A tempered seventh chord already has the tuned one's low note

A dominant seventh tuned exactly to the harmonic series has a property no tempered chord can have. Every partial of every note is a whole multiple of one frequency, so the whole sound repeats, exactly, at that frequency’s period. A tuned seventh chord does not beat; it buzzes used that fact to explain why such a chord has no slow beating: any two of its partials either coincide or differ by at least the series’ fundamental. It ended on the obvious next question. A sound that repeats at 65 hertz is, by the usual account, a sound with a pitch at 65 hertz. Is that pitch heard? Does tuning a seventh chord put a note underneath it that the tempered chord lacks?

The expectation was yes. A tempered chord is not periodic at all: its third is fourteen cents sharp of 5 : 4 and its seventh thirty-one sharp of 7 : 4, so no lag brings every partial back into phase at once. It seemed to follow that the tempered chord could have no single low periodicity to hear, and that tuning would supply one.

The computation says otherwise, and the difference is worth stating carefully. The tempered chord repeats at the same place as the tuned one. Not exactly, and not quite at the same frequency, but closely enough that a pitch mechanism looking for the strongest periodicity in the sound finds it there in both. What tuning adds is exactness, not a note.

The account of pitch that needs no template #

There are two families of explanation for a residue pitch, and which harmonics carry the pitch set out the spectral one: the ear resolves a handful of low partials and finds the fundamental whose harmonic series they fit. That account needs a tolerance, a statement of how far a partial may sit from a harmonic and still count, and a tolerance is exactly what cannot be assumed here. Tempering moves partials by amounts on the order of any plausible tolerance, and the answer would then be a property of the constant chosen.

The temporal account needs no tolerance. It asks at what delay the sound most resembles itself, which is the autocorrelation of the waveform, and it reads the pitch as the reciprocal of that delay. A pitch with nothing to match is the evidence that some such mechanism exists: a click train filtered so high that no partial is separable still has a pitch, and loses it when the clicks are jittered, which changes their timing and nothing else. For a sum of steady partials the autocorrelation can be written down exactly — the power-weighted sum of a cosine at each partial’s frequency — so it needs no sampling, no window and no free parameter beyond the weighting given to each partial, which is swept below.

Three further pieces complete the model. The partials are those of a struck string at 80 decibels, sixteen per note falling as one over their number and dying faster the higher they are, the same strings every essay on the ceiling has used. Partials above five kilohertz are removed, because the auditory nerve stops following the waveform somewhere between four and five, and a temporal account cannot use what the nerve does not carry. And the search runs only over delays shorter than a thirtieth of a second, because Pressnitzer, Patterson and Krumbholz found that a periodic sound slower than about 30 hertz stops carrying a melodic pitch at all and is heard as a flutter. A periodicity below that floor is not a candidate note, whatever its height.

A repeat at the octave below the bass #

The first figure is the tuned chord’s answer, and it is as clean as the theorem promised. At a delay of 15.29 milliseconds every partial of all four notes has completed a whole number of cycles, and the autocorrelation reaches 1. That delay is the period of C2, an octave below the chord’s lowest note, which is where the series 2 : 5 : 6 : 7 over C3 puts its fundamental. The bass note’s own period, 7.64 milliseconds, is not special at all: the bass repeats there, but E4 and B♭4 are each halfway through a cycle, and the chord as a whole scores only 0.24.

So the pitch this account gives the tuned chord is not its bass. It is the note an octave below the bass, which nobody plays. That is the note the root an ear supplies found a template supplying under a just major triad, and the kind of note an organ’s resultant stop builds on purpose from two pipes for a note neither makes, arrived at here with no template: the chord’s notes are the second, fifth, sixth and seventh members of one series, and the period of the whole is the period of the series.

The tempered chord was expected to have nothing comparable. It has almost the same thing. Its autocorrelation has its highest peak at 15.21 milliseconds, height 0.82, and its lowest-note period scores 0.25 exactly as the tuned chord’s does. The shape of the two curves is nearly the same across the whole range of pitch. The tempered chord’s peak is a little lower and a little early, which is to say its best periodicity is at 65.75 hertz rather than 65.41, nine cents sharp of C2.

