Intervals and chords

A combination-tone bass needs a forte

A scale in just thirds draws a diatonic bass line through its difference tones, and in sixths the cubic product draws one. Given the two published level laws, with their constants swept, the thirds' bass is not heard at all below primaries of about 66 dB and is heard whole only from 71 to 81. The cubic products are a different kind of object: the primaries mask them decibel for decibel as they rise, so no dynamic changes whether they are heard. Most of the thirds' inner line never is, and the sixths' bass needs a forte and a gentle law.

15 min read 6 figures The ear is not a microphone What the listener supplies Small whole numbers

Assumes: The bass line under a passage in thirds · The third sound magnifies cents, not hertz

The bass line under a passage in thirds found that a major scale harmonised in parallel just thirds puts a difference tone under every pair, and that the eight difference tones are a diatonic bass — C, A, C, F, G, F, G, C — made of the scale’s own notes. In sixths the same bass is drawn by the other product, the cubic one, because the cubic product of a pair is the difference tone of the same pair inverted. And under the thirds the cubic products draw an inner line with a quarter tone in it.

Every one of those lines was a list of frequencies. The essay ended on the quantity all four essays about the ear’s products had stopped at: which of the two products is actually there at a given dynamic, and how far above what a listener can hear. The published measurements give each product a level law, and each law has a constant nobody should trust to two figures. So the laws go in as stated and the constants are swept.

The thirds' difference-tone line, note by note, against the dynamic. Each note of the line the difference tone f₂ − f₁ draws under a scale in just thirds, as its level above the higher of the threshold of hearing and the primaries' masking, against the level of the primaries. The product sits 50 dB below the primaries at 60 dB and grows twice as fast as they do. C2 above the limit from 73.5 dB; A1 above the limit from 76 dB; C2 above the limit from 73.5 dB; F2 above the limit from 70 dB; G2 above the limit from 68.5 dB; F2 above the limit from 70 dB; G2 above the limit from 68.5 dB; C3 above the limit from 66 dB. — Fig. 1 Each note of the difference-tone bass under a scale in just thirds, as its level above the higher of the threshold of hearing and the primaries’ masking, against the level of each primary. At the middle constant, with the product 50 dB below the primaries when they are at 60 dB, the top note of the line is heard from 66 dB, the lowest note, A1, only from 76 dB, and below 66 dB not one note of the bass is above its limit.

Two level laws, and what each has to clear

The quadratic difference tone, $f_2 - f_1$ , is the product Tartini heard and the one that draws the thirds’ bass. Measurements from Zwicker’s onward agree on its behaviour in outline: at moderate levels it is weak, tens of decibels below the primaries, and it grows about twice as fast as they do, two decibels for every decibel the primaries rise. The distance below the primaries at a reference level is the constant, and it is swept here from 40 to 60 decibels at primaries of 60.

The cubic product, $2f_1 - f_2$ , behaves the other way. Goldstein’s measurements, and most since, found its level nearly independent of the primaries’: it keeps a roughly constant distance below them, growing one decibel per decibel, and it falls steeply as the interval between the primaries widens. Here that is two constants — the distance below the primaries at a ratio of 1.2, swept from 15 to 25 decibels, and how many decibels further below it sits for every tenth of ratio wider, swept from 5 to 20. Narrower than 1.2 the distance is held, because the law is a fall as the interval widens and says nothing about a gain as it narrows.

Each product then has two limits to clear. The first is the threshold of hearing at its own frequency, which matters enormously for the difference tone: the thirds’ bass runs from 54 to 131 hertz, where the threshold is between 21 and 42 decibels — the steep low end of the curve that makes a subito piano four seconds longer in the bass. The second is the masking the two primaries cast at the product’s frequency, which is what a product sitting just below the notes that made it has to rise above. The masking pattern is the one the loudness figures use, falling 27 decibels per Bark below a masker and more gently above it.

The thirds’ bass is a forte phenomenon

The difference tones under the thirds sit two octaves below the passage, far from the primaries’ masking, so the only limit they meet is the threshold of hearing. That makes their behaviour simple, and the figure above is the whole of it: each note’s sensation level rises two decibels for each decibel of the primaries, and each crosses its threshold at a dynamic set by its frequency.

Where each note of the thirds' difference-tone line is first heard, constant by constant. The lowest level of the primaries at which each note of the thirds' difference-tone line under a scale in just thirds clears both the threshold of hearing and the primaries' masking, for 3 settings of the published constants. 40 dB below at 60: 68.5, 71, 68.5, 65, 63.5, 65, 63.5, 61; 50 dB below at 60: 73.5, 76, 73.5, 70, 68.5, 70, 68.5, 66; 60 dB below at 60: 78.5, 81, 78.5, 75, 73.5, 75, 73.5, 71. A dash is a note never heard at any level up to 110 dB. — Fig. 2 The level of the primaries at which each note of the thirds’ difference-tone bass is first heard, for three settings of its constant. With the product 40 dB below the primaries at 60 dB the line is heard whole from 71 dB; at 50 dB below, from 76; at 60 dB below, from 81. Its lowest note, A1 at 54.5 Hz, is always the last to appear.

