Three harmonics of the bass arrive before the bass

The essay that priced the two level laws put two published level laws into numbers and came to a discouraging answer. A major scale harmonised in just thirds draws a diatonic bass line through its difference tones, and at any dynamic softer than about 66 decibels that line is not there at all: the product sits under the threshold of hearing at its own frequency, which is a low frequency, where the threshold is expensive. Most of the inner line a listener is supposed to hear, on those numbers, is never delivered.

Every product in that calculation was made from two pure tones. A violin is not a pure tone, an oboe is not, and a voice is not. Give each of the two notes a spectrum and the ear no longer receives one pair of primaries; it receives every pair — partial three of the lower note against partial two of the upper, and thirty-five other combinations — and each pair makes its own products.

For a just interval that turns out to be an unusually tidy object, and the tidiness is the whole of this essay.

One line of arithmetic #

Write the interval as $p : q$ and let $f_0$ be the fundamental it implies, so the lower note is $p f_0$ and the upper is $q f_0$. Partial $j$ of the lower note sits at $j p f_0$ and partial $k$ of the upper at $k q f_0$, and their difference is

$$k q f_0 - j p f_0 = (kq - jp) \, f_0$$

which is an integer multiple of $f_0$ for every pair $(j, k)$. Every difference tone the ear manufactures out of a just interval played on complex tones is a harmonic of one absent note, and it is the same absent note in every case — the note the essay that found the two products changing places at the fifth identified as the interval’s own implied root. Six partials on each side give nineteen distinct multiples, spread from the fundamental itself up past twenty times it.

That is why the fifth essay’s problem might be solvable. Its difficulty was the threshold of hearing at 65 hertz, which is about 37 decibels; the threshold at 523 hertz is about 4. A product that lands eight harmonics up has more than thirty decibels of head start.

What the crowd costs to make #

The head start is not free, and the accounting is the reason this essay is a calculation rather than an observation.

A high-order product is made from high partials, and high partials are quiet. The eighth multiple of $f_0$ comes from partial three of the lower note against partial four of the upper, which are 9.5 and 12 decibels below their own fundamentals. The quadratic law makes matters worse than that: a difference tone’s level grows about twice as fast as its primaries’, so twelve decibels off the primaries is twenty-four off the product.

So the trade is thirty-two decibels of cheaper threshold against twenty-four decibels of quieter primaries, and it comes out eight decibels ahead. Not by much, and only for some of the orders — the ones near the primaries in frequency are heavily masked by them rather than limited by the threshold, and several of the low-numbered multiples are masked out entirely.

The net of all that is the question, and it has to be run rather than reasoned.

Three, not one #

With the constants, on a just major third whose implied fundamental is C2, the first product clears at 70 decibels, the crowd names C2 at 71, and the product at C2 itself clears at 74.

Two of those numbers are nearly the same and the reason is worth stating, because it nearly is the answer. The first product to get through does not name anything: a single component is a single component, and two are hardly better. Two tones a fifth apart are the second and third harmonics of one note, the fourth and sixth of the note an octave below, the sixth and ninth of the one below that, and nothing in the pair chooses between them. At 70 decibels the crowd consists of the eighth and eleventh multiples, and the template that fits them best names 103 hertz rather than 65 — a wrong answer with an empty slot in it.

A residue needs three, and the third arrives at 71. From there the fit is exact and unambiguous: the eighth, ninth and tenth multiples of one fundamental have no empty slots between them and admit no cheaper explanation.

An octave lower the margin opens up. With the implied fundamental at C1, 32.7 hertz, the crowd names it at 75 decibels and the component at 32.7 itself does not arrive until 85 — a ten-decibel rescue, which is most of the dynamic range between mezzo-piano and forte. That is the right direction: the lower the bass the interval implies, the more expensive its own frequency is and the more the crowd’s higher landings are worth.

At the top of the range drawn the two coincide. An implied fundamental of C3 is cheap enough at its own frequency that nothing is gained, and the fifth essay’s single-component calculation is the whole story there.

So the answer to the question that essay left is yes, and by three to ten decibels, with the gain concentrated in the register where it was needed. A passage in thirds at a dynamic that delivers no difference tone at the bass’s own frequency can still deliver a bass, by delivering that bass’s eighth, ninth and tenth harmonics instead.

The quiet product names a different note #

There is a second crowd and it behaves completely differently, which the fifth essay’s own results predict and which turns out to matter more than the first.

The cubic product of partial $j$ against partial $k$ sits at $(2jp - kq)f_0$, also a multiple of the implied fundamental. Its loudest members are the pairs where $j = k$, and for those the multiple is $(2p - q) \cdot j$ — so the cubic crowd is an exact harmonic series on $(2p - q) f_0$ rather than on $f_0$.

