A major triad's combination tones are its own notes

Two notes a just interval apart are two harmonics of a fundamental neither contains, and the tone on the root changes hands at the fifth showed that every combination tone the ear makes from them is a harmonic of that fundamental too. A chord of three notes is three such pairs sharing one fundamental, and its combination tones are nine harmonics of it. Some of those harmonics are notes of the chord.

A just major triad on middle C is harmonics 4, 5 and 6 of C2. The cubic product of the third and the fifth, 2f₁ − f₂, is harmonic 2 × 5 − 6, which is 4: the root. The cubic product of the root and the third taken above the pair, 2f₂ − f₁, is harmonic 2 × 5 − 4, which is 6: the fifth. The ear makes, out of the chord, two tones that are the chord’s own outer notes, at exactly their frequencies, and adds them to the notes that were already there.

That is the kind of result that sounds like numerology until the reason is found, and the reason is short.

The middle note is the mean of the outer two #

Write the three frequencies of any three-note chord as a, b and c, low to high. The cubic product of the lower pair taken above it is 2b − a; the cubic product of the upper pair taken below it is 2b − c. The first is c exactly when 2b = a + c, and the second is a under the same condition. So a chord’s cubic products land on its own notes precisely when its middle note is the arithmetic mean of the outer two in hertz — when the three notes are evenly spaced in frequency rather than in pitch.

Consecutive harmonics are evenly spaced in frequency by definition: they are one unit apart. So every run of three consecutive harmonics has the property, and 4:5:6 is one. The major triad is the only common triad that can be built from three consecutive harmonics — as 4:5:6 in root position and as 3:4:5 with its fifth in the bass — and that is the sense in which the series is not a chord and yet produces this one.

A minor triad is 10:12:15. Its middle note is twelve units and the mean of the outer two is 12.5, so the landing fails by one unit: the cubic product of the lower pair lands a unit below the fifth and the cubic product of the upper pair a unit below the root. A suspended fourth, 6:8:9, has a middle note of eight against a mean of 7.5 and misses by one unit of its own. An augmented triad, 16:20:25, misses by one unit of its own. None of them is three consecutive harmonics, and none of them lands.

Tempered, the landing misses by one number #

Equal temperament moves the third of the major triad up by 13.7 cents and the fifth down by two, which on middle C leaves E4 2.82 hertz above the mean of C4 and G4. So the middle note is no longer the mean of the outer two, and the amount by which it is not — doubled, since both products use 2b — is the amount by which both products miss.

The cubic product of E4 and G4 is at 267.26 hertz and the chord’s root is at 261.63. The cubic product of C4 and E4 taken above the pair is at 397.63 and the chord’s fifth at 392.00. Both are 5.63 hertz too high, because each exceeds the note it missed by the same quantity, 2E − C − G. A tempered major triad of pure tones offers a beat of 5.6 per second on middle C, between two tones the ear made and two tones the air carried.

That rate is not the beat of any pair of notes in the chord. A tempered major third on C4, played with partials, beats at 10.4 per second between C’s fifth partial and E’s fourth; the minor third above it beats at 17.8 between E’s sixth partial and G’s fifth; the fifth beats at 0.88 between C’s third and G’s second. The triad’s product rate is none of these. It is the two thirds’ beat rates added together and divided by five, since the two thirds beat in opposite senses — a rate that exists only when all three notes are sounding, and that beats are arithmetic cannot find in any single pair.

The temperament in which the products land #

Just intonation lands the products and equal temperament misses by 5.6 hertz, and between the two lies a family of tunings in which the miss can be tuned away without making every interval just. A fraction of a comma set the meantone temperaments of three centuries on a single line, indexed by how far each narrows its fifths, and on that line the miss is a smooth function with one zero.

The zero is not where the third is pure. Quarter-comma meantone makes every major third exactly 5:4, but it narrows the fifth by 5.4 cents to do it, and a narrow fifth puts G below the place the mean needs it; the triad on middle C still misses, by 1.22 hertz. Narrow the fifths further and the third falls below pure while the fifth falls further still, until at a narrowing of 6.3 cents — 0.29 of a syntonic comma, with thirds 3.8 cents flat of just — the middle note is the mean again and the products land on the notes of every major triad the temperament can play.

That is close to a tuning with a name. Zarlino’s two-sevenths-comma meantone narrows each fifth by 6.1 cents, and in it the miss on middle C is 0.23 hertz, one beat every four seconds. Third-comma meantone overshoots, missing by 1.08 hertz the other way. The landing picks out a point on the meantone line nobody designed for it, and the tuning that sits nearest was designed for something else — a compromise between the fifth’s error and the third’s.

