Intervals and chords

The bass line under a passage in thirds

A major scale harmonised in parallel thirds gives the ear a difference tone under every pair, and in five-limit just intonation those tones are a diatonic bass line — C, A, C, F, G, F, G, C — made of the scale's own notes. Tempered, the same line moves only by whole tones, a neutral third and a fourth stretched to 650 cents, and wobbles by up to 84 cents from note to note. In sixths the bass is drawn by the other product, because the cubic product of a pair is the difference tone of the same pair inverted.

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

Assumes: The tone on the root changes hands at the fifth · A major triad's combination tones are its own notes

A melody doubled a third below is one of the most ordinary textures two voices make. Every pair in it is an interval with a difference tone, and the tone on the root changes hands at the fifth found that for a just third that tone is the fundamental the third implies. A passage of thirds is therefore a succession of implied fundamentals, one per pair, and the ear puts a physical component at each of them. Played through in order, those components are a third line under the two that were written.

Nobody writes that line and nobody plays it. Whether it is a line anybody would write is a question the arithmetic can answer, because each of its notes is fixed by the pair above it.

A scale in parallel thirds has a line underneath it that nobody plays. A major scale on C4 harmonised in parallel diatonic thirds, with the difference tone f₂ − f₁ of each pair drawn as a third line. In five-limit just intonation that line is C2, A1, C2, F2, G2, F2, G2, C3. In equal temperament it moves to C♯2, A1, B1, F♯2, A♭2, E2, F♯2, C♯3, departing from the just line by +67, +33, −82, +69, +65, −80, −84, +67 cents. — Fig. 1 A C major scale harmonised in parallel diatonic thirds, the two voices at the top, with the difference tone of each pair drawn as a line underneath. In five-limit just intonation the line is C2, A1, C2, F2, G2, F2, G2, C3; the dashed line is the same passage in equal temperament, labelled with how many cents each note of it has moved.

In just intonation, the line is a bass

The just scale here is the five-limit major scale — C, D, E, F, G, A, B as 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 — and its diatonic thirds are the thirds that scale actually makes rather than a table’s ideal ones. C–E, F–A and G–B are 5:4. E–G, A–C and B–D are 6:5. D–F is 32:27.

Under C–E the difference tone is C2, two octaves below. Under D–F it is A1. Under E–G, C2 again; under F–A, F2; under G–B, G2; under A–C, F2; under B–D, G2; and under the returning C–E, C3. Every note of the line is a note of the C major scale, at the scale’s own just tuning, and read as a bass under the thirds above it the progression is a tonic, a six-four on the supertonic, the tonic again, and then subdominant, dominant, subdominant, dominant, tonic.

That is not a subtle harmonisation, and it is not a bad one. It chooses the subdominant under A–C rather than the submediant, and the dominant under B–D rather than the leading-note triad, which is to say it prefers major triads — because a difference tone of a 6:5 is the root of the major triad the minor third belongs to, not the root of a minor triad containing it. A major triad’s combination tones are its own notes is the same preference seen from inside a single chord.

Where the line sits

A difference tone of a third sits two octaves and a bit below the pair, and that places the line in a specific part of the range. A passage in thirds around middle C puts it around C2, at 65 hertz, which is the bottom of a cello. The same passage an octave up, as two sopranos or two flutes might play it, puts it around C3, at 131 hertz — the register a written bass line would occupy under a treble duet. Two violins in thirds on their A strings put it on A2.

The register matters for whether the line exists for a listener at all. The quietest thing audible at 131 hertz is a tone of about 21 decibels and at 65 hertz one of about 37, so the same product made two octaves lower has to be sixteen decibels stronger to be heard. The higher a passage in thirds is played, the more likely its bass is audible and the more nearly it sits where a bass belongs; the lower it is played, the further its line sinks below both the threshold and the texture.

The Pythagorean third names the odd note

One note of the line is not the root of a major triad, and it is under the one third that is not 5:4 or 6:5.