That is a different note for every interval. A fifth, $3:2$, gives $2 \times 2 - 3 = 1$, and its cubic crowd names the bass itself. A major third gives 3, a twelfth above the bass. A minor third gives 4, two octaves above. A minor sixth gives 2, an octave above.

And because the cubic product keeps a nearly constant distance below its primaries instead of growing at twice their rate, it is available where the difference tone is not. Three members of a major third’s cubic crowd are audible at forty decibels — thirty-one below where the difference-tone crowd first names the bass, and thirty-four below where the component at the bass’s own frequency arrives.

So the answer to the fifth essay’s question has two halves and the second is the surprising one. At a forte, a passage in just thirds delivers a crowd that names its bass. At a piano it delivers a crowd too, loudly and unambiguously, and that crowd names a note nineteen semitones higher.

Two ghost lines, and they are not the same tune #

A scale harmonised in thirds alternates major and minor thirds, and the two send the crowd to different places — nineteen semitones up and twenty-four. The consequence is that the quiet line is not the loud line transposed.

The loud line is C, C, F, G, F, G, C — it falls to the subdominant and returns. The quiet line is G, C, C, D, F, G, G, and it climbs almost throughout. The contour is not preserved, because the displacement changes step by step with the quality of the third.

So a player who softens a passage in thirds does not lose the inner voice; they change it. That is a strange claim and it should be read as what it is — a consequence of two published level laws applied outside the range they were measured in, described in the section below. What makes it worth stating is that it is not a matter of degree: the two lines have different notes, not merely different loudnesses, and nothing in the previous five essays would have suggested a dynamic could rewrite a melody.

What temperament does to the crowd #

The arithmetic at the top of this essay needs the interval to be just. Move the upper note by the thirteen and a half cents equal temperament puts on a major third, and $k q f_0 \cdot 2^{d/1200} - j p f_0$ is not a multiple of anything.

What happens is not the collapse that might be expected, and it is more interesting. Six of the twelve audible products — the ones made by a partial against the same-numbered partial of the other note — form an exact harmonic series of their own, because $k q \cdot 2^{d/1200} - k p$ is $k$ times a constant whatever $k$ is. The other six sit between 26 and 51 cents off that series and belong to nothing.

The series the tempered crowd forms sits on 62.83 hertz where the interval implies 65.41: 69.5 cents flat, more than half a semitone. That number is not new here. The essay that measured the third sound’s gearing established that a difference tone moves $q/(q-p)$ times as many cents as the interval that made it, which for a major third is five; thirteen and a half cents of temperament, magnified five times, is 68.4. The crowd inherits the gearing entire, because every member of it is built from the same displaced upper note.

So a tempered third does not lose its bass. It supplies one at the wrong pitch, with a scatter around it — and the wrong pitch is wrong by well over the quarter-tone at which a listener would call it a different note. That is a sharper version of what the essay that drew the ghost line when it reported the tempered ghost line wobbling by up to 84 cents from step to step: the line is not merely out of tune, it is out of tune coherently, and the coherence is what would let a listener hear it as a pitch at all.

The interval that lands on the bass is the one that cannot be heard #

There is one more inversion in the numbers and it is the neatest thing on this page.

The cubic crowd’s target is $(2p - q)f_0$, so the interval whose quiet crowd lands on the bass itself is the fifth. It is also the interval whose quiet crowd is hardest to hear. Three members of a major third’s cubic crowd are audible at forty decibels and three of a minor third’s at thirty-five — and a fifth needs seventy-three.

Two mechanisms push the same way. The cubic product falls steeply as its interval widens, which is the second of the two published laws and is why a fifth’s is much fainter than a third’s. And a fifth’s crowd lands on $f_0$ and its low multiples, where the threshold of hearing is at its most expensive, while a third’s lands three or four times higher, where it is cheap. The interval that aims at the right note aims it at the worst place and fires it weakly.

That bears directly on a practice these essays were written to explain. Eighteenth-century string players were told to tune by listening for the third sound, and the instruction is usually quoted about thirds, because a third is where a tuning error is largest and where the the essay that measured the third sound’s gearing does the most work. The arithmetic says something more specific: at the dynamic two players would actually use to tune — not a forte, since a tuning note is held quietly and listened into — the crowd a third delivers is loud and sits a twelfth above the bass, and the crowd a fifth delivers sits on the bass and is barely there at all.

