Harmonic series — where it appears
Named by 52 essays across 5 fields — each of them below, with the objects they name alongside it.
The note that is not there
A telephone reproduces nothing below about three hundred hertz, and a bass line down at eighty comes through it perfectly. The pitch that is heard is not a frequency present in the sound, and one nineteenth-century experiment settles what it is instead.
A string does everything at once
A plucked string does not vibrate at one frequency. It vibrates at all the whole-number multiples of one frequency simultaneously, and nearly everything in music theory is downstream of that fact.
The piano is tuned wrong on purpose
Every well-tuned piano has a sharp treble and a flat bass, by up to a third of a semitone at the extremes. It is not an error, it is not a compromise about keys, and it follows from one property of a steel wire that can be computed from its diameter and its length.
A tube that skips every other partial
Stop one end of a cylinder and half its modes vanish. That single fact about where the pressure has to be decides that a clarinet sounds hollow, that it plays an octave below its length suggests, and that it must cover nineteen semitones with fingers before it can overblow — while every other woodwind covers twelve.
A cone is not a cylinder
A saxophone has a reed at a closed end, exactly as a clarinet does, and it overblows at the octave rather than the twelfth. If the previous essay's argument were about reeds that would refute it. It is about geometry, and a cone closed at its apex has the complete harmonic series for a reason that takes one line of algebra and is genuinely surprising.
Where the hammer lands
Strike a string at exactly one over n and the nth partial is silent, because the hammer has landed on that mode's node and cannot move it. A piano's hammers strike between a seventh and a ninth of the way along, which puts the seventh partial — the most dissonant member of the series — at or near a null. That is a design decision made in wood, and it takes one line of trigonometry.
The bow makes a corner
A bowed string is not a string swinging. It is two straight lines meeting at a single sharp corner that travels round the string once per period, triggering the slip that keeps it going. The force on the bridge is therefore a sawtooth, and a sawtooth is every harmonic at exactly one over n — which is why a bowed string is the most nearly perfect harmonic series in the orchestra.
A bell has no fundamental
The note a listener names when a church bell is struck is not any partial the bell has. Its nominal, twelfth and double octave sit at 2, 3 and 4 times the prime, which is a harmonic series on a pitch an octave below the loudest thing in the sound — and that pitch is supplied by the listener. Founders have been tuning it by ear since the fifteenth century.
What a drum is doing instead
An ideal membrane's modes are zeros of Bessel functions — 1, 1.59, 2.14, 2.30 — which support no common fundamental, so a drum rings without a note. A timpani is that problem solved — the kettle's air and the radiation load drag four modes onto 1, 1.5, 2, 2.5, and the pitch a timpanist tunes is the fundamental those four imply and none of them is.
Where to put the third
Take three pitch classes, four octaves to put them in, and score all twenty-seven arrangements. The smoothest is root, fifth an octave up, third two octaves up — which is partials one, three and five of the harmonic series — and the roughest, at every register tried, is the chord in close root position at the bottom of the range. Every orchestration manual states that rule and none of them derives it.
Which harmonics carry the pitch
A missing fundamental is inferred from a pattern, and a pattern has to be legible before it can be matched. Counting how many harmonics of a note land in separate auditory filters prices the inference — and the answer at the bottom of a bass guitar's range is none of them.
The root an ear supplies
A major triad's notes fit 4:5:6 with nothing missing and a fundamental two octaves below the bass. A minor triad's fit two different series with two different answers a major sixth apart, and the model cannot choose between them. The ambiguity theorists argued about for two centuries is a computable quantity, and the spectrum invented to remove it is one no object produces.
A chord is a register
The same three pitch classes are five times rougher at the bottom of a piano than in the middle, and the arrangement that minimises the roughness over a low bass turns out to be the bass's own fifth and sixth partials. The orchestration rule about low thirds is not a convention. It falls out of the width of a critical band, exactly.
The register where the series becomes a scale
A melody needs steps, and a natural trumpet has only the harmonic series. The gap between one partial and the next falls as the series rises — an octave, a fifth, a fourth, a third — and does not reach a whole tone until the eighth. That single arithmetic fact decides what two centuries of brass writing could be: fanfares below, and melody only in a band one octave wide with nine notes in it, three of them badly out of tune and every one of them lipped into place by a player who had no valve to press.
