A fifth on a piano is not a fifth a second later

Nine rungs of this ladder have drawn the harmonic series, and every one of them has drawn one note. A string does everything at once; a stiff string’s partials are not quite the series; the series is not a chord; it has three tops belonging to the ear and a fourth belonging to the maker; a flare makes it harmonic; a mouthpiece re-chooses it; seven lengths share one mouthpiece; and the top of a struck note’s series falls while the note lasts, which is the fifth top and belongs to the clock.

Every one of those is a note by itself, in silence. What a listener has in front of them is a texture — several notes at once, each with its own series, each series with its own top coming down at its own rate. The question that puts is not hard to state. Two notes an interval apart are an interval because certain of their partials meet; do those partials still exist by the time both notes have been heard?

The answer is yes for a while and then no, and the interesting part is what the collection can say about the while. Both halves of the arithmetic are already here. The coincidence half decides which partials of two notes meet: partial $p$ of the lower against partial $q$ of the upper, where $p/q$ is near the interval’s ratio. The decay half decides which are still present: a struck string loses each partial at a rate rising with partial number, so the series shortens from the top while the note sounds.

Neither half has ever been asked in the presence of the other. Putting them together settles a question about counterpoint that has a definite answer and has never been priced: whether the consonance of an interval on a struck instrument is the same quantity a second after the strike that it was at the strike.

It is not, and the way it fails is not the way the failure was expected to go.

The two halves, and why they had not met #

The coincidence half is old. Two notes a fifth apart in equal temperament have their third and second partials within two cents of each other; a major third brings the fifth partial against the fourth, fourteen cents apart; a minor second brings the fifteenth against the fourteenth. That table is what makes an interval an interval rather than a pair of pitches, and it is what two sections singing a tempered third beat against even when every singer in both of them is in tune.

Every entry in it is a statement about frequencies, and frequencies are all it contains. The fifteenth partial of a note is in that table whether the note has fifteen partials or four.

The decay half is newer and is entirely about whether the partial is there. Each partial of a struck string has its own loss, rising with partial number because air drag, internal friction and the loss to the bridge all rise with frequency. On the standard first approximation the loss rises in proportion to the partial number, so partial $n$ dies $n$ times as fast as the fundamental, and a note with a six-second fundamental has a sixteenth partial gone in well under a second.

Set the two side by side and a coincidence becomes an object with a lifetime. The $p{:}q$ coincidence of an interval exists while partial $p$ of the lower note and partial $q$ of the upper are both above the threshold of hearing, and it ends at the earlier of the two moments at which one of them falls below it. Nothing in that sentence needs a listener, a corpus or a recording. It needs a threshold curve, a loss exponent and the interval’s own ratio.

The consonance hierarchy falls out of a decay law #

The octave keeps its 2:1 for 3.16 seconds. The fifth keeps its 3:2 for 2.14. The fourth keeps its 4:3 for 1.60, the major third and the major sixth keep theirs for 1.26, the minor third for 1.04. Then the tritone at 0.87, the minor sixth at 0.74, the major second and the minor seventh at 0.64, and at the bottom the minor second and the major seventh at 0.38.

That is the consonance hierarchy of common-practice European theory, in order, arrived at from a hearing threshold and an exponent.

It should not be over-read, and the reason it comes out is not mysterious. A simple ratio is a ratio between small numbers, small numbers are low partials, and low partials are the ones a decay leaves alone longest. What the ordering shows is that the hierarchy’s usual justification — the simplicity of the ratio — and a fact about how long a struck note lasts are the same fact seen twice, because both are statements about how far up the series the interval’s evidence lies. A perfect consonance is durable because its evidence is cheap.

The size is worth stating in musical rather than physical units. At a moderate tempo a crotchet is two thirds of a second. The octave’s coincidence survives about five of them; the fifth’s about three; the minor second’s about one half. What is dying is not the roughness — that is the next section, and it does something else — but the evidence of identity: the partials that make a fifth a 3:2 rather than a pair of pitches. A dissonance on a piano loses the partials that say which interval it is inside the note that states it, and a perfect consonance does not. That is a claim about the instrument rather than about the interval, and it is one no figure computed on a sustained tone can make.

The other half of consonance runs the other way #

So far the arithmetic agrees with the tradition. The second half does not.

Roughness is the collection’s other account of consonance and it is not the same account. The Plomp–Levelt sum does not ask whether two partials coincide; it asks how far apart they are relative to the critical band, and it is largest when they are close and not equal. A coincidence contributes nothing to roughness, because two partials at the same frequency are one. What produces roughness is the near-misses, and the near-misses are mostly high up, where partials are dense in hertz.

