Acoustics & Music Theory
why it sounds like that
Concept

Unison — where it appears

Two or more sources sounding what is meant to be one pitch. Nothing about it is trivial: a piano's three strings are deliberately a cent or two apart for the sustain, and a choir's sixteen are fifteen cents apart because people are, which is what makes a section sound like a section.

Named by 5 essays across 5 fields — each of them below, with the objects they name alongside it.

The fluctuation stays; the rate goes. A unison of n voices with a spread of 15 cents, averaged over 5 draws. The depth of the amplitude fluctuation does not fall as voices are added — a choir is no steadier than a duet — but the fraction of that fluctuation in any single modulation component falls from 77 per cent at two voices to 24 at 32. Two voices make one beat and it can be counted; 16 make 120 and none of them is a rate. That is why a choir cannot be tuned by nulling anything.

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 this site measured describes only ensembles that hold their pitch still.

timbre · The voice 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 Which interval gives a tuner the deepest null. A tuner listening to an interval p:q is listening to the lower note's p-th partial against the upper note's q-th, and how deep the beat's trough goes is decided by those two amplitudes rather than by the interval. For a string spectrum, whose partials fall as one over n, the minor third pairs partial 6 against partial 5 at a ratio of 1.18 for a dip of 21.8 decibels; the major third pairs partial 5 against partial 4 at a ratio of 1.25 for a dip of 19.1 decibels; the fourth pairs partial 4 against partial 3 at a ratio of 1.32 for a dip of 17.2 decibels; the fifth pairs partial 3 against partial 2 at a ratio of 1.52 for a dip of 13.8 decibels; the major sixth pairs partial 5 against partial 3 at a ratio of 1.65 for a dip of 12.2 decibels; the minor sixth pairs partial 8 against partial 5 at a ratio of 1.67 for a dip of 12.0 decibels; the octave pairs partial 2 against partial 1 at a ratio of 2.00 for a dip of 9.5 decibels. The best is the minor third at 21.8 and the worst is the octave at 9.5, which is the reverse of the order a tuner is usually taught to trust: the deepest null in the list is on the interval whose coincidence sits highest in the spectrum, where adjacent partials are nearly equal in strength.

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.

intervals · Beating Where a section's fluctuation stops being a beat, on 220 hertz. Two rates up the spectrum of a 220 hertz note sung by a section whose voices are spread by 15 cents. The rising line is the beat rate between a typical pair of them, which grows with the partial because a mistuning in cents is a difference in hertz that scales with frequency; it reaches the 15 hertz at which a beat stops being a beat by partial 5.5, at 1217 hertz. The flat line is the amplitude modulation the vibrato imposes through the formants, which is 6.0 hertz at every partial because the vibrato modulates every partial by the same number of cents at the same rate. The two are equal at 487 hertz. Above 1217 hertz the beating has become roughness and the only fluctuation left is the vibrato's — and that frequency is the same one an octave up, where it is partial 2.8 instead.

The rate that does not rise with the partial

Twelve earlier essays give every vibrato the same six hertz, and the measured spread is 5.5 to 7.5. Putting the two fluctuations a choir contains on one axis shows why the rate matters: the beating between mistuned voices rises with the partial and leaves the range a listener follows as fluctuation at 1,217 hertz, while the vibrato's own modulation is six hertz at every partial. Above that frequency a section fluctuates by vibrato alone — and if every singer had the same rate, it would barely fluctuate at all.

instruments · The voice 14 players summed, against the one at their average onset. Each thin line is one player's rising envelope, started at its own moment, with a spread of 30 milliseconds about the beat and a 90-millisecond attack. The heavy line is the section: nominally identical sources add incoherently, so their powers add and the sum is the root of the mean of their squares, drawn here as a fraction of the section's own peak. The dashed line is the single player who started at the section's average onset. The section reaches the 6 dB below peak criterion at 18.1 milliseconds and that player at 27.6, a difference of 9.5. The section is early because the players who started first are already sounding while the average one is still building, and nothing a late player does can make the sum quieter.

Twelve violins are more punctual than one

Every essay until now treats a part as one player, and an orchestral part is a dozen. Sectioning does two things at once and only one of them was expected: it pulls the part's heard moment forward, by four milliseconds against a map spanning twenty-six, and it makes the part's arrival more accurate by very nearly the root of the number of players. So the map of required leads applies to an orchestra better than it applies to a quartet, and the case where it fails is three trumpets rather than fourteen violins.

rhythm · Perceptual-centre

Named alongside it

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

BeatingChorus effectCritical bandwidthIncoherent additionModulationPartialRoughnessVibratoAmplitudeAttack transientDetuningEnsemble timing

All concepts