The series is not a chord
The major triad is derived from the first six partials of the harmonic series in every textbook that derives it from anything, and the derivation works: the first six contain a major triad and nothing else. Take the first four instead and there is no third at all; take seven and there is a dominant seventh with a flat seventh thirty-one cents below any keyboard note; take sixteen and there are eight pitch classes. Six is the last stopping point that gives the answer, roughness prefers three, four or ten depending on the register, and no criterion in the arithmetic selects six over any of them.
The series has three tops
How far up the harmonic series can an ear go? The question has three answers and they are an order of magnitude apart. Consecutive partials stop being separately resolvable somewhere around the eighth, and where exactly depends on the fundamental. They stop being a semitone apart at the seventeenth, at every fundamental, because the ratio does not know what it is measured in. And they stop being distinguishable in pitch at all between the thirty-fourth and the hundred and fortieth. Every claim about where the series runs out is a claim about which of the three was meant.
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.
The hand that changes the bore
A conjecture refuted here left a question with a number on it: the natural trumpet's eleventh partial is 48.7 cents flat, the lips can move it 23.1, and 25.6 cents are owed by something that is not an embouchure. There is exactly one thing a player can change about the bore while playing, and putting the hand in the bell buys those 25.6 cents at a cost of 3.8 decibels — because the aperture that tunes the instrument is the aperture that radiates it.
The fourth top is the maker's
The series has three tops and all three are the ear's: where consecutive partials stop being resolved, where their spacing falls below a semitone, and where it falls below what the ear can hear at all. The fourth is how far up a player can go, and it is the only one somebody chose. Every modern brass bell puts the boundary at about the ninth partial across a family whose tubes differ by a factor of four — and the baroque natural trumpet, whose bell is small, puts it at the seventeenth, which is exactly the top of the clarino register.
Only two shapes make a series
A tube's length sets where its modes are and its shape sets whether they are a series at all — and of every shape a maker could choose, exactly two give evenly spaced modes. Neither of them is the shape of a trumpet. Webster's equation says how far the others miss by, and the answer is a quarter of a semitone for the obvious ones and four cents for the family brass bells actually belong to.
What the mouthpiece is actually for
A cup and a throat look like a comfort fitting and are a component. Put a real one on the front of a real bore in the horn equation and the whole mode series slides sideways by about a seventh of its own spacing — which is exactly the distance between a series whose second resonance is 1.85 times the spacing and one whose second resonance is the second harmonic. The flare decides whether the modes are evenly spaced; the mouthpiece decides which harmonic each one is.
The flare that makes a series harmonic
Every figure until now treats a partial as n times a fundamental, and the whole of brass instrument design is the business of making that true. A plain cylinder is 127 cents from a harmonic series and a plain cone is 21; the best flare found by sweeping is 4.6, and only two per cent of the swept surface comes within five cents of it. The shape is forced rather than chosen.
The bell is tuned for the cup
Sweeping a brass bell's two flare parameters found a shape that makes the resonance series harmonic to four and a half cents. Bolt the mouthpiece that instrument is actually played with onto it and the series is twenty-five cents out — worse than a plain cone. The flare and the cup are not two independent choices, and the best bell for a maker to build is one that is deliberately wrong.
A bar's partials are the odd numbers, squared
A string's wave equation is second order in space and a bar's is fourth, so a string's frequencies go as its wavenumbers and a bar's go as their squares. Both objects fit a nearly arithmetic sequence of wavenumbers between their ends; squaring one is the whole of why a marimba key has no note in it. The ratios are 1 : 2.756 : 5.404 : 8.933, which are the odd squares over nine, uniformly 12.9 cents flat, and the 12.9 cents is one root that refused to be 3π/2.
One cup and seven lengths
A trumpet is seven tubes with one mouthpiece serving all of them, and the harmonicity of its resonance series is a different number in every position — 19.5 cents open, 24.8 with all three valves down. The bare bore is flat across the same seven lengths to within eight-tenths of a cent, so none of it is a length problem. It is the cup, standing still while the series walks past it, and pressing all three valves does to the series exactly what fitting a mouthpiece 56 per cent of the catalogue depth would do.
The top that falls while the note lasts
Four earlier essays have asked where the harmonic series stops, and all four answered with a number computed for a tone that never ends. A struck C3 has 152 audible partials at the strike and eight after two-thirds of a second, so the ear's own resolution limit governs the first ten per cent of the note and the decay governs the rest. Playing ten decibels louder buys the ear a further tenth of a second, and doubling its reign would take fifty-eight.
