wrong: splitting a mistuning between the two notes gives exactly the mean of what each end gives alone, to a thousandth, at every one of the twenty-one intervals in the scale. What is not the mean is which end carries it — and the pull turns out to be largest not on the shortest notes but on notes of about an eighth of a second, where the likelihood is a quarter of a semitone wide.
A boundary beside a fifth
A closed form established earlier says an n-category division costs n·σ·√(2/π) whatever the widths are, so a boundary costs the same wherever it is put and the step pattern cannot matter. Its own last section named the assumption that produces the null: one σ, applied to every boundary in the octave. Let σ follow how securely each interval is held and the formula becomes a sum over boundaries rather than n copies of one — the diatonic's error rises from 5.1 per cent to 11.8, and where the degrees are put matters again.
The best seven of the twelve
Once the noise is allowed to differ from boundary to boundary, a scale can be chosen to minimise identification error — and the choice is a search over four hundred and sixty-two sets rather than an argument. Run, it returns a cluster of semitones around the tonic and around the fifth, and puts the diatonic major at rank 376 of 462, in the worse fifth of the ranking. A criterion whose optimum is a scale nobody has ever played is a criterion that is not what scales are chosen for, and the reason it fails is legible in the model rather than in the music.
The unequal scale that is easier to name
The whole of the earlier result was that a scale's step pattern cannot matter, and every tradition it drew agreed with the equal division of its own size to three decimal places. With one noise per boundary the comparison is live again, and the tempered diatonic beats seven equal steps by eleven per cent — while the pentatonics gain nothing and the maqam scales gain two tenths of one per cent. Only one of the two security models produces the effect, and it is the one that was not measured on listeners raised inside the tradition it is being used to explain.