Which means the note a player tuning by the third sound was nulling was probably not the note the treatises describe. It was a real pitch, it moved with the gearing, and it was as good a target for the purpose; it was simply an octave and a fifth above the fundamental everyone was writing about. Nothing here can establish that — it is a claim about what somebody heard two hundred years ago, from a model of what would have been audible — but it is a testable prediction about a passage anybody can play now.

Which computation produced the numbers #

Each note is given $n$ partials at one over $n$, normalised so the fundamental sits at the stated level. Every pair of partials is treated as a pair of primaries, and the level that pair can deliver is taken from the weaker of the two — a product cannot be built out of a partial that is not there. The two level laws are the fifth essay’s, with the same constants: the quadratic difference tone at $L - q_0 - (60 - L)$ and the cubic product at a nearly fixed distance below the primaries that widens steeply with the ratio.

A product is audible when it clears both the threshold of hearing at its own frequency and the masked threshold each of its two primaries casts there. Where several pairs make the same frequency, the loudest is taken, because that is what arrives.

Whether the audible set names a fundamental is decided by the residue template: rising harmonic numbers are assigned to the components in order, a fundamental is fitted by least squares, and candidates are ranked by how many harmonic slots they predict and nothing occupies. The crowd is said to name $f_0$ when the winning candidate is within thirty cents of it.

The cubic law is used the same way and its constant is swept the same way; the cubic crowd’s thirty-decibel advantage over the difference-tone crowd narrows to twenty and never closes, because the two laws differ in their slope against level and not only in their offset.

The constants are swept rather than trusted, as the fifth essay swept them, and the sign of the answer does not move. Taking the difference tone’s reference distance from thirty decibels to sixty moves every onset up by twenty decibels together and leaves the margin at three: the crowd names the bass at 61, 66, 71 and 76 decibels as the constant hardens, and the component at the bass’s own frequency arrives at 64, 69, 74 and 79.

Where the model stops #

The level laws were measured on pure tones, at moderate levels, on two primaries at a time. Applying them thirty-six times over inside one ear assumes the products superpose and that a partial pair’s contribution does not depend on what the other pairs are doing. The nonlinearity that makes the products is not linear in that sense by construction, so this is an approximation whose error is unknown rather than small.

And the products make products. If the ear’s nonlinearity turns two components into a third, it turns that third and a fourth into a fifth. Second-order terms are ignored here entirely, and since every component in the just case is a harmonic of the same fundamental, every second-order product would be too — which would strengthen the finding rather than weaken it, and is not counted.

The spectrum is one over n and flat across the compass. A real instrument’s roll-off varies with register and with dynamic, and a louder note is a brighter note on almost every instrument, so the crowd grows with the dynamic faster than this model says.

What the picture cannot show #

It cannot show whether a listener attends to any of it. A residue pitch forming in the arithmetic is a statement that the information is present. The eighth, ninth and tenth harmonics of an absent bass, sitting among two notes’ own partials and at a fraction of their level, are a needle in a considerable haystack, and nothing here models the attention that would have to find it.

Nor what the two notes’ own partials do to the template. The fit above is run on the products alone. In the ear they arrive alongside twelve partials of the two played notes — which, in the just case, are also harmonics of the same fundamental. That should help; it might instead make the whole complex fuse into one note with a bass, which is a different percept from a bass under two notes, and the spectrum that will not fuse is where the question of which happens belongs. The same ambiguity is what a just major triad’s own products run into when they land on notes the chord is already playing.

And it cannot show a real passage. Every figure here is one interval held. A passage in thirds changes chord several times a second, and a residue pitch needs some tens of milliseconds to establish; whether a bass that takes that long to form survives a moving line is the question the whole account has been circling and none of its essays has touched.

Still open: whether the crowd survives the passage #

The one thing the arithmetic here cannot supply is time. Every number above is a steady state: two notes held, a crowd of products in equilibrium, a template fitted to what is audible. A passage in thirds at a moderate tempo gives each interval two or three tenths of a second.

What that would need is not a new model but a clock on this one. A residue pitch has an establishment time, measured in the tens of milliseconds for a strong stimulus and longer for a weak one; the products themselves follow their primaries with no delay worth counting; and the crowd changes wholesale at every chord change, because the implied fundamental moves by a whole tone or a fourth. Running the fit over a moving window as the eight steps of the scale go past — with the window as the free parameter, and the tempo as the dial — would say at what tempo the bass stops being a line and becomes a blur.

That is worth doing because the answer bears on the claim the whole account rests on. A Tartini bass that exists only for held intervals at a forte is a laboratory object. One that survives a scale in thirds at a walking tempo is the thing eighteenth-century players said they were listening for when they tuned by it.