Acoustics & Music Theory
why it sounds like that
Intervals and chords

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.
14 min read 5 figures The ear is not a microphoneSmall whole numbersWhat the listener supplies

Assumes: The ghost bass drops a twelfth at a forte · Three harmonics of the bass arrive before the bass

The ghost bass drops a twelfth at a forte put every product of every pair of partials of a just interval into one set and asked what pitch the set implies. For a major third, 5 : 4, the answer depended on the dynamic. Softly, only the cubic products clear their thresholds, and they are an exact harmonic series on three times the fundamental the ratio implies — a twelfth above the bass. Loudly, the difference tones fill in the low harmonics and the fit falls to the fundamental. The ghost’s pitch drops nineteen semitones as the passage gets louder.

That essay said plainly which simplification carried the result. The crowd was fitted on its own, with the two played notes’ partials set aside, and they are not set aside in an ear. As a major triad’s combination tones are its own notes found for pure tones, for a just p : q interval the partials of the lower note are multiples of p times the fundamental and those of the upper are multiples of q times it — the same fundamental every product is a multiple of. So the full evidence a listener receives is three sets of multiples of one number, and the played notes contribute exactly the low harmonics that the soft crowd lacks. The essay declined to run that fit because it depended on a decision it could not make: whether a pitch mechanism treats the loud notes and the faint products as one object or as a figure and a ground.

Both fits can be made on the same components, and the difference between them is the whole of the earlier result.

Put back beside its notes, the crowd names the bass at every dynamic. A just major third on complex tones, drawn on one axis of harmonic numbers of the fundamental its ratio implies: the partials of the two played notes, and the products of those partials that clear threshold, at 55 and 80 dB. At 55 dB the products alone name 3 times the fundamental with 0 empty slots; the products and the notes together name 1 times it with 4 empty slots, and the notes alone name it with 9. At 80 dB the products alone name 1 times the fundamental with 0 empty slots; the products and the notes together name 1 times it with 0 empty slots, and the notes alone name it with 9. The soft reading on a higher note exists only when the loud notes are set aside. Taken together, the products do not decide which fundamental is named; they decide how many holes its template has.
Fig. 1 A just major third on complex tones, drawn on one axis of harmonic numbers of the implied fundamental, at 55 and 80 dB: the played notes’ partials, and the products that clear threshold. Beneath, the template each reading fits. At 55 dB the products alone name three times the fundamental with no empty slot; the notes and products together name the fundamental itself, with four empty slots. At 80 dB both name the fundamental with none.

Two readings of one set of components

The interval is a major third, C4 against E4, with the fundamental its ratio implies at C2, 65.4 hertz. Each note has six partials falling as one over their number, and the products are the earlier essay’s: every difference tone and every cubic product of every pair of partials, kept when it clears its own threshold under the masking the primaries cast.

The ground reading fits a harmonic template to the lowest eight audible products and ignores the notes. That is exactly the earlier essay’s fit, and it reproduces its staircase.

The object reading fits the same template to the lowest eight members of the union of products and the notes’ own partials. Nothing else differs.

Both are the collection’s residue fit: it assigns rising harmonic numbers to the components, fits a fundamental by least squares, and among candidates that fit within thirty cents prefers the one that leaves the fewest harmonic slots unexplained inside its span. Since every component is a whole multiple of 65.4 hertz, every candidate is a whole multiple too, and the only question is which one.

The ghost bass drops when the passage gets louder. The note the whole crowd of products names, as a multiple of the fundamental the interval implies, against how loudly the interval is played. a major third: 2.9999999999999996 times the fundamental below 70 decibels and 1 times above it, a drop of 19 semitones; a minor third: 4 times the fundamental below 62 decibels and 2 times above it, a drop of 12 semitones; a fourth: 2 times the fundamental below 72 decibels and 1 times above it, a drop of 12 semitones. Softly, only the cubic products clear their thresholds, and they are an exact series on (2p − q) times the fundamental with no gaps in it. Loudly, the difference tones fill in the low harmonics, no template on the higher note can explain them, and the fit falls. Nothing about the interval has changed; the listener is simply being given a different subset of the same harmonic series.
Fig. 2 The earlier essay’s figure, for comparison: the multiple of the implied fundamental that the products alone name, against the dynamic, for a major third, a minor third and a fourth. Each is a staircase that steps down as the interval gets louder.