The reason is the size of the mistunings against the size of the delay. A partial whose frequency is off by δ hertz drifts out of phase by δ times the lag. The tempered E4 is 2.6 hertz sharp of 5 : 2 over C3, and over 15.3 milliseconds that is a twenty-fifth of a cycle, which costs its fundamental about three per cent of its contribution. The seventh is further out, 8.3 hertz, an eighth of a cycle at its fundamental and correspondingly more at each partial above it, since the drift of the k-th partial is k times the fundamental’s. Those are real losses and they add up to the 0.18 the tempered chord is short by. But they are losses from a peak that is still there. Twelve-tone temperament moves the notes of a dominant seventh by amounts too small, at a delay this short, to destroy the chord’s periodicity. It blurs it.

One chord of three #

The earlier essay set three chords side by side at the ceiling and asked whether tuning adds a note the tempered chord lacks. The same measure run on all five of the chords it compared gives a sharper picture of what “the tuned chords” have in common, because they turn out to have very little.

The tuned dominant seventh stands alone at 1. The half-diminished seventh tuned to its mirror, which is equally a harmonic series on paper, is at 0.48 — below the tempered diminished seventh’s 0.59. It is periodic, but its period is 460 milliseconds, the period of 2.18 hertz, because the mirror chord’s notes stand in the ratio 60 : 70 : 84 : 105 and their common fundamental is C3 divided by sixty. Nothing inside the range of pitch is special for it, so its best periodicity there is whatever near-coincidence of phases happens to occur, at F♯1, eighteen cents flat. The earlier essay found that the mirror chord escapes beating by arithmetic luck; it has no pitch-range periodicity to trade for it either.

The three tempered chords all sit between 0.37 and 0.59, and their best periodicities land on notes with no simple relation to their written roots: the diminished seventh on B0, sixteen cents sharp, the tempered dominant seventh in its first-inversion spacing on B0, the half-diminished seventh on F1. None of them reaches the tempered dominant seventh’s 0.82 from the first figure, because that figure drew the dominant seventh at the tuned chord’s spacing, in root position with the upper notes opened — and in that spacing, tempered or not, the chord is close to a series.

That qualification carries the whole result. A tempered chord has a strong low periodicity when its notes are near the members of one series, and the dominant seventh in root position is the four-note chord whose tempered notes are nearest. The periodicity belongs to the chord and the spacing, and tuning is the last few per cent of it.

The tempered chord gets closer as it decays #

A struck chord is not a steady spectrum. Its high partials die in a fraction of a second and its low ones ring on, and the high partials are where a mistuning costs most phase, because a partial’s drift over a delay is its harmonic number times the note’s. So the tempered chord should become more periodic as it sounds, and it does.

At the strike the tempered chord’s periodicity is 0.82. A fifth of a second later it is 0.92, by half a second 0.95, and from a second onward it sits at 0.96. For most of the note’s audible life the tempered dominant seventh repeats at its low period with an exactness within four per cent of the tuned chord’s.

The place it repeats moves as it does so. At the strike the peak is nine cents sharp of C2; within a fifth of a second it is sixteen cents sharp, and it stays there. The direction follows from which notes are mistuned. The bass C3 is exactly twice C2, and all of its partials are exact multiples of C2 whatever the temperament; the E4 implies a fundamental fourteen cents sharp, the B♭4 one thirty-one sharp, and the G4 one two cents flat. The peak is a compromise between these claims, weighted by the power each note puts into the partials still sounding. At the strike the bass’s many upper partials hold the compromise down near C2. As they die, what remains is dominated by the low partials of all four notes, and the two sharp notes pull harder.

This is the opposite of the timing the beating had. The trace of the beating found the tempered chord’s beating concentrated in its first few hundred milliseconds and gone by the time the chord reached its ceiling, because the pairs that beat were high partials. The same high partials are what spoil the periodicity. When they die, the beating stops and the periodicity sharpens, together. A tempered dominant seventh on a struck instrument begins as a fluttering sound with a blurred low note and settles, within half a second, into a nearly steady one whose low note is a sixth of a semitone sharp of where the tuned chord would put it.