At the kindest constant the whole bass is heard from primaries of 71 decibels, and at the harshest from 81. At the middle one, no note of the bass is heard below primaries of 66 decibels, and the full line of eight arrives only at 76. Two violins playing mezzo-forte in a quiet room put something like 65 to 75 decibels at a listener nearby, so on these laws the bass line under a passage in thirds is absent at piano, fragmentary at mezzo-forte and whole at forte.

The order in which the notes arrive is the threshold’s order, not the line’s. The C3 under the top pair of the scale, at 131 hertz, is the first note heard at every constant; the A1 under the second pair, at 54.5, is the last, two and a half decibels after the C2s either side of it. So a passage played through a crescendo acquires its bass from the top down — the upper notes of the line first, the note that makes it a descending bass last.

The margins either side of those onsets are worth stating, because a note that clears its threshold by a decibel is not a note anyone reports. At the middle constant, with primaries at 65 decibels, the line’s best note is still a decibel short of threshold and its worst twenty-two decibels short. At 80 decibels every note is above threshold, the weakest, A1, by eight decibels and the strongest, C3, by twenty-nine. Because the difference tone grows two decibels for every one the primaries do, those fifteen decibels of crescendo move the whole line thirty decibels — from nothing at all to a bass comfortably above threshold, inside the span a player covers between mezzo-forte and forte.

That makes the difference-tone bass a property of the dynamic as much as of the tuning. The bass line under a passage in thirds showed the line is diatonic only in just intonation; the arithmetic here adds that it exists only at forte. A performance in just thirds at piano has no Tartini bass, and the ear’s product that would draw it is thirty or forty decibels short of being heard.

The cubic products do not care how loud the passage is

The cubic products sit close under the primaries, and there the second limit takes over.

A cubic product and the masking over it rise together. The cubic product B♭3 under the pair 294 and 349 Hz in a scale in just thirds, against the level of the primaries: the product's own level, the masked threshold the primaries cast at its frequency, and the threshold of hearing there. The product and the masking both rise one decibel per decibel, so from 40 to 90 dB the product stays 6.0 dB below the masked threshold. Its audibility is decided by its distance below the primaries and by nothing a player can change by playing louder. — Fig. 3 One note of the thirds’ cubic line — B♭3, under the pair D4 and F4 — against the level of the primaries: its own level, the masking the primaries cast at its frequency, and the threshold of hearing there. The product and the masking rise together one decibel per decibel, and the product stays 6.0 dB below the masked threshold from 40 dB to 90.

The product’s level rises one decibel for each decibel of the primaries, because that is what the published law says of the cubic product. The masked threshold at its frequency also rises one decibel per decibel, because the masking a tone casts below itself falls at a fixed 27 decibels per Bark whatever its level. Two lines rising at the same rate never cross. Once the primaries are loud enough that their masking exceeds the threshold of hearing — true of this pair from the quietest level drawn — the product sits a fixed number of decibels above or below its limit, and no dynamic a player can choose moves it.

That is a structural result rather than a numerical one, and it survives every setting of the constants. What the constants decide is which side of the limit each product is on; what they cannot do is make a product audible at one dynamic and inaudible at another, as long as masking is the limit.

Where each note of the thirds' cubic line is first heard, constant by constant. The lowest level of the primaries at which each note of the thirds' cubic line under a scale in just thirds clears both the threshold of hearing and the primaries' masking, for 9 settings of the published constants. 15 dB below, −5 per 0.1: never, never, 26, 28.5, 27, 23, 21.5, 24; 20 dB below, −5 per 0.1: never, never, never, never, 32, 28, 26.5, 29; 25 dB below, −5 per 0.1: never, never, never, never, never, never, never, 34; 15 dB below, −10 per 0.1: never, never, 26, 31, 29.5, 23, 21.5, 26.5; 20 dB below, −10 per 0.1: never, never, never, never, never, 28, 26.5, 31.5; 25 dB below, −10 per 0.1: never, never, never, never, never, never, never, 36.5; 15 dB below, −20 per 0.1: never, never, 26, never, never, 23, 21.5, 31.5; 20 dB below, −20 per 0.1: never, never, never, never, never, 28, 26.5, 36.5; 25 dB below, −20 per 0.1: never, never, never, never, never, never, never, never. A dash is a note never heard at any level up to 110 dB. — Fig. 4 The level of the primaries at which each note of the thirds’ cubic inner line is first heard, for nine settings of the two constants. A dash is a note never heard at any level up to 110 dB. Where a note is heard, it is heard from a low level — between 21.5 and 36.5 dB — and from then on at every dynamic; at the middle setting five of the eight notes are never heard at all.

