The tone on the root changes hands at the fifth

The third sound magnifies cents, not hertz treated a difference tone as a gauge: something whose position says how far an interval is from just. It is also a note. Tartini built a whole theory of harmony on the claim that the note it names is the bass the interval implies, and the claim can be checked, because the note is fixed by the same two numbers that fix the interval.

Put a just interval p:q on a fundamental of one unit that neither note contains. The lower note is harmonic q, the upper note harmonic p, and every combination tone the ear makes from them is a whole-number combination of p and q — so every one of them is a harmonic of the same fundamental. The difference tone f₂ − f₁ is harmonic p − q. The cubic product below the pair, 2f₁ − f₂, is harmonic 2q − p. The cubic product above it, 2f₂ − f₁, is harmonic 2p − q.

That is three integers per interval, and laid out for the whole octave they make a pattern nobody would draw from a table of ratios.

Up to the fifth, the difference tone is the fundamental #

Read down the square markers first. For the minor second, the major second, both thirds, the fourth and the fifth, the difference tone is harmonic 1: the fundamental itself. That is not a coincidence of these ratios but the definition of them. All six are superparticular — each is n + 1 over n — so p − q is always one, and the difference between the two notes is always the unit they are both multiples of.

So the claim Tartini’s harmony rested on is exactly true for the intervals he was mostly talking about. A just major third C4–E4 has its difference tone at C2, two octaves below its lower note; a just minor third E4–G4 has its difference tone at C2 as well, a major third and two octaves below its lower note, which is where the root of the C major chord that minor third belongs to sits. A fourth G3–C4 and a fifth C4–G4 also give C, at C2 and C3. Every consonance smaller than a sixth sounds the root of the triad it is part of.

The sixths and sevenths are not superparticular, and on them the difference tone drifts off the fundamental. The major sixth 5:3 has it on harmonic 2, still the root’s pitch class an octave up. The minor seventh 9:5 has it on harmonic 4, still the root’s pitch class. But the minor sixth 8:5 puts it on harmonic 3 — the fifth of the implied root, not the root — and the major seventh 15:8 puts it on harmonic 7.

Above the fifth, the cubic product takes over #

Now read the diamonds. The cubic product below the pair is harmonic 2q − p, and for the fifth that is one: 2 × 2 − 3. For the major sixth it is one, 2 × 3 − 5. For the minor seventh one, 2 × 5 − 9, and for the major seventh one, 2 × 8 − 15. The intervals for which it is one are the ratios (2q − 1):q — three to two, five to three, seven to four, nine to five, fifteen to eight — which is exactly the family of intervals from the fifth to just below the octave.

So the product on the fundamental changes hands, and it changes hands at a precise place. Below the fifth only the difference tone is the root; above it only the cubic product is; at the fifth they coincide on the same note, since 3 − 2 and 2 × 2 − 3 are both one. The fifth is the only interval inside the octave where both of the ear’s two distortions agree about which note is underneath.

The minor sixth is the single exception to the second half of the rule, and it is an exception of a mild kind. Its cubic product is harmonic 2, which is the root’s pitch class an octave above the fundamental rather than the fundamental. Of the ten intervals drawn, then, every one has a product on its root’s pitch class, nine have one on the fundamental itself, and the one that does not is the minor sixth.

At the fifth two products are one note, and temperament splits it #

The coincidence at the fifth is not only a curiosity of the numbering. p − q equals 2q − p only when p is one and a half times q, so the fifth is the one interval inside the octave at which two different combination tones fall on the same frequency. In just intonation they are a single component. Temper the fifth and they separate.

Two components 0.88 hertz apart beat at 0.88 hertz, and that number is not arbitrary. The gap between f₂ − f₁ and 2f₁ − f₂ is 2f₂ − 3f₁, which is exactly the rate at which the upper note’s second partial and the lower note’s third would beat if the tones had partials — the fifth’s own beat, the one the unison is the coarsest thing in the room found betrays a mistuning three times faster than a unison does. A tempered fifth of two pure tones, which have no partials to beat, still offers the beat of a tempered fifth, generated inside the ear from two products that just intonation would have laid on top of each other. On C3 it is 0.44 a second and on C5 1.77, doubling every octave like any other beat.

That is one explanation of a fact beats are arithmetic does not cover: that mistuned consonances between pure tones are heard to beat at all. It is not the only explanation. A pair of sinusoids near 3:2 also has a waveform whose repetition drifts at the same rate, and an ear reading the timing of the waveform would hear the same beat without any distortion. The two accounts predict the same number, so the number cannot choose between them; what would choose is cancelling the products with added tones, and seeing whether the beat survives.