Every step toward 7 : 4 helps, which is not true of the beating #

The contrast with beating is sharpest when one note is moved continuously. The earlier essay moved the seventh from tempered to forty cents flat, with the third and fifth already tuned, and found the beating rising as the seventh approached 7 : 4, peaking a cent either side of it at more than three times its tempered value, and falling to nothing only in a notch well under a cent wide. Running the same sweep through the autocorrelation gives a different shape entirely.

The periodicity improves at every step. With the seventh tempered it is 0.85; ten cents flat, 0.90; twenty, 0.96; twenty-five, 0.99; at the seventh partial, 31.17 cents flat, exactly 1; and it falls away only slowly past it, to 0.995 at thirty-five and 0.975 at forty. It is within a hundredth of exact over a band about ten cents wide.

So the two quantities disagree about what a partly tuned seventh is. At twenty-five cents flat the chord is 0.99 periodic — its low note is nearly as exact as it will ever be — and its slow beating is 0.65 of the maximum, more than twice what it was tempered. By one measure the chord is almost in tune. By the other it is worse than when it started.

That is a real conflict, and it says something about how a seventh is tuned by ear. The beat a tuner can actually use found that a tuner needs a beat slow enough to count, and the beating in the seventh sweep supplies exactly that: slow, growing, and then gone. The periodicity supplies nothing a tuner can use. It rises smoothly into a broad plateau, and a listener attending to it would hear the low note settling long before the seventh arrived, with no sign of when it had. The beating is the tuning cue; the low note is what the tuning is for.

The same sweep answers a narrower question the first figure raised. With the third and fifth already tuned and the seventh tempered, the periodicity is 0.85 against the fully tempered chord’s 0.82. Tuning the third and fifth buys three hundredths. The seventh buys the other fifteen. Almost the whole of the difference between the tempered and the tuned chord’s low note is the seventh.

How much the answer depends on the weighting #

The one free choice in the model is how much each partial counts. The autocorrelation of the pressure wave weights each partial by its power, which lets the strong low partials dominate. A cochlear channel compresses before any periodicity is extracted, so a summary autocorrelation built from channels gives the weak high partials proportionally more say — and the high partials are where the tempered chord is least periodic. The obvious worry is that the tempered chord’s 0.82 is an artefact of letting the fundamentals outvote everything else.

It is partly an artefact, and the figure says how much. With pressures raised to the power 0.6, the tempered chord’s periodicity at the strike falls from 0.82 to 0.69; at 0.3, to 0.62. Compressing the weak partials up costs the tempered chord a quarter of its exactness in the attack. It costs it much less later: by two seconds the three weightings put it between 0.94 and 0.96, and by three all are at 0.95 or above, because by then the partials that were being given more weight have decayed below hearing and are not in the sum at all.

What does not depend on the weighting is the location. Under every weighting the tempered chord’s strongest periodicity in the range of pitch is at C2, a few cents sharp. The tuned chord is at exactly 1 under every weighting, because an exact repeat is exact however its components are counted. So the conclusion survives in its strong form — the tempered chord has the same low note — and in its quantitative form only with a range: the tempered chord begins somewhere between 0.62 and 0.82 as periodic as the tuned one, depending on a compression this essay cannot fix, and ends within a few per cent of it regardless.

Close position puts the note below the range of pitch #

Everything so far has been the open spacing at the ceiling, where the series fundamental is an octave below the bass. In close position — C3, E3, G3, B♭3, tuned 4 : 5 : 6 : 7 — the series fundamental is two octaves below the bass, and at the bottom of the keyboard that is below the 30-hertz floor.

Over C3 the close chord’s fundamental is C1, 32.7 hertz, just inside the range; the tuned chord repeats there at 1 and the tempered at 0.79. Over A2 the fundamental is 27.5 hertz and outside it, and over every lower bass the same. There the tuned chord still repeats exactly — that is arithmetic — but at a rate the ear does not hear as a note, and within the range of pitch the best it offers is a periodicity of 0.25, no better than a random chord’s. This is the regime the earlier essay’s title named. A tuned seventh chord in close position low on the keyboard does not have a low note; it has a flutter, at the series fundamental, the whole chord waxing and waning together.