Every product of a just interval is a harmonic of the note it implies. Each interval drawn as two harmonics of a fundamental it does not contain — the lower note is harmonic q and the upper harmonic p — with its three combination tones placed on the same numbering: the difference tone at p − q, the cubic product below the pair at 2q − p, and the one above at 2p − q. major third 5:4: 1, 3, 6; minor third 6:5: 1, 4, 7; Pythagorean minor third 32:27: 5, 22, 37; major sixth 5:3: 2, 1, 7; minor sixth 8:5: 3, 2, 11; Pythagorean major sixth 27:16: 11, 5, 38. — Fig. 2 The thirds and sixths a five-limit major scale contains, each as two harmonics of a fundamental neither note has, with its combination tones on the same numbering. The major and minor thirds put their difference tone on harmonic 1; the Pythagorean minor third D–F puts it on harmonic 5. Among the sixths it is the cubic product that sits low, on harmonic 1 for the major sixth and harmonic 5 for the Pythagorean one.

D–F in the just major scale is 32:27, a comma narrower than a 6:5, and 32:27 is not superparticular: its difference tone is harmonic 5 of a fundamental far below. In pitch that is A1, the scale’s own just A. So the line gets a note that is diatonic but is not a root, and the chord it implies with the pair above it is a D minor triad over its fifth.

The comma that makes D–F narrow is the same one a second comma, arriving by a different road found separating four pure fifths from a pure major third, and it is the reason a just major scale has a bad minor third on its second degree at all. Its effect on the line is specific: the one Pythagorean interval in the scale is the one pair whose third sound is not the root of anything. A singer who tuned D–F as a pure 6:5 would move either D or F by a comma, and the line under it would move to B♭1, the root of a B♭ major triad that is not in the key.

Tempered, every note of the line moves

Equal temperament widens the major thirds by 13.7 cents and narrows the minor thirds by 15.6, and the difference tone gears each of those by five or six. The line under a tempered passage in thirds is the line above displaced note by note.

The tempered line underneath wobbles by up to a quarter of a tone. For each pair of a major scale in parallel thirds, how far equal temperament moves the difference tone from where just intonation puts it, as the wide bar, beside how far it moves the third itself, as the thin one. C: +67.4 against +13.7; D: +33.3 against +5.9; E: −82.4 against −15.6; F: +69.3 against +13.7; G: +65.4 against +13.7; A: −80.4 against −15.6; B: −84.3 against −15.6; C: +67.4 against +13.7. — Fig. 3 How far equal temperament moves each note of the line from where just intonation puts it, as the wide bars, beside how far it moves the third above it, as the thin ones. The major thirds’ notes rise by 65 to 69 cents, the minor thirds’ fall by 80 to 84, and the note under D–F rises by 33.

The displacements alternate with the quality of the third. C–E, F–A and G–B put their notes 65 to 69 cents sharp; E–G, A–C and B–D put theirs 80 to 84 cents flat; D–F, whose just form was already a comma narrow, is moved only 33 cents sharp. Read as pitches, the tempered line is C♯2 a third of a semitone flat, A1 a fifth of a semitone sharp, B1 sharp, F♯2 flat, A♭2 flat, E2 sharp, F♯2 sharp and C♯3 flat. Five of its eight notes are nearer a note outside the scale than any note in it, two are nearest the wrong note of the scale, and only the note under D–F is still nearest the note just intonation gave it.

The steps of the line suffer more than its notes, because adjacent notes move in opposite directions. In just intonation the line’s steps are minor thirds of 316 cents, whole tones of 204 and a fourth of 498. In equal temperament the same steps are 350, 200 and 650 cents. The fourth from C to F under the move from E–G to F–A is stretched by 152 cents, to a size between a tritone and a fifth; the rise from A to C is shrunk to a whole tone; and the fall from C to A is widened to a neutral third. The tempered line has exactly three step sizes, 200, 350 and 650 cents, and two of them are not intervals of the scale.

The tuning makes that inevitable. The ghost under a tempered major third is the lower note times a fixed factor, the ratio of a tempered major third less one, and under a tempered minor third it is the lower note times the corresponding factor for the minor third. The two factors stand in a ratio of 1.374, which is 550 cents. So every step of the line is the lower voice’s own step while the thirds keep their quality, and that step with 550 cents added or taken away whenever the quality changes — a semitone becomes 650, a whole tone becomes a descending 350.

The quiet product draws something else

A passage in thirds has a second product under every pair, the cubic 2f₁ − f₂, and at a moderate dynamic it is the one the ear makes more readily. It draws a different line.