Both accounts are computed on the same struck dyad here, with the same partials, the same threshold and the same loss law. They come apart at once.

At the strike the fifth is the smoothest of the eleven intervals above C3, which is what every dissonance curve on this site says it should be. It is passed by the major seventh after 0.14 seconds, by the major sixth after 0.22, by the minor seventh after 0.24 and by the minor sixth after 0.30. By then the note is at roughly a quarter to two fifths of the roughness it had at the strike, which is well inside its audible life. This is not a rearrangement that happens in the dying embers of a note; it happens during the first third of a second.

Six pairs of intervals out of fifty-five finish in the opposite order from the one they began in. That is a small number and it is concentrated in one place: five of the six involve the fifth, which is the only interval to move more than one rank. The dissonant end of the list barely stirs — the major second is the roughest interval at the strike and the roughest interval two seconds later.

Which computation produced the number #

The roughness at each moment is dissonancePair over two decayed spectra. Each note’s partial $n$ starts at 80 decibels less $20\log_{10}n$ for a $1/n$ source, loses $60(t/T_{60})n$ decibels by time $t$ with $T_{60}$ six seconds, and is dropped entirely once it falls under the threshold of hearing at its own frequency. What is left of the two notes is fed to the same double sum every roughness figure in this collection uses, with the pairs inside one note excluded as they are everywhere else. The unit is Pa², so a number here is comparable with another number here and with nothing in the unlevelled curves.

Two things about that recipe carry the whole result and both are worth saying plainly.

A partial below threshold is not a quiet partial; it is not there. This is the convention a loud chord is a smaller chord established for level and it is what makes the decay a change of spectrum rather than a change of volume. A partial that has gone contributes no roughness at all, not a little.

A decay is not a fade. This is the sentence the whole rung turns on. The ranking of intervals by roughness is level-independent above about C3 — that rung tested twelve intervals at middle C between thirty and ninety decibels and not one of them moved, because turning a note down multiplies every partial by the same factor and multiplying a whole curve reorders nothing. A decay does not multiply every partial by the same factor. It multiplies partial $n$ by a factor that depends on $n$, so it tilts the spectrum rather than lowering it, and a tilt is exactly the operation a ranking cannot survive.

So the two rungs are consistent and they say opposite-looking things for a reason that is arithmetic rather than perceptual: the ranking survives level and does not survive time.

Two nearly bare fundamentals have no ratios left #

What the tilt converges on is the mechanism. After a second a C3 struck at 80 decibels has six partials left; after two it has three. Two notes reduced to a fundamental and a whisper of a second partial are, for the purposes of a roughness sum, two sine tones — and the roughness of two sine tones is a single-humped function of their separation in hertz with no minima anywhere.

The amplitudes the same note has a second later are not a guess and are not chosen for the drawing: they are what the decay law leaves, $1/n$ multiplied by ten to the power minus $n$ over two, which is a fundamental with a sixth of a second partial on it and nothing else worth the name. Swept through the same octave against the same model, that spectrum draws a different curve.

That is why the fifth loses. A fifth’s smoothness in the first picture is bought entirely with coincidences: its third partial sits on the upper note’s second, its sixth on the fourth, its ninth on the sixth. Take the partials away and the payment stops. What is left is two fundamentals 65 hertz apart at C3, and 65 hertz at 130 hertz is well inside a critical band — so the bare fifth is a rough pair. A major seventh’s two fundamentals are 116 hertz apart and are not.

The consonance of an interval stops being about ratios and becomes about hertz, and the changeover takes a third of a second. The first picture is the one the whole subject teaches. The second is the one a piano presents for most of every note it plays.

Only the bass loses its fifth #

That mechanism carries its own limit, and it is the limit a third is rougher in the bass established at the third rung of the consonance ladder. A critical band is a fixed width in hertz and an interval is a ratio, so two fundamentals a fifth apart leave the band as soon as the pitch is high enough.

At C5 and C6 the fifth is the smoothest interval at the strike and the smoothest interval a second later, and nothing moves at all: up there two bare fundamentals a fifth apart are far outside one band, so what the decay leaves is a smooth pair whatever the interval. At C4 the fifth falls from first to third. At C3 it falls from first to fifth. At C2 it was never first — down there the low notes of a chord are rough whatever they are and the fifth starts fifth.

One quantity moves in the same direction at every pitch, and it is the one the debt this rung pays did not expect. The spread between the smoothest and the roughest interval widens as the note decays, everywhere. At C2 it goes from 1.41 to 2.16 times, at C3 from 1.77 to 4.87, at C6 from 10.2 to 930. Consonance and dissonance do not converge as a struck note fades. They separate, because a smooth interval’s roughness is being taken away faster than a rough one’s — the smooth interval’s roughness lived in the high partials, which go first.

