Acoustics & Music Theory
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Combination tone — the series

13 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A tempered interval moves its difference tone several times further than itself. For every interval inside the octave tuned to twelve equal steps, how far the difference tone f₂ − f₁ lands from where the just interval would put it, in cents, with the interval's own departure from just drawn as the thin bar beside it. minor second −198.0 (the interval −11.7); major second −35.5 (the interval −3.9); minor third −96.0 (the interval −15.6); major third +67.4 (the interval +13.7); fourth +7.8 (the interval +2.0); fifth −5.9 (the interval −2.0); minor sixth −36.7 (the interval −13.7); major sixth +38.8 (the interval +15.6); minor seventh −39.8 (the interval −17.6); major seventh +25.0 (the interval +11.7). The major third's product is +67.4 cents out and the minor third's −96.0, and the largest error is the minor second's, at −198: a product moves p/(p − q) times as far as the interval p:q that made it.

    The third sound magnifies cents, not hertz

    Tartini's third sound is said to be a few cents off on a tempered interval. It is sixty-seven cents off on a major third and ninety-six on a minor third, because a difference tone moves p/(p − q) times as many cents as the interval p:q that made it. In hertz it moves exactly as far as the note that moved, and no further — so what the magnifier is worth is the ear's finer resolution at the low frequency where the product lands, which is a factor of two for a long note and nothing at all for a short one.

    part 1 · intervals
  2. 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. minor second 16:15: 1, 14, 17; major second 9:8: 1, 7, 10; minor third 6:5: 1, 4, 7; major third 5:4: 1, 3, 6; fourth 4:3: 1, 2, 5; fifth 3:2: 1, 1, 4; minor sixth 8:5: 3, 2, 11; major sixth 5:3: 2, 1, 7; minor seventh 9:5: 4, 1, 13; major seventh 15:8: 7, 1, 22. The shaded column is the fundamental itself. The difference tone sits on it for every interval up to the fifth, the cubic product for the fifth and every interval above except the minor sixth, whose products are its fundamental's octave and twelfth.

    The tone on the root changes hands at the fifth

    Every combination tone of a just interval is a harmonic of a fundamental neither note contains, and which harmonic is fixed by the ratio. The difference tone lands on that fundamental for every interval up to the fifth; the cubic product lands on it for the fifth and every interval above except the minor sixth. So the loud product names the root of a narrow interval and the quiet one names the root of a wide one — and a just major seventh's difference tone is a note seven harmonics up that no keyboard has.

    part 2 · intervals
  3. A major triad's combination tones, against its own notes. The three notes of a major triad on C4 in root position, voiced C4–E4–G4, as tall lines, and every combination tone its pairs make, as short ones: difference tones lowest, cubic products taller. In just intonation 2 cubic products land exactly on a note of the chord, and none comes within forty hertz of one. In equal temperament no cubic product lands on a note of the chord, and the nearest miss is 5.63 hertz.

    A major triad's combination tones are its own notes

    Play a just major triad of pure tones and two of the ear's cubic products land exactly on its root and its fifth. The reason is a condition rather than a coincidence — a chord's cubic products fall on its own notes when its middle note is the mean of the outer two in hertz — and it holds for the major triad in root position and in the six-four, and for no minor triad in any position or tuning. Equal temperament misses the landing by one number, 5.6 hertz on middle C, which is a beat that belongs to no pair of notes in the chord.

    part 3 · intervals
  4. 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.

    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.

    part 4 · intervals
  5. 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.

    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.

    part 5 · intervals
  6. Every product of every pair of partials is a harmonic of one absent note. A 4 : 5 interval on 261.6 and 327.0 hertz, just, each note carrying 6 partials at one over n, with the fundamental at 80 decibels. Every pair of partials makes its own difference tone, and there are 19 distinct frequencies among them. All of them are exact multiples of 65.41 hertz — the note neither instrument is playing — because partial j of a p·f₀ note and partial k of a q·f₀ note differ by (kq − jp)·f₀ whatever j and k are. The heavy line is what each product has to clear — the threshold of hearing at its own frequency, or the masked threshold the two primaries cast there, whichever is higher. 11 of the 19 get through.

    Three harmonics of the bass arrive before the bass

    Every product priced until now was between two pure tones, and nothing that plays thirds is pure. Give each note a spectrum and the ear receives every pair of partials — and for a just interval p:q every one of their products is an exact multiple of the same absent fundamental. That crowd lands where the threshold of hearing is tens of decibels cheaper, so it names the bass at 71 decibels where the component at the bass's own frequency needs 74, and at 75 against 85 an octave lower. Tempered, the crowd still forms and names a note seventy cents flat.

    part 6 · intervals
  7. The ghost bass drops when the passage gets louder. The note the whole crowd of products names, as a multiple of the fundamental the interval implies, against how loudly the interval is played. a major third: 2.9999999999999996 times the fundamental below 70 decibels and 1 times above it, a drop of 19 semitones; a minor third: 4 times the fundamental below 62 decibels and 2 times above it, a drop of 12 semitones; a fourth: 2 times the fundamental below 72 decibels and 1 times above it, a drop of 12 semitones. Softly, only the cubic products clear their thresholds, and they are an exact series on (2p − q) times the fundamental with no gaps in it. Loudly, the difference tones fill in the low harmonics, no template on the higher note can explain them, and the fit falls. Nothing about the interval has changed; the listener is simply being given a different subset of the same harmonic series.