A fifth on a piano is not a fifth a second later
Nine essays draw one note in silence, and a listener is given a texture. Put two struck notes an interval apart and consonance comes apart into two quantities that had agreed while nothing moved: the partial coincidence that names the interval outlives the strike in exactly the order common-practice theory ranks its intervals, and the roughness that scores it reorders itself inside a third of a second, with the fifth overtaken by four intervals every treatise calls harsher.
The tone on the root changes hands at the fifth
Every combination tone of a just interval is a harmonic of a fundamental neither note contains, and which harmonic is fixed by the ratio. The difference tone lands on that fundamental for every interval up to the fifth; the cubic product lands on it for the fifth and every interval above except the minor sixth. So the loud product names the root of a narrow interval and the quiet one names the root of a wide one — and a just major seventh's difference tone is a note seven harmonics up that no keyboard has.
An inversion lasts as long as its outer sixth
A struck interval keeps the partial coincidence that names it for a time set by its ratio, and a chord is three intervals at once. Voiced over one bass and struck on a piano, a triad keeps the evidence of all three only as long as its weakest one lasts, and for an inversion that is the sixth on the outside: a major sixth lasts as long as a major third, a minor sixth dies first. So the major six-four and the minor sixth chord are the most durable voicings of their triads and the major sixth chord and the minor six-four the least — and unlike a dyad, a triad's inversions keep their order by roughness through almost the whole decay.
An open triad lasts as long as its tenth
Close, every triad's weakest link is a third or a sixth. Raise its middle note an octave and the link becomes a compound interval, and compound intervals do not last alike: a major tenth, 5:2, lives as long as a major sixth, while a minor tenth, 12:5, needs the twelfth partial and lives 0.47 seconds. So the spacing orchestration manuals recommend for a major chord in the bass is the longest-lived voicing a struck triad has, and the same spacing halves the life of a minor chord — and the six-four's advantage reverses.
A minor triad can be spaced to last
Two spacings of each triad, drawn side by side, say that opening lengthens a major chord and halves a minor one. Drawn over every arrangement of the three pitch classes within two octaves of a fixed bass, the asymmetry disappears: a minor triad reaches 1.26 seconds over C3 and so does a major one, with two spacings of each tied at the top. The short-lived chords were never the minor ones. They were the ones with a minor tenth on the outside, and a minor triad has three spacings that avoid it.
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.
The ghost bass drops a twelfth at a forte
Both crowds arrive at once and every member of both is a multiple of the same absent fundamental, so a listener is never given a choice between them — only a different subset of one harmonic series at every dynamic. Softly, the subset is an exact gapless series on three times the fundamental. Loudly, the difference tones fill in the low harmonics and no template on the higher note survives them. Between 62 and 72 decibels, depending on the interval, the note the crowd names falls by an octave or a twelfth, and the two qualities of third cross at different levels.
The played notes already name the ghost bass
The products of a just third's partials, fitted on their own, name a note a twelfth above the bass when the interval is soft and drop to the bass when it is loud. Put the two played notes back beside them and the drop disappears: the notes and their products name the bass at every dynamic, because the notes' own partials are harmonics of it already. What the dynamic changes is not which note is implied but how complete its harmonic series is — nine holes from the notes alone, four when soft, none when loud.
A seventh chord cannot be spaced to last like a triad
A struck triad keeps the partial coincidences of all its intervals for at most 1.26 seconds, whichever way it is spaced. Run the same census over every inversion and spacing of five kinds of seventh chord and none gets near: the dominant, minor and half-diminished sevenths top out at about two thirds of a second, the major seventh at 0.42. The diminished seventh, which theory calls the least stable of them, lasts longest at 0.86 — because it is the only one with no tone or semitone among its pitch-class distances, and the worst distance a chord contains sets a ceiling no spacing can lift.
A tempered third cannot be one sound with its own ghost
The essay before this one fitted one harmonic template to a just third's played partials and the products of those partials together, and it worked at every dynamic. Tempered, it does not work at all — and not by naming a different note. There is no template. The lower note's partials sit exactly on their harmonics, the upper note's are every one fourteen cents out, and the products are twenty-three cents out the other way, a span of thirty-seven cents across a tolerance of thirty. The window is nine cents wide when the third is soft and two when it is loud, so equal temperament's fourteen falls outside it — while the fifth's two cents fall inside. The consonances a temperament leaves nearly alone are exactly the ones that stay single sounds.
