Tuning by ear — where it appears
Named by 23 essays across 4 fields — each of them below, with the objects they name alongside it.
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.
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.
Somebody has to pay the comma
A choir has no frets and no keys, so it can tune every chord exactly — and a common progression sung that way arrives a comma flat, every circuit. The freedom a keyboard lacks turns out to be a freedom about where the error goes rather than whether there is one, and the two costs always add to the same number.
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.
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 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.
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.
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.
The third sound magnifies cents, not hertz
Tartini's third sound is said to be a few cents off on a tempered interval. It is sixty-seven cents off on a major third and ninety-six on a minor third, because a difference tone moves p/(p − q) times as many cents as the interval p:q that made it. In hertz it moves exactly as far as the note that moved, and no further — so what the magnifier is worth is the ear's finer resolution at the low frequency where the product lands, which is a factor of two for a long note and nothing at all for a short one.
The cello cannot hear its own tempering
Narrowing a quartet's fifths to meet a piano is one number, 1.96 cents a fifth, and it is a different beat on every string: once every two thirds of a second on the violin's A–E and once every four and a half seconds on the cello's C–G. Set by ear for two seconds a fifth, the violin's E lands within two thirds of a cent and the cello's C within 5.7 — which is as large as the Pythagorean error the tempering was meant to remove. The string whose tuning is most wrong is the string whose tuning is least certain, and a cellist tuning down the chain cannot tell pure from tempered.
An open string pulls the quartet flat
Once the tuning note has stopped, a quartet corrects toward itself and nothing holds its pitch. But four of its pitches do not move: the open strings, on a Pythagorean chain from C 5.9 cents flat to E 2.0 sharp, each ringing when a stopped note shares its pitch class. Give that sympathy a weight of a hundredth of a correction and it beats the random walk within a movement. The quartet settles flat in every major key — by 0.9 cents in E, 2.5 in C and 2.6 in A♭ — and in A♭ major the cellist's tuning scatter moves the whole ensemble by 1.7 cents.
A guitar tuned by harmonics hides a comma
A quartet's stopped notes go wherever its players put them, so its open strings can sit on a pure chain. A guitar's stopped notes are fixed by frets that are already an equal temperament, and its six open strings — four fourths and a third — close two octaves exactly a syntonic comma short when tuned pure, since (4/3)⁴·(5/4) is 4 × 80/81. Tuning by harmonics decides where the comma goes, not whether: a pure third puts it in the octaves of an open E major chord, 17.6 cents narrow; a B from the low E's twelfth puts it in the G–B third, 21.5 cents wide.
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.
One tuning has no comma to place
Standard guitar tuning is four fourths and a third, and tuned pure it closes two octaves exactly a syntonic comma short — so tuning by ear decides where the comma goes rather than whether. An open tuning replaces the interval list. D A D G A D's fifth, two fourths, a tone and a fourth multiply to exactly four, so its chain closes and there is nothing to place: its six strings sit within four cents of the frets. The open-chord tunings close too, and pay for it with one string fifteen cents flat — which every chord barred straight across inherits exactly, and which is why every such chord is precisely as pure as the open one.
The quartet settles at two pitches, not four
The quartet's open strings have been treated as five fixed pitches on one chain, and they are not: the violin, the viola and the cello each tuned a G string by ear and the three are expected to sit two and a half cents apart. Giving each player their own strings, with their own scatter, and pulling each toward only their own, changes the ensemble's settled pitch by a hundredth of a cent. What it does change is systematic rather than random: a violin has an E string and no C, the lower instruments have a C and no E, so the quartet splits by section by a tenth of a cent in every key.
A tuning is right for some chords and wrong for the rest
An earlier essay asked which tuning minimises a piece's total departure from just, and guessed the answer would be a few cents off a named one. Searched over all six strings, two of three pieces get back the tuning they were already in — because their chains close and there is nothing to improve. The third lands on a tuning nobody names, three and nine tenths of a cent better than anything a player would have tried, and it is not a compromise: two of its three chords are exactly just and the third is abandoned by thirteen cents.
Counting beats moves the price of a chord, not the tuning
The search that found a guitar piece's best tuning counted every cent of error alike, and the obvious objection is that a cent of a third beats faster than a cent of an octave. Counted in beats instead, every piece gets exactly the same tuning back. What changes is how much of the piece the abandoned chord has to hold before it is rescued — thirty beats instead of forty-four, arriving in three steps instead of one.
The chord a tuning gives up is a fingering
A guitar piece's best tuning makes two of its chords exactly just and abandons the third by thirteen cents, and no way of counting error — cents, beats, squared cents, or only the beats slow enough to hear — brings the third chord back. Fingered as a barre chord in the shape the other two already use, it comes back completely, and the whole piece is exactly just. The conflict was never between chords. It was between hands.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
BeatingOpen stringPartialEqual temperamentIntonationJust intonationTemperamentChain of fifthsDecayPianoSyntonic commaCritical bandwidth