The inner line under the thirds comes out mostly silent. At the middle setting — 20 decibels below the primaries at a ratio of 1.2, ten more for each tenth wider — only the three notes under the top of the scale are ever heard, and they are heard from primaries of 26.5 to 31.5 decibels, which is to say at every dynamic anyone plays. The other five, including the quarter-tone B♭ that made the line strange, are masked at every dynamic.

The pattern across the sweep has the same shape. Kinder constants add notes from the top of the line downward, and the two lowest notes, G3 and the quarter-tone B♭3, are never heard at any setting; harsher constants remove notes until at 25 decibels below the primaries at most one survives. The inner line with a quarter tone in it is, at every plausible constant, the part of the passage’s distortion that is not there.

The sixths’ bass needs a forte and a gentle law

The cubic products under a passage in sixths sit much further below their primaries — a sixth’s cubic product is two thirds of an octave below its lower note or more — and they land in the bass, from 87 to 196 hertz. Two things change at once. The masking no longer reaches them, so the threshold of hearing is their limit again and the dynamic matters. And the interval is wide, 1.6 to 1.69, so the steep fall with ratio takes most of their level away before anything else does.

The sixths' cubic line, note by note, against the dynamic. Each note of the line the cubic product 2f₁ − f₂ draws under a scale in just sixths, as its level above the higher of the threshold of hearing and the primaries' masking, against the level of the primaries. The product sits 20 dB below the primaries at a ratio of 1.2 and 10 dB further below for every 0.1 the ratio widens. F2 never above the limit; G2 never above the limit; C3 never above the limit; A2 never above the limit; C3 above the limit from 88 dB; F3 above the limit from 77 dB; G3 above the limit from 75 dB; F3 above the limit from 83.5 dB. — Fig. 5 Each note of the cubic bass line under a scale in just sixths, as its level above its limit against the level of each primary, at the middle constants. The four lowest notes are never above their limit at any dynamic up to 110 dB; the upper four are heard from 75, 77, 83.5 and 88 dB.

At the middle constants the four lowest notes of the sixths’ bass are never heard, and the upper four only from primaries of 75 to 88 decibels — a fortissimo line with its bottom missing.

Where each note of the sixths' cubic line is first heard, constant by constant. The lowest level of the primaries at which each note of the sixths' cubic line under a scale in just sixths clears both the threshold of hearing and the primaries' masking, for 9 settings of the published constants. 15 dB below, −5 per 0.1: 68, 65.5, 56.5, 64.5, 60, 52, 50, 55; 20 dB below, −5 per 0.1: 73, 70.5, 61.5, 69.5, 65, 57, 55, 60; 25 dB below, −5 per 0.1: never, 75.5, 66.5, 74.5, 70, 62, 60, 65; 15 dB below, −10 per 0.1: never, never, never, never, 83, 72, 70, 78.5; 20 dB below, −10 per 0.1: never, never, never, never, 88, 77, 75, 83.5; 25 dB below, −10 per 0.1: never, never, never, never, never, 82, 80, 88.5; 15 dB below, −20 per 0.1: never, never, never, never, never, never, never, never; 20 dB below, −20 per 0.1: never, never, never, never, never, never, never, never; 25 dB below, −20 per 0.1: never, never, never, never, never, never, never, never. A dash is a note never heard at any level up to 110 dB. — Fig. 6 The level at which each note of the sixths’ cubic bass is first heard, for nine settings of the two constants. With a gentle fall of 5 dB per tenth of ratio the whole line is heard from 68 or 73 dB at the two smaller distances, and all but its lowest note at the largest; with a fall of 10, only its upper half or less, from 70 to 88.5 dB; with a fall of 20, not a single note at any level.

Swept, the sixths’ bass is the most fragile of the three lines and the one whose existence depends most on a constant. If the cubic product falls gently as the interval widens, the line is heard whole, or all but its lowest note, from forte up. If it falls at twice that rate, only the notes nearest the passage are heard, and only at fortissimo. At four times that rate the line does not exist at any dynamic.

The published direction is a steep fall, and that is the direction that removes the line. The tone on the root changes hands at the fifth found the cubic product naming the root of the wide intervals and the difference tone the root of the narrow ones, and that finding is about which harmonic each product is; the level law says that for intervals as wide as sixths the product naming the root is the one least likely to be heard.

What inverting a duet does

The earlier essay drew one conclusion from the fact that a pair’s cubic product is its inversion’s difference tone. Inverting a passage does not move the bass it implies, it moves the mechanism that supplies it — a composer who swaps the voices changes which of the ear’s distortions is doing the work and leaves the implied harmony where it was.