This is not evidence that anyone chose two-sevenths comma for its combination tones, and nothing on this page could make it evidence. What it adds is a third criterion to the two the meantone line is usually argued on: a regular temperament can make its fifths nearly pure, or its thirds pure, or its major triads’ products land, and the three are different points.

The minor triad misses in tune #

The major triad lands in just intonation and misses by a beat in equal temperament. The minor triad has no tuning in which it lands, and the amount by which it misses is not small.

A just C minor triad misses by one unit of 10:12:15, and that unit is C4 divided by ten: 26.16 hertz. The minor triad’s products miss its own notes by exactly the frequency of the fundamental its ratios imply — a just A♭ more than three octaves below middle C, a note that is not in the chord. It is one of the two fundamentals the root an ear supplies found the minor triad offering a pattern-matcher, a major sixth from the other and with no way to prefer either; the distortion account meets the same foreign fundamental, as the distance by which its products fail to close.

Twenty-six hertz is not a beat. The beat a tuner can actually use is slow, and a difference of twenty-odd hertz between two tones inside one auditory filter is heard as roughness rather than as a swelling. So in tune, a minor triad of pure tones offers a roughness between its products and its notes where a major triad offers nothing, and tempering does not remove it — it widens it to 31 hertz.

That is a difference between the two triads that is not a difference of interval content. Both contain a major third, a minor third and a fifth. What differs is the order of the thirds, and the order decides whether the middle note is the mean. Three is the largest agreeable number recorded that the two triads are asymmetric in a way nobody has fully explained away; this is one more asymmetry, and it is exact rather than statistical.

Across four chords and three positions #

The condition says what to expect of any chord in any inversion without drawing it: land when the middle note is the mean, miss by the shortfall otherwise.

Inversion changes the numbers because it changes which harmonics the chord is. The first inversion of the major triad, E4–G4–C5, is 5:6:8 — harmonics five, six and eight, with seven missing — so the middle note is six against a mean of 6.5 and the landing fails by a whole unit, which on this voicing is 65 hertz — far enough that no product comes within forty hertz of a note. The second inversion, G–C–E, is 3:4:5, three consecutive harmonics, and lands.

The minor triad lands in no position. Its first inversion is 12:15:20 and its second 15:20:24, and neither is consecutive. The suspended fourth misses by more than forty hertz in every position. The augmented triad misses by sixteen to thirty-three hertz in every position and tuning, which is the edge between a beat and a roughness — and the augmented triad is the one of the four whose notes are evenly spaced in pitch, which is exactly the spacing that does not help.

The six-four is the inversion that lands #

That the second inversion lands and the first does not is the finding on this page that cuts hardest against practice.

The first-inversion triad — the sixth chord — is the ordinary, stable, freely usable inversion in common-practice harmony, and the second inversion is the one that has to be prepared, resolved, and kept away from the end of a phrase. The inversion that cannot end a phrase already set three measurements against that practice: the six-four is the smoothest of the three positions by roughness, the best explained by a virtual-pitch model, and not distinguished from the first inversion by support from the bass’s harmonics. Every one of them favours the chord the practice restricts.

This is a fourth, and it points the same way. The six-four is the only inversion of the major triad whose products are its own notes, because 3:4:5 is a run of harmonics and 5:6:8 is not. Whatever the rule against ending on it is, it is not that the chord is acoustically less settled than the sixth chord, since by this measure it is more settled than the sixth chord. That essay’s conclusion — that the prohibition belongs to a two-voice skeleton in which a fourth over the bass is a dissonance — survives unchanged, and gains another measurement it has to overrule.

Four consecutive harmonics land everywhere #

The condition generalises past three notes. In a chord of consecutive harmonics every inner note is the mean of its neighbours, so every adjacent triple lands, and a chord of four consecutive harmonics has twice as many products on its own notes as a triad does.

The four-note chord in question is 4:5:6:7 — a major triad with a seventh harmonic on top, the dominant seventh with its seventh 31 cents flat of a keyboard’s. On middle C its cubic products land on all four notes: the third and fifth give the root and the seventh, the fifth and seventh give the third, and the root and third give the fifth. The equal-tempered dominant seventh lands on none of them, missing by 5.6 hertz on the lower pair of products and 11.8 on the upper pair. A five-limit seventh at 9:5 lands only two and misses by 13 hertz; one at 16:9 lands two and misses by 7.3.