The object reading never drops

Fitted with the notes, the crowd names the fundamental at every dynamic, for every interval tried.

The ghost bass drops only when the notes are set aside. The multiple of the implied fundamental that each reading names, against the dynamic: the products alone (dashed), and the products with the played notes (solid). major third: the products alone name 3 times the fundamental at 38 dB and 1 times it once loud; the products with the notes name 1 times it at every dynamic; minor third: the products alone name 4 times the fundamental at 36 dB and 1 times it once loud; the products with the notes name 1 times it at every dynamic; fourth: the products alone name 2 times the fundamental at 50 dB and 1 times it once loud; the products with the notes name 1 times it at every dynamic.
Fig. 3 The multiple each reading names against the dynamic, for a major third, a minor third and a fourth: the products alone dashed, the products with the notes solid. The dashed lines step down from three, four and two times the fundamental; the solid lines sit on one throughout.

For the major third the products alone name three times the fundamental at the softest dynamic that delivers any and once it above about seventy decibels. The minor third’s products name four times it softly — the note two octaves above the bass — and the fourth’s name twice it. With the notes added, all three name once the fundamental from the quietest level to the loudest. There is no switch.

The reason is visible on the axis at the top of the essay. The lower note of a major third is the fourth harmonic of the implied bass and the upper note the fifth, so the two played notes’ partials sit at harmonics 4, 5, 8, 10, 12, 15, 16 and 20. No template on three times the fundamental can explain a component at harmonic 4 or 5 — they are not multiples of three — so once the notes are in the set, the twelfth is not a candidate at all. The soft crowd’s perfect series on the twelfth was perfect only because the components that contradict it had been left out.

So the earlier essay’s drop is a property of the figure-and-ground reading specifically. It is not wrong; it is conditional, and the condition is now a sentence rather than an assumption. A listener who hears the played notes as the thing and the products as something else is offered a ghost that jumps a twelfth with the dynamic. A listener who hears all of it as one sound is offered the same bass at every dynamic.

What the products do instead

If the products do not decide which note is implied, they are not doing nothing, and the fit says what they do.

The products fill the holes in the bass the notes already imply. For a just major third, the number of empty harmonic slots the template on the implied fundamental has to leave unexplained among the lowest eight components, against the dynamic. The played notes alone leave 9. With the audible products added, 9 at 36 dB, 4 at 44 dB, 4 at 52 dB, 4 at 60 dB, 4 at 68 dB, 0 at 76 dB, 0 at 84 dB, 0 at 92 dB. The note named never changes; how well supported it is rises as the products fill the slots between the notes' own partials.
Fig. 4 For the just major third, how many empty harmonic slots the template on the implied fundamental leaves among the lowest eight components, against the dynamic. The notes alone leave nine at every level. With the products, four between 44 and 68 dB, one at 70, and none from 76 up.

The two played notes alone already imply the bass — every one of their partials is one of its harmonics — but they imply it with gaps. Their lowest eight partials span harmonics 4 to 20 and fill eight of those seventeen slots, so a template on the fundamental has nine holes in it. That is a weak implication: a pitch mechanism asked to hear a fundamental beneath eight widely spaced harmonics with nine missing is being asked to do a great deal.

The soft products fill some of the holes. The cubic crowd at 6, 9, 12, 15 and 18 lands between the notes’ partials, and the lowest eight members of the union — 4, 5, 6, 8, 9, 10, 12 and 15 — leave four holes from 44 decibels to 68. At about seventy the difference tones arrive at the low harmonics, and from 76 decibels the lowest eight components are harmonics 1 to 8 with nothing missing. The note named has not moved; its support has gone from a sketch to a complete series.