The open chord never falls out of the range over these ten basses, because its fundamental is only an octave below the bass: 32.7 hertz over C2, rising to 130.8 over C4. The tempered chord follows the tuned one throughout, from 0.81 at the bottom to 0.87 at the top. It improves slightly with register, and the reason is the phase-locking limit: a higher chord has fewer partials below five kilohertz to disagree. With the limit lifted to the top of hearing the rise disappears, and the tempered chord sits at 0.80 to 0.81 over every bass.

So spacing decides whether there is a low note to hear at all, and tuning decides only how exact it is. The first is a much larger effect.

What a quartet’s locked chord is doing #

The practice where this matters is sustained close harmony, and barbershop is its best-documented case. Barbershop singers tune the dominant seventh toward 4 : 5 : 6 : 7 and describe a chord that has locked as producing a sound none of the four is singing. The usual reading of that report is spectral: when the parts coincide, their shared partials reinforce one another and some high partial is heard as a separate note. The arithmetic here offers a second reading that does not compete with the first. A locked dominant seventh is exactly periodic at its series fundamental, which in the voicings barbershop favours is an octave or two below the bass singer, and an exact periodicity is the strongest form of the cue the temporal account of pitch reads.

But the same arithmetic says the tempered chord already carries most of that cue. A barbershop quartet singing a dominant seventh a sixth of a semitone out of lock is not missing a low note that the locked chord has; it has the same low note, blurred, slightly sharp, and — because a sung chord does not decay — blurred for as long as it is held. On a struck string the high partials die and the blur clears in half a second. On voices it never clears. That may be why the practice that cares most about the lock is a sustained one: the difference tuning makes to the periodicity is largest in exactly the conditions that never let the tempered chord’s high partials fall silent.

The claim is about that repertoire and its chord. It is not a claim about seventh chords in general, which the second figure shows are mostly not near a series at all.

What the picture cannot show #

Whether a listener reports the low note. The autocorrelation says where the periodicity is and how exact. It does not say whether a listener attends to it, hears it as a separate note under the chord, or hears it only as a quality of the chord — fusion, solidity, the “ring” choral singers describe. The note that is not there was heard because the harmonics that implied it were all a listener was given; a chord’s four played notes are much more salient than an implied fundamental below them, and nothing in this model ranks the two.

Which account of pitch is right. The temporal account was chosen because it needs no tolerance, and on this question the spectral account needs one badly. A template tolerant to thirty cents would hear the tempered chord’s C2 just as this does; one tolerant to ten would not. The result that the tempered chord already has the low note is a result of the temporal account, and it holds for a template account only if that template’s tolerance is at least as wide as the seventh’s mistuning.

The compression. The weighting was swept, not fixed, and the tempered chord’s exactness at the strike depends on it by as much as a quarter. Only the location of the periodicity and its later exactness are independent of it.

A sustained sound. Every number here is for struck strings whose high partials die. A sung or bowed chord keeps them, so the tempered chord’s periodicity stays at its strike value — whatever the weighting makes that — for as long as the chord is held.

Still open: whether the flutter under a close chord is heard #

Below the 30-hertz floor a tuned chord in close position still repeats exactly, and the question changes from pitch to modulation: whether the whole chord’s periodic waxing and waning at 16 to 28 hertz is heard as a flutter, as a roughness, or not at all. The pitch that does not wobble found that a periodicity smeared by vibrato can survive integration while the waveform moves, and the same question — how deep a modulation, at what rate, survives the ear’s temporal integration — decides whether the close chord’s slow repeat is audible. The computation is the envelope of the summed partials in each auditory filter, and the depth of its modulation at the series fundamental, compared with the threshold for detecting an amplitude modulation at that rate. It would say whether a tuned close chord low on the keyboard has a flutter the tempered chord lacks — which would be the first place in these chords where tuning adds something rather than sharpening what was there.