The level laws refine that sentence into a sharper one. The implied bass is the same in thirds and in sixths, note for note, but the mechanism that supplies it in thirds is heard whole from about 76 decibels and the one that supplies it in sixths needs at least as much and a favourable constant besides. A duet that moves from thirds into sixths at forte keeps its bass only if the cubic product’s fall with interval width is gentle; at the steeper falls the measurements report, it loses the bass at the moment of inversion, whatever the implied harmony.

So two voices that stop being two have a third voice under them in one texture and not the other, and the difference is not in the notes. It is in the ear, and it is in whether the passage is loud.

The laws and the limits the lines rest on

The primaries are the eight pairs of a major scale on middle C in parallel diatonic thirds or sixths, in five-limit just intonation, each note a pure tone at the level on the horizontal axis. Each product’s frequency is the one the third sound magnifies cents, not hertz computes, and the same integers that put a major triad’s combination tones on its own notes put the thirds’ products on the scale’s. Its level follows the stated law for its order: the difference tone at the primaries’ level minus the constant minus the primaries’ shortfall below 60 decibels, the cubic product at the primaries’ level minus the constant minus the fall for its interval’s width above a ratio of 1.2. Its limit is the larger of the ISO threshold of hearing at its frequency and the masked threshold each primary casts there, and a note is heard from the lowest primary level, in half-decibel steps, at which its level exceeds its limit.

What the laws do not carry

Pure tones. The measurements behind both laws are of two sine waves. Two violins have partials, the partials make their own products, and in just intonation all of them are harmonics of the same implied fundamental. A real passage in thirds puts energy on several harmonics of each implied root, and a residue pitch can supply a bass whose own fundamental component is below threshold. The arithmetic here prices the fundamental component and not the pitch.

One listener’s laws. The distortion products vary widely between listeners and fall with age and with any hearing loss. A sweep over three and nine settings covers the published spread in outline and says nothing about any one ear.

A masking pattern built for loudness. The spread of masking used here is the one the loudness figures use, fitted to broadband and tonal maskers at moderate levels. Its lower skirt is the steep part of the pattern and the best measured, which is the part the cubic products meet, but a pure-tone masker’s pattern is narrower in detail than a smooth slope.

And no room. A room’s reverberation smears the primaries in time and adds nothing at the products’ frequencies that the ear did not generate, so it should not help a product; it does lower the direct level at a distance, which moves every onset in this essay toward the loud end.

What a threshold cannot tell a listener

Whether a heard product is heard as a note. A difference tone ten decibels above threshold under a passage of quavers is a component too short to have a pitch, and a bass line made of them is a bass line only at a slow tempo. Clearing a threshold is necessary and it is not sufficient.

Whether players balance for it. A player who hears the combination tone — and string players and flautists tuning thirds often report hearing it — may adjust the thirds toward just intonation to steady it, and doing so moves the line by several times the adjustment. The pull runs against the one the note that is sharp because of where it goes describes, which widens a third under a leading note and so moves the bass away from the scale. Whether that makes the bass more or less audible depends on where the player was.

And what it adds to the harmony. A bass that is present at forte and absent at piano would make the same passage harmonically fuller when loud, which is a plausible description of how parallel thirds sound and not one any measurement here tests.

Tartini’s bass, again

Tartini used the third sound as a tuning aid for violinists and built a theory of harmony on it, and a violinist playing double stops close to the ear is the listener in the most favourable position this arithmetic knows: the instrument’s own two strings at very high level a few centimetres away. At that distance the primaries are far above the levels in the figures, and the difference tone under a third is well above threshold at any dynamic a player uses. The same passage heard from the hall is the passage in the figures, and there the bass is a forte effect.

That is a fair reading of the historical disagreement about whether the third sound is a musical fact or a laboratory curiosity. It is both, and which one depends on where the listener is standing and how loudly the passage is played — which is also why the reports of it come overwhelmingly from players rather than from audiences.

Still open: whether the partials rescue the bass

Every product here is between two pure tones, and every instrument that plays thirds has partials. The products of each partial of one note with each partial of the other are harmonics of the same implied fundamental when the interval is just, so a real passage puts the missing fundamental’s harmonics into the ear several times over, some of them at frequencies where the threshold of hearing is far lower than at the fundamental. The root an ear supplies describes how a pitch forms from harmonics whose fundamental is absent. Running the two level laws over every pair of partials of a violin-like spectrum would say whether a passage in thirds at piano — which delivers no difference tone at its fundamental — delivers enough of the bass’s upper harmonics above threshold for a residue pitch to form, and so whether Tartini’s bass is heard at a dynamic where its own component is silent.