The product on the root is the quiet one on wide intervals #

The two products are not equally available, and the ear makes its own sound described how they differ. The difference tone needs a loud pair and vanishes when the primaries are soft. The cubic product survives at moderate levels, but it is strongest when the two primaries are close — near a ratio of 1.2, which is between a minor and a major third — and falls away as they separate.

Put that beside the switch and the root-bearing product turns out to be the weak one at both ends. A third’s root is carried by the difference tone, which is audible only at forte; its cubic product, which is audible at mezzo, is harmonic 3 on the major third and harmonic 4 on the minor, so what a moderately loud C–E actually adds underneath is a G, not a C. A sixth’s or a seventh’s root is carried by the cubic product, which exists at moderate levels but is at a ratio of 1.6 to 1.9, far from where it is strong.

That is a statement about levels and none of the drawings here computes a level, so it has to be held as a direction rather than a result. The direction is clear enough to be worth stating: the ear names the root of a narrow interval when the interval is loud, and names the root of a wide one faintly at any dynamic.

How low the root lands #

Where each root-bearing product sits in frequency is fixed as simply as which harmonic it is. It is the lower note divided by q, so the larger the denominator of the ratio, the further below the interval the root is — an octave under a fifth, a twelfth under a fourth or a major sixth, two octaves and a third under a minor third, three octaves under a major second or a major seventh.

Spell every interval inside a C major chord around middle C and the registers follow. The thirds, the fourth, the major sixth and the minor seventh all put their root product on C2, at 65 hertz. The fifth and the minor sixth put it on C3, at 131. The major second and the major seventh put it on C1, at 33 hertz, which is the lowest note of a double bass with an extension. The minor second B3–C4 puts it at 16 hertz, which is not a pitch at all.

The threshold of hearing rises steeply over exactly that range. The quietest thing audible at 131 hertz is a tone of about 21 decibels; at 65 it is 37; at 33 it is 58; and below twenty hertz nothing is heard as a tone however strong. A distortion product is generated well below the level of the primaries that make it, so a product at 33 hertz has to clear a threshold nearly forty decibels higher than one at 131 before it exists for a listener.

So the register adds a second reason to the one above. The seconds and sevenths name their roots on notes that are at or below the floor of hearing, and the fifth, whose root sits an octave under its lower note, names it where the ear is most ready to receive it. The ordering is the ordering of two notes and a ratio — the smaller the numbers, the more audible the product that stands for them — arrived at through the threshold of hearing rather than through anything about agreeableness.

Geared backwards and harder #

The cubic product that carries the root of the wide intervals does not move the way the difference tone does. When the upper note of p:q rises, 2f₁ − f₂ falls, and in cents it falls p/(2q − p) times as far. For the fifth that is three, the same size as the difference tone’s gearing with the opposite sign. For the major sixth it is five, the minor seventh nine and the major seventh fifteen.

So equal temperament puts the root-bearing product of a major sixth 80 cents from the root the just interval implies, a minor seventh’s 151 cents, and a major seventh’s 186 — nearly a whole tone. Those are the wide intervals, and on them the product that names the root is the product that equal temperament moves furthest.

That reverses a habit. A player listening underneath a sixth for the root and hearing it flat would, reading the difference tone’s behaviour, raise the upper note. On the cubic product the same correction moves the root further flat still. Which way to move a finger depends on which product is being heard, and at a moderate dynamic on a sixth it is the backwards one.

Some products are notes no keyboard has #

The harmonics on the drawing so far have been ones a scale can name: 1, 2, 3, 4, 5, 6 and their doublings sit within a few cents of notes of twelve equal steps on the fundamental. The triangles and some of the squares land on harmonics that do not.

Eight products fall more than twenty cents from any keyboard note, and they fall on the harmonics the series is not a chord found to be the problem with deriving harmony from the series: the seventh, which is 31 cents flat of a minor seventh above the fundamental; the eleventh, 49 cents from anything; the thirteenth, 41 cents sharp. The major second’s cubic product is a septimal B♭ under a C–D; the minor third’s upper product is the same note; and so is the major sixth’s.

Only one of the eight is a difference tone, and it is the major seventh’s. A just major seventh C–B has its difference tone on the seventh harmonic of a C three octaves below — a B♭ a third of a semitone flat, sounding inside a chord that contains a B natural. Tartini’s claim that the third sound names the bass does not merely fail on this interval; it names a note the tradition has no staff position for.