The seventh has two places to be
Tune a dominant seventh's seventh as the seventh partial and its coincidences last 0.87 seconds instead of 0.66 — exactly the diminished seventh's figure, which is the highest any four-note chord in this census reaches. It gets there by levelling rather than by improving: the root's pair with the seventh gains a third of a second and the fifth's loses one, and all three pairs that contain the seventh arrive at the same number. The route is a valley. A seventh tuned six cents flat has lost the ninth-partial coincidence and not yet found the seventh, and lasts 0.46 seconds, which is worse than leaving it alone. And the same retuning halves the minor and half-diminished sevenths.
A ghost bass an octave lower needs notes twice as long
A scale played in just thirds has a bass underneath it that nobody plays — the difference tones of its thirds spell a diatonic line. Every essay on that bass has held its intervals still. A moving passage gives the bass a new pitch every note, and a low pitch is not there until it has run a few of its own cycles. With four cycles, a passage in thirds from C4 keeps its bass for 83 per cent of each note at three notes a second and loses it completely below 73 milliseconds a note. An octave down, every number doubles: the passage from C3 has no bass at all in its lowest note at semiquavers, and it is the scale's one Pythagorean third that sets the limit.
Four notes cannot get below the seventh partial
Tune a dominant seventh as 4 : 5 : 6 : 7 — every note moved, not only the seventh — and its coincidences last 0.868 seconds, which is the one-note retuning's figure to the millisecond. Tuning the third and fifth buys nothing at the top and costs seven voicings a fifth of their life. Every other chord of four notes a harmonic series holds does worse, and the reason is not temperament: no four pitch classes avoid a semitone, a tone and a tritone together, and none of those three can be tuned to coincide below the seventh partial. The half-diminished seventh, which the one-note retuning ruined, reaches the same ceiling read down the series.
The notes name the ghost bass before it has a cycle
A ghost bass that had to wait for four of its own cycles needed 61 milliseconds under a third from C4, and so vanished from fast passages. A pitch mechanism fitting the whole object — both notes' partials, which run their cycles many times faster, and the products — names the same bass in five. Seven of the eight thirds of a just scale have their bass for most of every note at any tempo a player reaches. The eighth, the scale's Pythagorean D–F, never has its own: the object hears it as a 6 : 5 and supplies the bass a D a comma lower would give, a semitone above the difference tone, until the difference tone arrives and breaks the fit.
A tuned seventh chord does not beat; it buzzes
Three four-note chords share a ceiling of 0.87 seconds on how long their partial coincidences last, and nothing in that measure could tell them apart. Their beating can. A chord tuned exactly to one harmonic series is one periodic sound, so no two of its partials can differ by less than the series' fundamental: the tuned dominant seventh has no slow beating at all, where the tempered diminished seventh has 125 beating pairs at the strike. Its roughness is another matter. Tuning removes the beats and changes the roughness by a few per cent, in either direction, and a seventh on its way to 7 : 4 beats more, not less, until it arrives.
A tempered seventh chord already has the tuned one's low note
A dominant seventh tuned 4 : 5 : 6 : 7 is one periodic sound, and its period is a note below the chord that nobody plays. The question left open was whether tuning adds that note. Run through the temporal account of residue pitch, the tempered chord turns out to repeat at the same place, a few cents sharp, at 0.82 of the tuned chord's exactness at the strike and 0.96 a second later. Tuning does not create the low note. It makes it exact, and it gets there by a route on which every step helps, which is not true of the beating.
A tuned close seventh chord low on the keyboard flutters at one rate
A dominant seventh tuned 4 : 5 : 6 : 7 in close position repeats exactly at a frequency two octaves below its bass, and below B2 that frequency is too slow to be a pitch. The question left was whether the repeat is heard as a flutter. In every auditory filter the chord reaches, its envelope pulses at that one rate, forty times over the threshold for detecting a fluctuation in the filter where it is strongest. The tempered chord fluctuates too, but a third as deeply and at rates that change from filter to filter. Tuning adds something here rather than sharpening it, and it adds it only within two cents of 7 : 4.
A leadpipe tunes the series, and its average does not
Every trumpet drawn in the essays before this one runs at full bore from the mouthpiece to the flare, and no trumpet does. Put a real taper in front of it and the wall losses it was predicted to cost come to a fraction of a per cent. What it does instead is pull the resonances onto a harmonic series, halving their average mistuning, while a cylinder of the same mean bore makes them worse.