That reframes the thing three harmonics of the bass arrive before the bass found — that the crowd names the bass at a lower dynamic than the bass’s own component needs. Read with the notes in, the crowd was never the source of the bass. The notes were, and the crowd is what makes their implication complete enough to hear.

The consequence for a passage is direct. The bass line under a passage in thirds found that a scale harmonised in just thirds draws a diatonic bass line through its difference tones, and the essays after it found that line fading in and out with the dynamic and jumping a twelfth when soft. Read as one object, every step of that passage names its bass at every dynamic, so the line is there at piano as well as at forte — thinner, with more of its series missing, but on the same notes. What a crescendo does to it is fill it in, not move it.

Across five intervals

The same two readings for a set of just intervals, at a soft and a loud dynamic.

Every interval's notes already name its bass, with holes the products fill. For five just intervals, the empty slots in the template on the implied fundamental: from the played notes alone, and with the products audible at 55 and 80 dB; beside each, what the products alone name. major third: the notes alone leave 9; with the products, 4 at 55 dB and 0 at 80 dB; the products alone name 3× and 1×; minor third: the notes alone leave 12; with the products, 6 at 55 dB and 1 at 80 dB; the products alone name 4× and 1×; fourth: the notes alone leave 6; with the products, 6 at 55 dB and 4 at 80 dB; the products alone name 2× and 1×; fifth: the notes alone leave 3; with the products, 3 at 55 dB and 2 at 80 dB; the products alone name 1× and 1×; major sixth: the notes alone leave 8; with the products, 6 at 55 dB and 2 at 80 dB; the products alone name 1× and 1×.
Fig. 5 For five just intervals, the empty slots in the template on the implied fundamental — from the notes alone, and with the products at 55 and 80 dB — with what the products alone name at each dynamic. The notes and products name the fundamental in every case.

Every interval’s notes name its implied fundamental, and they do it with a number of holes set by the ratio. A fifth, 3 : 2, leaves only three: its notes are the second and third harmonics, and their partials crowd the low series. A fourth, 4 : 3, leaves six; a major sixth, 5 : 3, eight; a major third nine; a minor third, 6 : 5, twelve. The count grows with the size of the numbers in the ratio, which is a restatement of why the small ratios are the stable ones in the series is not a chord — a small ratio puts its notes low in a series, where the harmonics are close together.

The products help unequally. For the minor third they halve the holes when soft, from twelve to six, and leave one when loud. For the major sixth, eight becomes six and then two. For the fourth they do almost nothing softly and leave four even when loud, because the fourth’s products fall largely on harmonics its notes already supply. For the fifth, three becomes two at the loudest. The intervals whose notes imply their bass weakly are the ones whose products do the most filling, with the fourth as the exception, and the fifth barely needs them.

The products alone, beside each row, show where the earlier essay’s staircase came from, and why the tone on the root changes hands at the fifth. The major third’s soft crowd names three times the fundamental and the minor third’s four; the fourth’s names twice it softly; the fifth’s and the sixth’s name the fundamental whatever the dynamic. The intervals whose soft crowd names a higher note are exactly the ones for which 2p − q, the multiple the cubic products sit on, is greater than one.

How many components the fit is allowed

The one free choice in the object reading is how many of the lowest components the template is fitted to, and the result is only interesting if it survives changing it. So the soft major third, minor third and fourth were refitted with anything from three to twelve components.

The object reading names the fundamental at every size from three to ten, for all three intervals, soft and loud. What changes is the count of holes, and it changes in the direction the resolved-harmonics argument would want. Fitted to its three lowest components the soft major third is harmonics 4, 5 and 6 — a complete stretch of the series, no holes at all — and the fundamental is as well supported as a missing fundamental can be from three components. With four to six components the major third has one hole; with eight, four; with ten, five; with twelve, nine. The minor third runs from one hole at three components to ten at ten, and the fourth from one to eight.