That also marks the edge of the claim cleanly. The difference tone names a note of the scale for every interval but the major seventh, and the root for every interval up to the fifth. The seventh-harmonic product appears exactly where the ratio stops being small, and a theory of harmony built on the third sound runs out of notes at the same place a theory built on the series does, for the same reason.

Where the distortion and the pattern disagree about the root #

The ear supplies a root to an interval by a second route, and it is not a distortion. The root an ear supplies fits a harmonic template to the notes present and reads off the fundamental of the best fit — the residue, the note that is not there. For a just interval p:q that fundamental is the unit, always. The template names harmonic 1 for every interval in the octave.

The combination tones name harmonic 1 for nine of the ten intervals, by one route or the other. The minor sixth is where the two mechanisms come apart.

On the minor sixth the template says the root is at 100 hertz and the distortion says there is energy at 200 and at 300. The 200 is the root’s octave and a listener might fold it into the same pitch class; the 300 is the fifth above it. The two routes an ear has to a bass under an interval agree on every interval in the octave but this one, and the minor sixth is a consonance by every modern count — so the disagreement does not sort consonances from dissonances, and whatever the minor sixth’s product under a melody does, it does not do it because the interval is harsh.

The arithmetic and what it assumes #

Everything on this page follows from one assumption, and it is worth saying which. The intervals are just. A just interval is a pair of harmonics, so its products are harmonics too, and the question which harmonic has an integer answer. Tempered intervals are not pairs of harmonics of anything small, their products land between harmonics, and how far between is the gearing.

The just ratios used are the five-limit ones a table of intervals usually gives: 16:15, 9:8, 6:5, 5:4, 4:3, 3:2, 8:5, 5:3, 9:5 and 15:8. The minor seventh has a second common just form, 16:9, whose products are harmonics 7, 2 and 23, so its difference tone would be a septimal note and its cubic product the root’s octave — a different row with the same switch in it. The tritone is left out because 45:32 has no small products at all.

The positions of the products assume the ear’s nonlinearity generates the three lowest-order terms and nothing else. It generates higher orders too — 3f₁ − 2f₂ is routinely measured — and each of those is another harmonic of the same fundamental, so the pattern of which products are on the root would gain entries without losing any.

What the harmonic numbering cannot say #

It cannot say what is heard. A product on harmonic 1 is a physical component in the cochlea at the fundamental’s frequency, not a guarantee that a listener hears a bass. Whether it is heard depends on its level, and its level depends on the dynamic and on the ratio, neither of which the numbering contains.

It cannot say whether the root is heard as the root. A difference tone at C2 under a C4–E4 is a tone at the right frequency, and a listener may hear it as a low buzz, as a pitch, or not at all, and may assign it to the chord or to nothing. The claim that it functions as a bass is a claim about harmony and not about the cochlea.

And it says nothing about timbre. Real notes have partials, every partial of each note makes combination tones with every partial of the other, and those products are also harmonics of the same fundamental when the interval is just. So a pair of real notes fills the harmonic series underneath itself with distortion products at many levels, of which the three drawn here are the lowest-order ones between the fundamentals. That is a larger picture and the same arithmetic.

Whose theory this tests #

The theory is Tartini’s, published in the middle of the eighteenth century, in which the third sound of a consonance is treated as the bass the consonance implies. The tests here are the part of it arithmetic can reach. For the intervals up to the fifth the claim is exactly true in just intonation; for the sixths it is true of the quieter product; for the minor sixth it is off by a fifth; for the major seventh the difference tone names a note outside the scale. Rameau’s fundamental bass, which is a different construction and the one that won, has no such trouble with the minor sixth, because it was never derived from distortion — and it has its own trouble with the minor triad, which the root an ear supplies set out.

Still open: a chord’s products against a chord’s own notes #

Every interval here has been a pair. A triad is three pairs, and their products are harmonics of the triad’s own fundamental — which raises a question the dyads cannot: whether some of those products land not merely on the root’s harmonics but on the notes of the chord itself. For a just major triad, 4:5:6, the cubic product of the fifth and the sixth harmonics is harmonic 4, the root itself, and the cubic product of the fourth and fifth above the pair is harmonic 6, the fifth itself — so products of the chord land exactly on notes of the chord. What the minor triad does, what the inversions do, and what equal temperament does to a product that was sitting exactly on a note, are arithmetic on the same integers.