Acoustics & Music Theory
why it sounds like that
Concept

Ensemble tuning — where it appears

The operation by which one player's pitch reaches sixty: a note given, matched by ear, and passed on. Its precision is a limen rather than a standard, and the arrangement of who listens to whom multiplies it by the square root of how many matches deep a player sits.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Two ways to match a tuning note, and they are not the same size. How finely one player can put their A on another's, at 440 hertz on a note of 2 seconds, by each of the two criteria available. Judging one pitch is good to 4.0 cents; comparing two of them adds two errors in quadrature and is good to 5.7. Nulling the beat between them is a different operation altogether — a mistuning of 2.0 cents makes a beat of 0.50 hertz, which shows one full cycle inside the note — and it is 2.9 times finer. The beat criterion is available only when the two tones sound together and share a partial. A player tuning to a note that has already stopped has the coarse one, and so does a singer with nothing to beat against.

An orchestra is given a note

Every earlier essay on pitch standards draws a standard as a number an ensemble is at, and no ensemble is at a pitch. It is handed one, by one player, on one note, and everything else is matched to it by ear — so a standard reaches an orchestra through a limen nobody quotes. Matching by comparing two pitches is good to 5.7 cents at A; nulling the beat between them is good to 2.0, and which of the two is available depends on how long the oboe holds the note. The crossover is at about seven tenths of a second, which is shorter than an oboe's A and longer than a plucked one.

tuning · Pitch standard Who listens to whom decides how far apart an orchestra ends up. The spread an ensemble of 60 arrives at, for four ways of passing the tuning note around, with one match good to 2.0 cents. Matching the giver directly leaves every player one match away and a spread of 2.0 cents whatever the size; passing it along a line leaves the last player 59 matches away and a spread of 15.1. Orchestral practice is the middle one — principals to the oboe, sections to their principals — which is two matches and 2.8 cents, and is within half a cent of the best arrangement available at any ensemble size. A convention nobody derived sits one step off the optimum of an arithmetic nobody wrote down.

Who listens to whom when an orchestra tunes

One match is good to two cents and an orchestra is sixty of them, arranged in an order that nobody chose deliberately. Passed along a line, the errors accumulate and the last player is fifteen cents from the first; given to everybody at once, nobody is more than two. The convention every orchestra uses — principals to the oboe, sections to their principals — is two matches deep, costs 2.8 cents whether there are four players or a hundred, and sits within a cent of the best arrangement that exists.

tuning · Pitch standard The interval an orchestra tunes on is the least sensitive one it plays. The smallest mistuning each interval betrays, on notes of 2 seconds with the lower note at 440 hertz, taking one full beat cycle as the criterion. A unison shows 1.97 cents; a fifth shows 0.66, a major third 0.39, a minor third 0.33. The ratio is exact and it is the interval's own upper term: the lowest coincidence of a p:q interval sits at p times the lower note's frequency, so the beat runs p times faster. The dashed line is what the same players manage by comparing two pitches instead, at 5.7 cents — coarser than every interval on the axis by between three and seventeen times.

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.

tuning · Pitch standard The ensemble agrees with itself more and more, about a pitch that is moving. Runs of 16 players each correcting toward the mean of their neighbours, with no term anywhere pulling them back to the note they were given, over 480 corrections. The shaded band is the root-mean-square displacement across all eight runs — the envelope a random walk has — and it grows from 0.84 cents a quarter of the way through to 2.01 at the end, which is the square-root growth a random walk has. Four individual runs are drawn inside it and the furthest of the eight over the top, ending at 4.16 cents. Meanwhile the spread AMONG the players falls from 2.8 cents to 0.5. A consensus with no anchor cannot hold a pitch, and it also cannot lose one quickly: a movement's worth of corrections is a few cents rather than the semitone unaccompanied choirs are said to fall by.

A consensus with nothing to hold it

Once the oboe has stopped, no reference is left in the room. Each player corrects toward what they hear around them, which is other players correcting toward them — and a consensus dynamic has a fixed point at every common value, so it pulls the ensemble together and nothing pulls it anywhere in particular. Simulated, the players' spread falls from 2.8 cents to 0.5 while the ensemble as a whole random-walks. The size is the result and it is small: two or three cents over a movement, which is a tenth of what unaccompanied choirs are said to lose.

tuning · Pitch standard

Named alongside it

The objects these essays reach for when they reach for this one.

BeatingCentsPitch standardDifference limenJust intonationError propagationPartialSyntonic comma

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