That is the reverse of what a sceptic of the object reading might expect. The fewer components a pitch mechanism uses, the more completely the soft notes and their products imply the bass, because the lowest few components of a third and its crowd are consecutive harmonics — the played notes at 4 and 5 and the first cubic product at 6. The holes come from reaching up the series into the region where the notes’ partials are spaced out and the soft crowd is thin. The only size at which the fit fails is twelve components for the minor third, where the highest partial is past the twenty-fourth harmonic the fit allows, and that is a limit of the search rather than a change of answer.

The ground reading does not change with the size either: three, four and two times the fundamental for the three intervals at every size tried. The difference between the readings is not a matter of how much of the spectrum is listened to. It is entirely a matter of whether the notes are in the set.

This is also the version of the result that connects to what is known about pitch from harmonics. Which harmonics carry the pitch found the dominant region for a residue pitch low in the series, in the first few resolved harmonics, and the note that is not there began from the observation that a fundamental is heard beneath consecutive harmonics even when absent. A soft just third supplies exactly that: three consecutive harmonics of its implied bass, two from the notes and one from the ear.

Which computation produced the numbers

The notes are C4 and E4, or the corresponding just intervals over the same implied fundamental of 65.4 hertz, each with six partials at one over their number. The products and their thresholds are the earlier essay’s: each pair of partials’ difference tone and cubic product at levels set by the weaker partial, compared against the threshold of hearing and the masking the primaries cast. When a harmonic number is reached by several pairs the loudest is kept. The residue fit uses the lowest eight components, harmonic numbers up to 24 and a tolerance of thirty cents, and counts empty slots between the lowest and highest harmonic assigned.

The played notes’ partials are all taken as audible. At the levels here they are tens of decibels above threshold, and a partial that is part of a played note is not in doubt in the way a product is.

What the fit takes for granted

That a pitch mechanism fits a template to the lowest few components. Eight is the earlier essay’s choice, made because the resolved region of the cochlea holds only a few. Fit more and the played notes’ higher partials dominate the set even further; fit fewer and at some dynamic the soft crowd might again supply most of the lowest components. The object reading’s stability is robust to adding components and not guaranteed against removing them.

That empty slots are the right measure of support. A template with four holes and one with nine name the same note in this fit, and whether a listener hears the implied bass at all depends on more than the count — on which harmonics are present and how strong they are. The third sound magnifies cents, not hertz showed how sensitive the products’ positions are to tuning, and a tempered interval has no single fundamental for all three sets to share.

That the notes are steady. Real notes have attacks in which their partials arrive at different times, and the products of partials that have not yet arrived do not exist yet. The two readings may briefly disagree at the onset of a note and agree once it has settled.

What neither reading can settle

Which reading a listener uses. That is the question the earlier essay could not decide from arithmetic, and it still cannot. What has changed is that the two answers are now known and differ in a checkable way: one predicts a pitch that jumps with the dynamic and one predicts a pitch that does not.

Whether the implied bass is heard at all. A template with nine holes names a note for a least-squares fit. Whether a listener hears a low C under a soft major third in the middle register is a perception experiment; a missing fundamental is heard reliably only when enough low harmonics are present and resolved, and the counts above say how many are.

Whose thirds

Paired voices in thirds — two horns, two clarinets, two sopranos — are among the commonest textures in Western writing, and players and singers in close harmony have long reported hearing a low tone beneath a well-tuned third. The arithmetic here says that tone is implied by the two notes themselves and made more complete by the products, and that its pitch does not depend on the dynamic unless the listener is somehow separating the notes from the rest of the sound. That separation is exactly what a player attending to their own part might do and a listener attending to the whole might not, which would make the same third a different object to the player and to the audience.

Still open: the tempered third, where the three sets stop sharing a fundamental

Everything above depends on a just interval, for which the notes’ partials, the difference tones and the cubic products are all multiples of one frequency. In equal temperament that stops being true: the upper note is fourteen cents sharp of 5 : 4, its partials are no longer multiples of 65.4 hertz, and each product lands a different number of cents from any harmonic — the magnification the earlier essays measured.