    The ghost bass drops a twelfth at a forte

    Both crowds arrive at once and every member of both is a multiple of the same absent fundamental, so a listener is never given a choice between them — only a different subset of one harmonic series at every dynamic. Softly, the subset is an exact gapless series on three times the fundamental. Loudly, the difference tones fill in the low harmonics and no template on the higher note survives them. Between 62 and 72 decibels, depending on the interval, the note the crowd names falls by an octave or a twelfth, and the two qualities of third cross at different levels.

    part 7 · intervals
  8. Put back beside its notes, the crowd names the bass at every dynamic. A just major third on complex tones, drawn on one axis of harmonic numbers of the fundamental its ratio implies: the partials of the two played notes, and the products of those partials that clear threshold, at 55 and 80 dB. At 55 dB the products alone name 3 times the fundamental with 0 empty slots; the products and the notes together name 1 times it with 4 empty slots, and the notes alone name it with 9. At 80 dB the products alone name 1 times the fundamental with 0 empty slots; the products and the notes together name 1 times it with 0 empty slots, and the notes alone name it with 9. The soft reading on a higher note exists only when the loud notes are set aside. Taken together, the products do not decide which fundamental is named; they decide how many holes its template has.

    The played notes already name the ghost bass

    The products of a just third's partials, fitted on their own, name a note a twelfth above the bass when the interval is soft and drop to the bass when it is loud. Put the two played notes back beside them and the drop disappears: the notes and their products name the bass at every dynamic, because the notes' own partials are harmonics of it already. What the dynamic changes is not which note is implied but how complete its harmonic series is — nine holes from the notes alone, four when soft, none when loud.

    part 8 · intervals
  9. A tempered major third and its own ghost are not one sound. The multiple of the implied fundamental that each reading names, against how far the upper note is tempered from the just major third, at 55 and 80 dB. At 55 dB the notes and products together name the fundamental from −9.0 cents to +8.5 cents and no template at all outside that; the products alone name 3 times it and go on naming very nearly that across the whole range. At 80 dB the notes and products together name the fundamental from −2.0 cents to +2.0 cents and no template at all outside that; the products alone name 1 times it and go on naming very nearly that across the whole range. Equal temperament's major third is +13.7 cents, which is outside the window at both dynamics. The notes alone name the fundamental throughout, because tempering moves every partial of the upper note by the same amount and a template can follow it.

    A tempered third cannot be one sound with its own ghost

    The essay before this one fitted one harmonic template to a just third's played partials and the products of those partials together, and it worked at every dynamic. Tempered, it does not work at all — and not by naming a different note. There is no template. The lower note's partials sit exactly on their harmonics, the upper note's are every one fourteen cents out, and the products are twenty-three cents out the other way, a span of thirty-seven cents across a tolerance of thirty. The window is nine cents wide when the third is soft and two when it is loud, so equal temperament's fourteen falls outside it — while the fifth's two cents fall inside. The consonances a temperament leaves nearly alone are exactly the ones that stay single sounds.

    part 9 · intervals
  10. An octave down, the ghost bass needs notes twice as long before it is there at all. The share of each note of a C major scale in just thirds that has its difference-tone bass, against how long each note lasts, for the passage starting on C3, C4 and C5, with the bass heard only after 4 of its own cycles. Thirds from C3: 40 ms 0%, 60 ms 0%, 80 ms 3%, 100 ms 12%, 125 ms 22%, 150 ms 33%, 200 ms 50%, 250 ms 60%, 300 ms 67%, 400 ms 75%, 500 ms 80%, 750 ms 87%, 1000 ms 90%; no note has its bass below 147 ms. Thirds from C4: 40 ms 3%, 60 ms 20%, 80 ms 38%, 100 ms 50%, 125 ms 60%, 150 ms 67%, 200 ms 75%, 250 ms 80%, 300 ms 83%, 400 ms 88%, 500 ms 90%, 750 ms 93%, 1000 ms 95%; no note has its bass below 73 ms. Thirds from C5: 40 ms 38%, 60 ms 58%, 80 ms 69%, 100 ms 75%, 125 ms 80%, 150 ms 83%, 200 ms 88%, 250 ms 90%, 300 ms 92%, 400 ms 94%, 500 ms 95%, 750 ms 97%, 1000 ms 98%; no note has its bass below 37 ms. Every step down an octave halves the bass and doubles the note length it needs.