A scale in parallel thirds has a line underneath it that nobody plays. A major scale on C4 harmonised in parallel diatonic thirds, with the cubic product 2f₁ − f₂ of each pair drawn as a third line. In five-limit just intonation that line is G3, B♭3, C4, C4, D4, F4, G4, G4. In equal temperament it moves to G3, B♭3, C4, C4, D4, F4, G4, G4, departing from the just line by −23, −12, +37, −21, −25, +39, +35, −23 cents. — Fig. 4 The same passage in thirds with the cubic product 2f₁ − f₂ of each pair drawn instead. In just intonation the line is G3, B♭3, C4, C4, D4, F4, G4, G4, sitting just under the lower voice, and the B♭3 under D–F is 49 cents sharp of a keyboard B♭ — the eleventh harmonic of the pair’s fundamental.

The cubic line is not a bass. It lies a fourth or a major third under the lower voice rather than two octaves, in the register the voices themselves occupy, and it moves mostly by the steps the voices move by. Under a major third it is harmonic 3 — the fifth of the implied root — and under a minor third harmonic 4, the root’s double octave, so it alternates between those two functions as the thirds alternate.

Under D–F it is harmonic 22 of the Pythagorean third’s fundamental, which is a note a quarter tone between B♭ and B. At a moderate dynamic, the line a passage in thirds adds is an inner voice with a quarter tone in it, sounding a fifth below the written parts and in their register. At forte the bass appears underneath as well. Which of the two a listener attends to, if either, is a question about a voice being a stream — two parts closer than about five semitones fuse — and the cubic line is often closer than that to the lower voice.

In sixths, the two products trade places

Invert every third of the passage into a sixth, putting the lower voice an octave higher, and the question which product draws the bass has a clean answer.

The cubic product below a pair a–b is 2a − b. That is exactly the difference tone of b and 2a — the same pair with its lower note raised an octave, which is the inversion. So the cubic product of any interval is the difference tone of its inversion, and it lands where the inverted interval’s difference tone would. That identity is the whole of the switch at the fifth: an interval narrower than a fifth inverts to one wider than a fourth, so whichever product names the root of one names the root of the other.

A scale in parallel sixths draws its line through the quiet product. A major scale on C4 harmonised in parallel diatonic sixths, with the cubic product 2f₁ − f₂ of each pair drawn as a third line. In five-limit just intonation that line is F2, G2, C3, A2, C3, F3, G3, F3. In equal temperament it moves to E2, F♯2, C♯3, A2, B2, F♯3, A♭3, E3, departing from the just line by −80, −84, +67, +33, −82, +69, +65, −80 cents. — Fig. 5 The scale harmonised in parallel sixths, with the cubic product of each pair drawn underneath. In just intonation the line is F2, G2, C3, A2, C3, F3, G3, F3 — the roots of the major triads the sixths belong to, with the scale’s A under the Pythagorean sixth F–D.

The sixths’ bass is drawn by the cubic product, and it is again a diatonic line: F, G, C, A, C, F, G, F, roots of major triads except under F–D, where the scale’s Pythagorean sixth 27:16 names A. It is the thirds’ line started five degrees along, because each sixth here is the inversion of the third that begins a sixth above it.

So a passage in sixths reaches its bass through the product that survives at moderate dynamics and fades as the interval widens, and a passage in thirds reaches it through the product that needs volume. Neither has a bass at every dynamic for free.

The tempered line underneath wobbles by up to a quarter of a tone. For each pair of a major scale in parallel sixths, how far equal temperament moves the cubic product from where just intonation puts it, as the wide bar, beside how far it moves the sixth itself, as the thin one. C: −80.4 against +15.6; D: −84.3 against +15.6; E: +67.4 against −13.7; F: +33.3 against −5.9; G: −82.4 against +15.6; A: +69.3 against −13.7; B: +65.4 against −13.7; C: −80.4 against +15.6. — Fig. 6 How far equal temperament moves each note of the sixths’ cubic line from where just intonation puts it. The displacements are the thirds’ displacements in a different order — from 80 to 84 cents flat under the major sixths, 65 to 69 sharp under the minor ones, and 33 sharp under F–D — because the cubic gearing of a sixth is the difference-tone gearing of its inversion.

Tempered, the sixths’ line moves by the same amounts as the thirds’ line, and in the opposite sense to the sixths themselves: a tempered major sixth is 15.6 cents wide and puts its product 80 cents flat. The identity makes that unavoidable too. The steps of the tempered line are again 200, 350 and 650 cents.

A duet that moves between thirds and sixths

Real two-part writing rarely stays in one interval. It moves between thirds and sixths, often within a bar, and the identity above says what the line under it does at each change: the notes of the bass can stay coherent while the product that carries them changes.