The object fit then has to choose between a fundamental that explains the notes and one that explains the products, and the dynamic decides how many of each are in the lowest eight components. The prediction is that a tempered third behaves like the figure-and-ground reading after all, because its products and its notes disagree about the fundamental, and that the note named softly and loudly will differ by the tuning’s error multiplied up the series. Running the same two fits with the upper note detuned is one parameter in functions that already accept it, and what would come out is whether just intonation is what lets a third’s notes and its ghost be heard as one sound.

Part 8 of 8

One essay in the series on combination tone. The essays either side of this one:

The ghost bass drops when the passage gets louder. The note the whole crowd of products names, as a multiple of the fundamental the interval implies, against how loudly the interval is played. a major third: 2.9999999999999996 times the fundamental below 70 decibels and 1 times above it, a drop of 19 semitones; a minor third: 4 times the fundamental below 62 decibels and 2 times above it, a drop of 12 semitones; a fourth: 2 times the fundamental below 72 decibels and 1 times above it, a drop of 12 semitones. Softly, only the cubic products clear their thresholds, and they are an exact series on (2p − q) times the fundamental with no gaps in it. Loudly, the difference tones fill in the low harmonics, no template on the higher note can explain them, and the fit falls. Nothing about the interval has changed; the listener is simply being given a different subset of the same harmonic series. The ghost bass drops a twelfth at a forte Part 7 — 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.

Where this goes next

The ghost bass drops when the passage gets louder. The note the whole crowd of products names, as a multiple of the fundamental the interval implies, against how loudly the interval is played. a major third: 2.9999999999999996 times the fundamental below 70 decibels and 1 times above it, a drop of 19 semitones; a minor third: 4 times the fundamental below 62 decibels and 2 times above it, a drop of 12 semitones; a fourth: 2 times the fundamental below 72 decibels and 1 times above it, a drop of 12 semitones. Softly, only the cubic products clear their thresholds, and they are an exact series on (2p − q) times the fundamental with no gaps in it. Loudly, the difference tones fill in the low harmonics, no template on the higher note can explain them, and the fit falls. Nothing about the interval has changed; the listener is simply being given a different subset of the same harmonic series. 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. Every product of every pair of partials is a harmonic of one absent note. A 4 : 5 interval on 261.6 and 327.0 hertz, just, each note carrying 6 partials at one over n, with the fundamental at 80 decibels. Every pair of partials makes its own difference tone, and there are 19 distinct frequencies among them. All of them are exact multiples of 65.41 hertz — the note neither instrument is playing — because partial j of a p·f₀ note and partial k of a q·f₀ note differ by (kq − jp)·f₀ whatever j and k are. The heavy line is what each product has to clear — the threshold of hearing at its own frequency, or the masked threshold the two primaries cast there, whichever is higher. 11 of the 19 get through. 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. A tempered interval moves its difference tone several times further than itself. For every interval inside the octave tuned to twelve equal steps, how far the difference tone f₂ − f₁ lands from where the just interval would put it, in cents, with the interval's own departure from just drawn as the thin bar beside it. minor second −198.0 (the interval −11.7); major second −35.5 (the interval −3.9); minor third −96.0 (the interval −15.6); major third +67.4 (the interval +13.7); fourth +7.8 (the interval +2.0); fifth −5.9 (the interval −2.0); minor sixth −36.7 (the interval −13.7); major sixth +38.8 (the interval +15.6); minor seventh −39.8 (the interval −17.6); major seventh +25.0 (the interval +11.7). The major third's product is +67.4 cents out and the minor third's −96.0, and the largest error is the minor second's, at −198: a product moves p/(p − q) times as far as the interval p:q that made it. 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.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

Combination toneDynamicsHarmonic seriesJust intonationMissing fundamentalResidue pitch