So the chord in which every cubic product is a chord tone is the one with a seventh no keyboard has. Barbershop quartets are reported to tune their dominant sevenths close to 4:5:6:7 and to describe the result as a chord that “rings”. The partials of four consecutive harmonics coincide with each other everywhere as well, which is the usual and sufficient account of that sound; what this adds is that the ear’s own products coincide with the chord too, and that no other seventh chord in any common tuning has that property.

Up the keyboard, a beat becomes a roughness #

The miss is a frequency, so it scales with the chord.

A tempered major triad’s product beat is 1.4 per second on C2, 2.8 on C3, 5.6 on middle C and 11.3 on C5; it passes out of the beat band and into roughness between C5 and C6. So through most of the range chords are actually voiced in, the tempered major triad offers a countable beat between its products and its notes, slowest in the bass. The six-four voicing drawn, G4–C5–E5, misses by a little less than the root position on C4 does.

The minor triad lives mostly on the other side of the band. In just intonation it misses by 6.5 hertz on C2 and 13 on C3, which are beats, and by 26 on C4 and more above, which are not. So a minor triad in the bass beats at its products whether it is in tune or not, and in the treble it roughens. That is a register effect with the same shape a third is rougher in the bass found for intervals, running in the opposite direction: there the bass is where roughness lives, and here the bass is where the minor triad’s roughness slows into a beat.

What the landing assumes #

The frequencies of every product are arithmetic on the chord’s three frequencies, and nothing else is used. The condition — middle note the mean of the outer two — follows in one line from the definitions of the cubic products. Neither depends on a measurement.

What is assumed is that the ear generates the lowest-order products at all, and that it generates them from pairs of notes. Both are established for pairs of tones: the cancellation and emission measurements that the ear makes its own sound describes were made on pairs, and the cubic product is strongest when the upper primary is about 1.2 times the lower. The major triad’s two thirds are ratios of 1.25 and 1.2, which is where the cubic product is strongest — so the two products that land are the two the ear makes most readily. That is a satisfying agreement and not a computed level; nothing here says how loud a landed product is.

What the drawings of one chord cannot say #

The notes are pure tones. A real triad on any instrument has partials, and its pairs of partials coincide or beat long before any distortion product is involved — a tempered major triad played on strings beats between its partials at the rates given above, and those beats are far stronger than a product’s. The landing is a statement about what the ear adds to a chord, and on instruments with rich spectra what it adds is small beside what arrives.

A landed product adds nothing new. A product exactly on a note of the chord is energy at a frequency already present, so in just intonation the landing strengthens the root and the fifth without creating a sound; it becomes audible only as a beat once temperament separates it from the note. The claim that a just major triad sounds settled because its products land is a claim this arithmetic cannot support; what it supports is that the tempered triad offers a beat the just one does not.

The beat may not be heard through the products. A chord of sinusoids near 4:5:6 also has a waveform whose shape drifts at 2E − C − G, and an ear reading the timing of the waveform would hear the same rate with no distortion involved. The two accounts predict the same number. Cancelling the products with added tones and asking whether the beat survives would separate them, and has been done for pairs rather than for triads.

And no chord is only three notes for long. Doubling a note adds pairs, every added pair has products of its own, and a doubled third or fifth an octave away adds products that fall between the chord’s notes rather than on them. The census here is of close-position triads, which is the voicing the condition is cleanest for.

Whose chords #

The chords are the triads of common-practice harmony, and the inversions are the three positions that practice distinguishes. The finding that matters to that practice is the six-four’s, and it joins a list: on roughness, on virtual pitch and now on combination tones, the restricted inversion is at least as settled as the free one, and what restricts it is a contrapuntal rule about the interval over the bass rather than anything the three notes do together.

Still open: the line a passage of such intervals draws #

A triad’s products land on its notes because they are harmonics of its fundamental. A melody harmonised in parallel intervals is a succession of such pairs, each with its own fundamental, so its products trace a line of their own underneath the passage — one note per pair, moving as the pairs move. In just intonation that line is made of harmonics of a changing fundamental and may or may not be a line anybody would write; in equal temperament every note of it is displaced by the gearing the third sound gives. Whether the line a passage in thirds draws is a bass, which of the two products draws it, and what equal temperament does to it note by note, is the same arithmetic applied along a melody instead of within a chord.