A tuned chord's flutter binds it only if its notes start together
A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates at one rate in every auditory filter, and one rate everywhere looked like the ear's evidence that a sound is one thing. It is only half of that evidence. The filters also have to rise and fall at the same moments, and whether they do depends on when each note starts — which tuning does not decide. Spread the notes' onsets by two milliseconds and the tuned chord's filters move together no more than a tempered chord's.
A tuned chord stays in step for half a cent-second
A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates in every auditory filter at once, but only if its notes start within a millisecond of each other — and a tuned chord, it was said, keeps its phases for as long as it sounds. It keeps them only as long as it stays exactly tuned. Mistune its notes by a fraction of a cent and they slide against each other, and the chord drifts out of step in a time inversely proportional to the mistuning: about half a cent-second. A tenth of a cent holds it for four seconds, a cent for half a second. A chord that starts out of step never drifts into it.
A shared vibrato cannot hold a tuned chord together
A tuned close seventh chord binds its auditory filters together only while its notes stay within a third of a millisecond of each other, and mistuned by a fraction of a cent it drifts out of step in half a cent-second. A vibrato shared by every voice looked like a way to hold it: if all four notes swing together, they slide together. They do not stop sliding. A perfectly locked vibrato leaves the drift exactly as fast as before, because it changes every note's frequency by the same factor and leaves their differences alone; a vibrato locked ninety-eight per cent takes an exactly tuned chord out of step in six seconds, and one locked ninety per cent in one.
A chord comes back into step too briefly to bind
A vibrato that is shared imperfectly does not carry a tuned chord's notes ever further apart: each note's time is warped back and forth, and the chord passes back through step twice in every swing. Locked ninety-seven or ninety-eight per cent, it does so about ten times a second. But each return lasts on average less than one cycle of the chord's own flutter, which is the rhythm the ear's filters would have to share to show the binding — fewer than one return a second lasts that long. The chord is in step often and never for long enough.
The venturi is heard by every resonance
The narrowing makers put just inside a trumpet's leadpipe was predicted to be invisible to the written middle and to act only at the top of the compass, because short tapers had acted only there. It is not a taper. To every resonance in the compass a venturi a few centimetres long is one lumped object, described by a single number — its length times its depth — and by where it sits. Four centimetres at a fifth of the radius moves the middle notes by up to eight and a half cents, raising some and lowering others, while the series as a whole stays as well tuned as before.
Two octaves up, a sung chord can bind at every swing
A vibrato's swing between two notes is the same number of milliseconds in every register; the tolerance for being in step halves each octave, and so does the period of the chord's own flutter. The first two make a scaling law — an octave higher is the same chord with its vibratos disagreeing half as much — and the third breaks the tie. Matched for time in step, the higher chord's returns carry twice as many flutter cycles: over C2 fewer than one return a second is long enough to bind, over C4 with the vibratos locked ninety-nine and a half per cent nearly every return is.
A narrow vibrato needs no lock
A tuned close seventh binds and releases at the vibrato's swings only where its notes' timing swings by about the width of its onset tolerance, and the swing is the vibrato's width times how far the singers' vibratos disagree. Down to a ninety-five per cent lock the two trade one for one: a 71-cent vibrato locked ninety-nine and a half per cent is the same chord as a 7-cent one locked ninety-five. Below that the trade fails, in the singers' favour. With no lock at all the swing stops growing with the disagreement, and a vibrato of three cents over middle C — twelve over the bass — binds the chord at most of its swings with each singer's vibrato its own.
A straight tone drifts out of step and stays there
A tuned close seventh binds — its filters flutter together — only while its notes keep their timing within a fraction of a millisecond, and a narrow vibrato keeps them there because a sinusoid's timing offset swings inside a fixed range. A straight-tone singer's pitch is never quite straight: it wanders slowly, a cent or less, at random. That wander's timing offset is a random walk. Over C4 a wander of half a cent takes the chord out of step within a fifth of a second and leaves it in step four per cent of the time, where a three-cent vibrato holds it nearly half. What matters is the wander's slow part, and adding a vibrato on top never helps.
Named alongside it
The objects these essays reach for when they reach for this one.
Just intonationPartialDominant seventhResidue pitchBorePeriodicityRoughnessAuditory filterBrassCombination toneDecayResonance