Equal temperament — where it appears
Named by 16 essays across 4 fields — each of them below, with the objects they name alongside it.
Where to hide the comma, which is the only real question
Four tuning systems, four different places to put an error that cannot be removed. Each one is coherent, each was standard somewhere, and the choice between them decided what music could be written.
A fraction of a comma
The tuning systems of three centuries are usually presented as rival schemes with names and dates. They are points on a line. Narrow every fifth by the same fraction of a comma and both errors that matter are linear in that fraction — and the system named as the break with the tradition turns out to be a member of it, at one part in eleven.
What a temperament cannot do
Add up the twelve major thirds of any keyboard tuning whose chain of fifths closes and the answer is 4,800 cents. Every time, in every system, before a single fifth has been chosen. So no temperament has ever made the average third less wrong than any other one, and four hundred years of argument were about a distribution rather than about accuracy.
A comma is a polyrhythm that never closes
Seven earlier essays have rested on a common grid. Two cycles have one when their ratio is a ratio of whole numbers, and the fifth against the octave is not — provably, from unique factorisation. So there is no grid, no composite and no least common multiple, and what is left over instead is the sequence 2, 5, 12, 41, 53 with a comma attached to each.
The partial the lips cannot reach
A wind instrument plays between its valve's preferred frequency and its bore's, which suggested that the natural trumpet's notorious eleventh partial would therefore turn out to be a statement about the lips. It is not. Run the same Q-weighted mean over the whole series and the lips can move that partial twenty-three cents, which is a quarter of what it needs — and the Q that would fix it is the Q at which the bore stops choosing the note.
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.
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 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.
A major triad's combination tones are its own notes
Play a just major triad of pure tones and two of the ear's cubic products land exactly on its root and its fifth. The reason is a condition rather than a coincidence — a chord's cubic products fall on its own notes when its middle note is the mean of the outer two in hertz — and it holds for the major triad in root position and in the six-four, and for no minor triad in any position or tuning. Equal temperament misses the landing by one number, 5.6 hertz on middle C, which is a beat that belongs to no pair of notes in the chord.
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.
Three harmonics of the bass arrive before the bass
Every product priced until now was between two pure tones, and nothing that plays thirds is pure. Give each note a spectrum and the ear receives every pair of partials — and for a just interval p:q every one of their products is an exact multiple of the same absent fundamental. That crowd lands where the threshold of hearing is tens of decibels cheaper, so it names the bass at 71 decibels where the component at the bass's own frequency needs 74, and at 75 against 85 an octave lower. Tempered, the crowd still forms and names a note seventy cents flat.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Just intonationTuning by earOpen stringChain of fifthsIntonationSyntonic commaBeatingCentsCombination toneKey colourTemperamentDifference tone