220 Hz against 223 Hz
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. 185 of its 185 placements carry sound, built from the same numbers as the drawing, offering these buttons: 110 Hz, 10 cents apart, 220 Hz, 10 cents apart, 220 Hz, 5 cents apart, 440 Hz, 10 cents apart, 440 Hz, 5 cents apart, 6 cents apart at 440 hertz, which beats 1.5 times a second, 6 cents apart at 880 hertz, which beats 3.1 times a second, 880 Hz, 5 cents apart and 187 more.
Called by 37 essays
the blast radius of changing it
Keys that had characters, and could be measured
Eighteenth-century writers described E flat major as devotional and F sharp major as harsh, and modern readers file it under synaesthesia. On the instruments those writers used, the difference between the two keys was seventeen cents of third, and that is a quantity.
Beats are arithmetic that anybody can hear
Two tones a few hertz apart swell and fade at exactly their difference. It is the most direct evidence available that the ear does sums on what reaches it, and it is how every instrument in the world gets tuned.
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.
A tuner counts beats, and that is the whole method
Laying a temperament sounds like the most subjective job in music, and it is arithmetic. Every interval in the bearing octave has a target rate in beats per second, the rate follows from the ratio and the register, and a tuner who hits the numbers has produced equal temperament without ever thinking about a cent.
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.
A rhythm fast enough to be a chord
Three against two is a polyrhythm. Speed the same pattern up until the events arrive faster than about twenty a second and it is a perfect fifth. The ratio never changed, the figure never changed, and the only thing that moved is the rate — which makes rhythm and pitch one continuum with a perceptual boundary across the middle of it.
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.
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.
What a choir does that a soloist cannot
Two singers on one note produce one beat and it can be counted. Sixteen produce a hundred and twenty at once, and the amplitude still fluctuates by as much as it did — a choir is no steadier than a duet. What has gone is not the fluctuation but its rate: the modulation energy that sat in a single line at two voices is spread across a band at sixteen, with no line in it. That is the choral sound, and it is also why the just-intonation drift measured for tuned singing describes only ensembles that hold their pitch still.
Three strings, and the note that comes back
The usual account of a beat adds two independent sources together. A piano's strings are joined at the bridge and are not independent: the pair has normal modes, one of which drives the bridge and dies in a second and a half while the other cancels at the bridge and rings for twenty. A hammer striking a perfect unison excites only the first — so a perfectly tuned unison is the shortest note on the instrument, dying in one second, and the longest is at a deliberate detuning of about two and a half cents, at three. Past four and a half cents the two modes swap what they share and the sustain collapses.
The beat that is not in the air
Every sound this site synthesises reaches both ears identically, and that is the assumption none of its figures ever varied. Put 500 hertz in one ear and 504 in the other and nothing sums anywhere: each eardrum sees a steady sinusoid with no modulation on it at all. A listener still hears a four-per-second beat, which means the arithmetic is being done behind the ears rather than in the room. And it stops working above about a kilohertz — not where phase locking gives out at five, but where a head 17.5 centimetres across stops being able to name a direction, which is 762 hertz.
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 pair tuned apart on purpose
A piano's unison buys its sustain with a detuning of two and a half cents and loses it past four and a half. Read from the other side that is a design constraint on every instrument that wants the beating instead: past the bifurcation the sustain is gone and no further detuning costs anything more, which is why every chorused voice in the world — the celeste rank, the musette reeds, the paired gamelan — is built out of sources that do not share a bridge.
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.
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.
A beat has a depth, and six essays held it at one
Every beat figure so far adds two tones of equal amplitude, which is the single ratio at which the trough of a beat is a true null — and a null is what a tuner is actually listening for. Vary the ratio and the picture changes: at two to one the dip is nine and a half decibels, at ten to one it is under two, and the interval that gives the shallowest null of all is the octave.
The beat a tuner can actually use
There is a modulation index for every interval on every timbre, and the published threshold for detecting a fluctuation is a function of exactly that and its rate. Running one against the other turns a table of decibels into a window in cents — and reverses the ordering, because the interval with the shallowest dip has the widest window.
A beat is never one beat
Every beat counted until now has come from one pair of partials. A real spectrum has many, so a mistuned octave produces eight beats at once — and on a perfectly flexible string those eight are an exact harmonic series of the slowest. On a real piano string they are a cubic, half of them are not beats at all, and no single width of octave silences more than one. That is where the tuner's five octaves come from.