    A ghost bass an octave lower needs notes twice as long

    A scale played in just thirds has a bass underneath it that nobody plays — the difference tones of its thirds spell a diatonic line. Every essay on that bass has held its intervals still. A moving passage gives the bass a new pitch every note, and a low pitch is not there until it has run a few of its own cycles. With four cycles, a passage in thirds from C4 keeps its bass for 83 per cent of each note at three notes a second and loses it completely below 73 milliseconds a note. An octave down, every number doubles: the passage from C3 has no bass at all in its lowest note at semiquavers, and it is the scale's one Pythagorean third that sets the limit.

    part 10 · intervals
  11. The notes name their ghost bass long before the bass has run a single cycle's worth of its count. A just major third, C4 against E4, with every component — each note's first six partials, the difference tone and the cubic tone — entering a harmonic fit once it has run 4 of its own cycles. at 2.0 ms, 1 component: nothing fitted; at 2.4 ms, 2 components: nothing fitted; at 2.5 ms, 3 components: names 103.3 Hz; at 3.1 ms, 4 components: names 82.0 Hz; at 3.8 ms, 5 components: names 86.3 Hz; at 4.1 ms, 6 components: names 81.8 Hz; at 5.1 ms, 6 components: names 65.4 Hz, the ghost bass; at 6.1 ms, 6 components: names 65.4 Hz, the ghost bass; at 7.6 ms, 6 components: names 65.4 Hz, the ghost bass; at 12.2 ms, 6 components: names 65.4 Hz, the ghost bass; at 15.3 ms, 6 components: names 65.4 Hz, the ghost bass; at 20.4 ms, 6 components: names 65.4 Hz, the ghost bass; at 61.2 ms, 6 components: names 65.4 Hz, the ghost bass. The fit names the 65.4 Hz ghost from 5.1 ms; the bass's own 4 cycles take 61.2 ms.

    The notes name the ghost bass before it has a cycle

    A ghost bass that had to wait for four of its own cycles needed 61 milliseconds under a third from C4, and so vanished from fast passages. A pitch mechanism fitting the whole object — both notes' partials, which run their cycles many times faster, and the products — names the same bass in five. Seven of the eight thirds of a just scale have their bass for most of every note at any tempo a player reaches. The eighth, the scale's Pythagorean D–F, never has its own: the object hears it as a 6 : 5 and supplies the bass a D a comma lower would give, a semitone above the difference tone, until the difference tone arrives and breaks the fit.

    part 11 · intervals
  12. Quietly, or quickly, the scale's Pythagorean third is heard with the bass of the just third a comma away. The third D4–F4 of a just C major scale, a Pythagorean 32 : 27, fitted as a whole object with every component that has run 4 of its cycles and every product only if it is audible at the level played: what the fit names at the end of a note of each length, at each dynamic. 40 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 50 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 60 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 66 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 70 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 74 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 76 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz; 80 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz; 85 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz; 90 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz. 58.1 Hz is the bass of D–F with D a syntonic comma lower, a just 6 : 5; the Pythagorean third's own difference tone is 54.5 Hz and is audible from 76 dB, and once it has run its cycles the fit breaks.

    The dynamic chooses which D the scale has

    A just C major scale has one third that is not 5 : 4 or 6 : 5 — D–F, a Pythagorean 32 : 27 — and a pitch mechanism fitting the notes' partials hears it as the just minor third a comma away, with that third's bass, B♭1, beneath it. Only the Pythagorean third's own difference tone can contradict that, and it is audible only from 76 dB. So below a forte the ghost line under a scale in just thirds is C, B♭, C, F, G, F, G, C for the whole of every note, and the syntonic comma is resolved by the listener rather than the players; above it, and only in notes longer than 73 milliseconds, the B♭ is taken away.

    part 12 · intervals
  13. Only the D a just ii wants gives the chord a bass that survives its voicing. The bass a pitch mechanism fitting the whole chord — every note's partials and every audible difference and cubic tone — names under the ii chord of C major, with its D tuned three ways, in root position and both inversions, at 50 and 80 dB. the scale's D, 9 : 8: D F A at 50 dB G ±0, D F A at 80 dB none, F A D at 50 dB B +24, F A D at 80 dB F −2, A D F at 50 dB A −30, A D F at 80 dB none. a just ii's D, 10 : 9: D F A at 50 dB B♭ −4, D F A at 80 dB B♭ −4, F A D at 50 dB B♭ −4, F A D at 80 dB B♭ −4, A D F at 50 dB F +13, A D F at 80 dB B♭ −4. equal temperament: D F A at 50 dB G +7, D F A at 80 dB D −1, F A D at 50 dB G +7, F A D at 80 dB D −1, A D F at 50 dB A −23, A D F at 80 dB D ±0. B♭ is the bass the 10 : 9 D implies, since D–F–A is then 10 : 12 : 15 over it.

    The scale's ii chord borrows the dominant's bass

    Heard quietly, the pair D–F of a just C major scale brings the bass of the other D with it — B♭, the bass the 10 : 9 D implies — so the comma is resolved by the listener rather than by the players. Add the A that makes the pair a ii chord and that stops being true. The whole-object fit now hears the chord over G, as the top of a dominant ninth, and it does so because the template prefers simple harmonics to accurate ones; a different bass is heard in each inversion, and at a forte none at all. Only a ii built on the 10 : 9 D has one bass in every voicing and at every dynamic. Played quietly from a fixed scale, I–vi–ii–V–I is heard with the bass line C, F, G, G, C: the ii merged into the chord after it, and the progression home.

    part 13 · intervals

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