Generator

The critical band, measured in semitones

The width of the ear's frequency-analysis band at each pitch, converted from hertz into semitones. Two intervals drawn as horizontal lines cross the curves: below the crossing the interval fits inside one band and its notes are not resolved from each other, and above it they are.

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. 10 of its 10 placements carry sound, built from the same numbers as the drawing, offering these buttons: And at 880 Hz, Major third at 110 Hz — inside one band, The same third at 440 Hz.

Called by 9 essays

the blast radius of changing it

A note with its first partial removed. The spectrum of a 220 Hz tone with the lowest partial deleted, and the wave that remains. The wave still repeats 220 times a second, because the repeat rate of a sum of harmonics is fixed by the spacing between them and the spacing has not changed. The pitch heard is the one that is no longer in the sound.

The note that is not there

A telephone reproduces nothing below about three hundred hertz, and a bass line down at eighty comes through it perfectly. The pitch that is heard is not a frequency present in the sound, and one nineteenth-century experiment settles what it is instead.

intervals · Missing fundamental

A third is rougher in the bass

Consonance is usually presented as a property an interval has. It is not. The same major third is muddy two octaves below middle C and clean two octaves above it, the ratio never changed, and the frequency where it stops being muddy can be solved for.

intervals · Consonance

A source 45° off centre, and the path difference it makes. A head from above with a source to one side. The near ear is reached first; the far ear's path runs round the head, and the difference between the two is 13.1 centimetres, which at 343 metres a second is 381 microseconds. That number, and the level difference the head's shadow produces, are the whole of what the ear has to work with.

Two ears, and the whole of the difference is 655 microseconds

Direction is computed from two numbers — when a sound reaches each ear and how loud it is at each — and which of the two is usable is decided by the wavelength against the width of a head. The changeover frequency is not a design choice. It falls out of 343 metres a second and 17.5 centimetres, and it is why the mechanism of hearing where something is changes halfway up the piano.

perception · Localisation

The same intervals, played higher and higher. Sensory roughness for four fixed intervals as the pair is transposed up five octaves, computed from the same Plomp–Levelt model as the dissonance curve. Every one of them falls as it rises, and the small intervals fall furthest — so how consonant an interval is depends on where it is played, not only on what it is.

A chord is a register

The same three pitch classes are five times rougher at the bottom of a piano than in the middle, and the arrangement that minimises the roughness over a low bass turns out to be the bass's own fifth and sixth partials. The orchestration rule about low thirds is not a convention. It falls out of the width of a critical band, exactly.

intervals · The triad

3 tones, one power, and the interval between them. 3 tones of fixed total power, spread symmetrically about 440 hertz, drawn against the interval between neighbours. Piled on one pitch they are one sound of that power; separated by more than a critical band — 4.5 semitones here — they are 3 sounds whose loudnesses add, and the same power reaches 2.08 times the loudness at 5 semitones. The two lines are two models of the same rule and they disagree about how abrupt the change is, not about where it goes.

A chord is not as loud as its notes

The first essay on loudness said what a tone's loudness is and recorded that it had said nothing about a chord's. Here is the missing rule, and it has a musical consequence nobody would predict from it: the same three notes, at the same power, are twice as loud in the treble as in the bass — because the critical band that makes a low triad five times rougher also makes it one sound instead of three.

perception · Loudness

One pair, 12 beat rates. Two notes at 220 hertz, 15 cents apart, drawn as the beat rate between each pair of partials. The fundamentals beat at 1.91 hertz and the k-th partials at k times that, so the series of rates is a straight line and it crosses 15 hertz at the 8th partial. Below the line the fluctuation is counted; above it the same physical fluctuation is heard as roughness. 1 per cent of this spectrum's energy is on the roughness side. The bars are drawn at each partial's own amplitude, so a spectrum with little energy high up crosses the line where nothing is listening.

Every partial beats at its own rate

Five earlier essays have drawn one beat rate per figure, and every one of them is the rate between two fundamentals. Two real notes beat between all of their partials at once, the k-th pair beats k times as fast, and somewhere up the spectrum the rate passes the point at which a beat stops being a beat — so a chorused note is a beat at the bottom of itself and a roughness at the top, simultaneously, with a crossover partial that is arithmetic.

tuning · Beating

How far equal temperament puts each interval's coincidence out. For each interval, the pair of partials it brings together and how many cents equal temperament mistunes that coincidence by. A fifth's third-against-second is out by 2 cents and a major third's fifth-against-fourth by 14 — so the same temperament that is inaudible on a fifth produces, at 220 hertz, a beat of 8.7 per second between two sections singing a third, with every singer in both of them perfectly in tune.

A section against another section

The choir has been treated as a unison, and no choir sings only unisons. Two sections an interval apart beat between partials rather than between fundamentals — the third brings the fifth partial of one against the fourth of the other — and equal temperament puts that coincidence fourteen cents out. So two sections singing a tempered third beat at nearly nine per second with every singer in both of them perfectly in tune, and the same temperament is inaudible on a fifth.

timbre · The voice

Adding parts adds power, and very little loudness. Each part is played at the same level, and the chord is realised every way its parts allow and averaged over them, so the quantity is a property of the texture rather than of one arrangement. Going from 3 parts to 8 adds 4.3 decibels of power and 0.1 decibels of loudness, because the extra parts land in bands that are already occupied — the count of occupied critical bands FALLS from 6.0 to 3.9 as the parts crowd into the same register.

The dynamics are in the score already

Count the parts in each bar, realise them in their ranges, put every partial in its critical band, sum the loudnesses and run the result through the two smoothers built earlier. What comes out is a dynamic curve for a piece with no performance in it anywhere — and it says that doubling the number of parts inside a fixed register adds three decibels of power and about one of loudness, because the extra parts land in bands that were already occupied. Let the register widen with the parts and the same arithmetic gives eight phon, which is what a tutti actually is.

form · Loudness

Of 8 beats at A3, 3 can be attended to. Every member of the beat family a 2:1 mistuned by 6.0 cents makes on a real string at A3, placed by how separable its coincidence is from the partials beside it — in auditory filter widths, across — and by how fast it beats, up. The shaded region is the set a listener can receive: wider than one filter, and between 0.4 and 15 fluctuations a second. 4 of 8 members fall inside it, and once rates within a factor of 2 are counted as one modulation channel there are 3. The members that fail do so for two different reasons: the low ones sit under the roughness ceiling but their coincidences are buried in a filter that holds three partials, and the high ones are resolved and far too fast.

Three beats at most, and only in the middle of the keyboard

A mistuned octave on a real piano makes eight beats at once, and a listener attending to one of them is doing something that has a threshold. Two thresholds, in fact — a rate and a place — and once both are applied the eight become four at A3, one at A1 and one at A5. Every interval a tuner sets goes to zero at both ends of the compass and peaks at eight countable beats in the octave the bearing is laid in.

intervals · Beating

All figures · What can be heard