Under the thirds the bass is the difference tone, and under the sixths it is the cubic product. The two products behave differently with level. The difference tone needs a loud pair; the cubic product survives at moderate levels but is strongest when the two notes are close and weak when they are a sixth apart. So a passage at a moderate dynamic that moves from thirds into sixths hands its bass from a product that is barely there to one that is there but faint, and a passage at forte does the reverse, from a strong difference tone to a cubic product at a ratio far from its best.

Neither of those is computed on this page, because neither product’s level is, and the drawing gives only where the notes of the line are. But it does settle one thing a duet’s texture decides: inverting the 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 two distortions is doing the work and leaves the implied harmony exactly where it was.

What the line assumes

Every note of both lines is arithmetic on the two notes above it. The just scale is the five-limit major scale, and the claim that the difference-tone line is diatonic depends on it: a scale with every minor third tuned 6:5 would give B♭ under D–F, and a Pythagorean scale, with every third a comma out, would give a line whose notes are harmonics 5 and 17 of fundamentals nowhere near the key.

The passage is the scale in close parallel motion, one octave, every pair sounding for as long as the next. A real melody in thirds leaps and repeats, but the line under it depends on nothing but each pair, so a melody that uses the same thirds in a different order draws the same notes of the line in that order; the result does not depend on the passage being a scale, only on its intervals being the scale’s. That is the texture the result is cleanest for, and it is a real texture — counterpoint that forbids parallel fifths and octaves allows parallel thirds and sixths freely, and melodies doubled in thirds are common for exactly that reason.

What the line on the page cannot establish

That anyone hears it. A difference tone at 65 hertz under a pair at 260 and 330 is a weak component two octaves below the texture; whether it is heard as a bass, as a colour, or not at all depends on its level, and its level depends on the dynamic in a way nothing here computes. The claim is that the component exists and where it is, not that it functions as a part.

That tempo allows it. The third sound magnifies cents, not hertz found that a difference tone’s pitch is useless for a note much shorter than a second. A line of eight difference tones in a passage of quavers is eight components too short to have pitches, and a bass line made of them is a bass line only at a slow tempo.

That real instruments leave it alone. Two violins in thirds have partials, and the partials of each pair make their own products, all of them harmonics of the same fundamental in just intonation. So a real passage in just thirds puts energy on many harmonics of each implied root, and the line drawn here is its lowest member rather than the whole of it.

And what a player does to the thirds. Performers do not play a passage in thirds at either tuning. They tend to raise a note that is sharp because of where it goes and to narrow a major third under a leading note, and each such adjustment moves the line underneath by five or six times as much. The line under a real performance is somewhere between the two drawn here, and the drawing cannot say where.

Tartini’s bass, and where it holds

Tartini’s harmonic theory treated the third sound of a consonance as its bass. For a passage of just thirds this page is a test of that idea on the texture it was most obviously about, and the result is exact: the line of third sounds under a scale in just thirds is a diatonic bass made of the roots of major triads, with one diatonic non-root under the scale’s one Pythagorean third. It holds for the difference tone in thirds and for the cubic product in sixths. It fails for the cubic product in thirds, which is an inner line with a quarter tone in it, and it fails entirely under equal temperament, where every note of the bass is a sixth of a semitone or more from the scale and the steps are 200, 350 and 650 cents.

The comparison with the bass line a composer writes is instructive rather than flattering. A written bass under a scale in thirds rarely sits on the root of every pair, and rarely rocks between IV and V as this one does. What the third sounds offer is the harmonisation that makes every third the third of a major triad, which is one harmonisation among many and the one the root an ear supplies would also favour.

Still open: which product a real dynamic delivers

Four questions about the ear’s products have now been answered in arithmetic alone: how far they move, which harmonic they are, when they land on a chord’s own notes, and what line they draw under a passage. Every answer has ended on the same unmeasured quantity — which of the two products is there at a given dynamic, and how far above threshold. The published measurements give the cubic product’s level relative to the primaries as roughly constant with level and falling steeply as the ratio widens, and the difference tone’s as growing about twice as fast as the primaries’. Put into numbers, those two laws would say at what dynamic a passage in thirds acquires its bass, at what dynamic a passage in sixths loses its, and whether the inner quarter-tone line is ever louder than the threshold at the frequencies where it sits. That is a model with two published slopes and two stated constants, and the constants would have to be swept rather than trusted.