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.
Every member of a beat family is the same depth
A mistuned octave's beats have been counted on their rate and their place, with the depth recorded as the thing left out and a prediction that the shallow upper members would take the count from four to two. The depth turns out not to fall at all: on any power-law spectrum every member of a family has exactly the modulation index its interval's own ratio gives, at every register, on every wire. The count does fall to two, and the thing that takes it there is the criterion that essay was already using.
The unison is the coarsest thing in the room
An orchestra tunes on a unison and then plays intervals, and the two are not the same test. The lowest coincidence of a p:q interval sits at p times the lower note's frequency, so a mistuning of a given number of cents makes a beat p times faster — a fifth betrays it three times sooner than a unison, a minor third six. The ritual that opens a rehearsal is therefore the least sensitive measurement anybody will make all evening, and every chord afterwards is a finer one.
One fluctuation or two
Whether two members of a beat family are one thing or two was decided by asking whether their rates differ by a factor of two, and the factor was written down as a stand-in for a modulation filterbank nobody had run. Run, the bank gives 1.618 — the golden section, and not by accident, since a channel of quality one has its half-power points there. The stand-in was conservative rather than optimistic, and the recomputed counts do not change at a single register, because the criterion was never what was binding.
How hard the note was struck
The auditory filter is not a fixed shape: its lower skirt shallows by about 38 per cent of its reference value every ten decibels, so a loud note is analysed through a wider filter than a quiet one. Every share computed so far was quoted at a moderate level, and a tuner does not strike moderately. Recomputed, the mean share of a mistuned octave's filter falls from 0.55 at forty decibels to 0.07 at seventy, and the count of separable beats goes from two to none — which is a prediction too strong to be right, and the way it fails is the useful part.
Counted in the decay, or not at all
A mistuned octave struck hard delivers no countable beat at the strike, and the reconciliation offered for that was that a tuner listens to the decay. Computed through a real decay it fails on its own terms: the partials fall silent before the filter has narrowed enough to separate them. It succeeds only when the filter is broadened by the level inside it, which is what the published parameterisation was fitted against — and then the window opens at a twelfth of the note's life and shuts at a quarter.
A string that decays twice is counted early
A mistuned octave's beats were found countable only between a twelfth and a quarter of a note's life, on a note decaying once. A piano string decays twice, a fast prompt sound over a slow aftersound, and the prediction was that this would open the count sooner and close it later. It opens sooner — at a seventh of a second rather than a second — and closes exactly where a single decay struck twenty decibels softer closes, so at 80 dB it holds 8.8 beats instead of 11.8. The count now rises with the strike to 90 dB, and a tuner who strikes hard is right.
A firm touch buys beats until the aftersound sinks with it
A piano string decays twice, and a mistuned octave's countable beats were found to rise with how hard the note is struck — on the assumption that the aftersound always starts twenty decibels below the strike. It need not. The moment the count shuts is set by the level the aftersound starts at and nothing else, so a harder strike buys beats only if its aftersound does not sink with it. If the drop deepens by more than 0.85 decibels for each decibel of strike, the firm touch costs beats instead.
The octave is counted best over a unison left two and a half cents wide
A firm touch buys a mistuned octave's beats only if the aftersound's drop does not deepen with the strike, and the drop was left as a number to be measured. It has a mechanism, and the mechanism has been known since the essay on three strings: the unison's two coupled modes. Computed from them, the drop does not depend on the strike at all. It depends on how wide the tuner left the unison — sixty-five decibels at a quarter of a cent, twenty at three — and the octave's beats peak, at 6.8, over the same two-and-a-half-cent unison that makes the longest note.
The plane of the blow matters only where the unison is clean
A piano string vibrates in two planes, and the one parallel to the soundboard gives even a single string a second, slower decay. A hammer tilted off the vertical feeds it. That was the last way left for a firm touch to cost a tuner beats: if harder blows came down straighter. They would have to straighten by half for every seven decibels, and even then only on a single string or a unison tuned clean. Over a unison left two and a half cents wide, no tilt of the blow reaches the count.