The prediction the debt made was that the coincidences would run out and the interval would blur. The coincidences do run out. The blur does not happen: the intervals get further apart on the model’s own axis and end up in a different order.

What the picture cannot show #

The pictures here are roughness against time and they cannot say what a listener does with either axis.

The most important omission is that the ordering reverses while the quantity being ordered is collapsing. At C3 the fifth’s roughness at the crossing with the major seventh is 40 per cent of what it was at the strike, and two seconds in every interval is at about a hundredth of its opening value. Whether a listener can compare two roughnesses that small is not something this arithmetic answers, and the smallest difference in roughness a listener can report has never been measured as finely as the equivalent for pitch. The claim these figures support is that the model’s answer changes, not that the sensation does.

Nor is roughness the whole of dissonance even where it is measurable. Consonance is half learned put a number on the part that is not sensory at all, and a dissonance is what has to be resolved is about a syntactic obligation that has no acoustic correlate whatever. A major seventh whose roughness has fallen below a fifth’s is still a major seventh, still needs to resolve, and will still be heard as the more strained of the two by anyone who has heard tonal music.

And the model is a struck string with an exponent of one. Real piano strings sit between about 0.5 and 1.2, real dampers end notes long before the arithmetic does, and a pedal keeps other strings ringing sympathetically in ways nothing here contains. The exponent moves the timing and not the direction: a faster loss brings the crossings forward and a slower one pushes them back, and the crossings happen at any exponent, because they are produced by the tilt and not by its steepness.

Whose music, and when #

The repertoire this is a claim about is keyboard counterpoint on a struck instrument — the Well-Tempered Clavier, the fugues of Bach’s late collections, and the two centuries of piano writing that inherited their habits — and the specific device it prices is the suspension.

A suspension is a dissonance prepared as a consonance, struck, held, and resolved by step. The whole rhetorical point of it is that the dissonance is sustained: the tension has to persist across a beat for the resolution to relieve anything. On an organ or in a choir it does persist, because the sound is driven. On a piano, the arithmetic here says the dissonance the suspension states is losing the partials that make it a dissonance from the first millisecond, and if the suspension is held for a crotchet at a moderate tempo the note has spent two thirds of a second decaying before it resolves.

Two consequences follow and they run in opposite directions, which is why this is more interesting than it looks.

The dissonance fades. A minor second’s strongest coincidence is gone in 0.38 seconds and its absolute roughness has fallen to a fifth of its opening value in half a second. A suspension held for a whole bar is doing most of its work in its first beat.

And the consonance fades faster. The resolution a suspension arrives at is usually a third, a sixth or an octave, and the smooth intervals lose their roughness fastest of all, because their smoothness is made of high partials. So the contrast between the suspension and its resolution — the thing the device exists to produce — is largest at the moment of the strike and grows in ratio while shrinking in size.

That is not advice. It is an account of why the same counterpoint sounds like a different piece on a harpsichord, a piano and an organ, in a form that has a number in it: on a sustained instrument the ordering on the left of the ranking figure holds all the way through the note, and on a struck one it holds for about a seventh of a second.

Where this ladder goes next #

Ten rungs. A string does everything at once; a stiff string’s partials are not quite the series; the series is not a chord; it has three tops that are the ear’s; a fourth that is the maker’s; the shape that makes the series a series; that shape re-chosen with a mouthpiece in front of it; the same mouthpiece against seven lengths; the fifth top, which is the clock’s; and now the sum, where two of those clocks run under one interval and consonance comes apart into two quantities that had never been asked to agree.

What is owed after this is the third note. Everything above is a dyad, and the object common-practice harmony is actually built from is a triad — three notes, three intervals, three clocks, and a coincidence structure that is not the union of the three pairwise ones. A triad has coincidences that belong to no pair in it: the fourth partial of the root, the third of the third and the second-and-a-bit of the fifth all land in the same neighbourhood, and whether that neighbourhood is still populated at the moment a listener judges the chord is exactly the question this rung answered for two notes. What it would settle is whether a chord’s inversion has a lifetime — whether a six-four is a different object a second after the strike than a root position, which is a claim about voicing on a piano that this collection has the machinery for and has never made. The coincidence table generalises to three notes without any new arithmetic, the decay law is unchanged, and the roughness sum already takes any number of parts. It needs arithmetic and nothing else — no corpus, no recording and no listener — and the first thing to check is whether the three-note coincidence hierarchy still comes out in the traditional order, because with three notes the traditional order is a ranking of inversions rather than of intervals and there is no reason in advance to expect the arithmetic to reproduce it twice.