A stiff string slows the fourths and quickens the fifths
Every rate in a bearing plan is computed from whole-number partials, and a piano string's partials are not whole numbers. The correction is usually summarised as 'every real beat is faster than the table'. It is not: stiffness quickens the beat of every narrow interval and slows the beat of every wide one, and on a small upright the tempered fourth stops beating altogether. Both errors push the same way, so a tuner who counts the printed table leaves every interval wide, and a fifth and a fourth counted that way make an octave stretched to the 4:2 type or a little past it.
The fourth–fifth check cannot hear the note it checks
A tuner tests a temperament by sounding the fourth below a note and the fifth above it, inside one octave, and comparing their beats. Both beats run to the same partial of the middle note, so moving that note speeds or slows both together and leaves their difference untouched. The check is a test of the octave, and of one particular octave: the 4:2. Measured with each note's own stiffness, the printed table laid fourth first builds exactly that octave on any piano; laid fifth first it overshoots, and the check pulls it back by a third of a cent.
The thirds and the fourths check different octaves
Every interval counted to its printed beat rate on a stiff string is left wide, by an amount set by the squares of the partial numbers that meet in it. Add the links of a chain and the sum lands on an octave type: a fourth and a fifth make the 4:2, a minor third and a major sixth make the 6:3, and three major thirds come just short of the 6:3. So the two most familiar checks in a bearing are exact tests of two different octaves, and on a stiff piano no octave passes both. Each fails on the other's octave by a beat a second at A3.
The octave that fails both checks least
A tuner who cannot pass both the fourth–fifth check and the third–sixth check settles on an octave that fails them equally, and a listener compares beat rates by ratio. Weighed that way, the compromise sits a ninth of the way from the 4:2 octave to the 6:3 on every note of every piano, because the fourth beats twelve times slower than the minor third and that ratio belongs to equal temperament, not to the strings. On a grand both checks then pass to a fifteen-per-cent ear up to middle C, and above it no octave passes both.
The progression of thirds votes for the bearing, not the octave
Added as a third vote beside the two contiguous checks, the progression of thirds was expected to push a stiff piano's octave wider. It pulls it narrower, because the octave it asks for is not set by the strings at all: it is whatever stretch the bearing already carries, plus an eighth of a cent. In an equal bearing that drags the compromise back onto the 4:2 and costs the top of the bearing its pass; give the bearing a fifth of the octave's own stretch and the third vote falls silent.
A bearing counted correctly is stretched by its route, not its strings
A tuner who sets a temperament octave by counting each fourth and fifth to its printed beat rate leaves every one of them a little wide on a stiff string, and the bearing ends up carrying a stretch nobody chose. Whether that stretch is enough to quiet the progression of thirds across the octave's edge turns out to be the wrong question. The stretch belongs to the order of the plan, not to the piano: one ordinary plan carries five times what the progression needs, another a fraction. And either plan leaves the thirds inside the bearing beating in a zigzag, faster then slower then faster, by up to eighteen per cent a step on a grand — which is why the thirds have to set the bearing rather than check it.
A bearing set by its thirds carries five eighths of its octave
A bearing counted from the printed fourths and fifths leaves its thirds zigzagging round the printed progression and takes whatever stretch its route gives it. Set the other way — every note solved at once so that each major third beats faster than the one below by exactly the printed step — the bearing bends smoothly upward and carries 63 per cent of its octave's own stretch, on a grand, on an upright and on everything between. That is more than three times what the octave's three checks need in order to agree, so following the thirds reconciles them with room to spare. The price is paid by the fourths and fifths, every one of which then beats slow.
A stiff string gives the bearing a comma of its own
The twelve fourths and fifths inside a piano's temperament octave are not twelve checks. Laid end to end they are one closed circle, and on stiff strings their printed rates fail to close it by 2.22 cents. Every disagreement between a bearing's tests follows from that one number: which interval never gets counted decides the stretch, and the zigzag in the thirds is the seam. Spread it round the circle and every fourth and fifth beats nine per cent slow of the table, and the thirds come right without being asked.
One walk round the bearing measures the piano and the tuner together
A tuner who counts twelve fourths and fifths to the printed table and comes home flat has measured something. Divide the closing beat's rate by three and correct the whole table by that share, and the worst interval improves from a hundred per cent off to about twenty, on any piano. It keeps the table with one number. And the number cannot tell a stiff string from a slow count: a tuner counting nine per cent slow gets the right bearing from the wrong table, and one correction removes that luck along with the error.