A tuned seventh chord does not beat; it buzzes

Four notes cannot get below the seventh partial proved that no chord of four pitch classes, however tuned, keeps all of its partial coincidences audible longer than the seventh partial allows: 0.87 seconds over C3, for struck strings of the kind these essays use. Three chords reach it. The tempered diminished seventh reaches it by accident, because its tritones are close enough to 7 : 5. The dominant seventh reaches it when the whole chord is tuned 4 : 5 : 6 : 7. The half-diminished seventh reaches it when tuned to the same series read downwards.

Within that measure the three are identical, and the essay said so. A lifetime counts whether a coincidence happens and how long it lasts, not what the partials that do not coincide are doing meanwhile. It named the quantity that might separate the chords: the beating among partials that do not coincide, which is where they differ most. The diminished seventh is a lattice of four equal minor thirds, none of them a small-number ratio. The dominant seventh tuned to the series should have no beats among its low partials at all. The lead was that summing that beating over the chord’s life might put a tuned seventh chord above the diminished seventh rather than beside it.

It does, completely and for a reason that is a theorem rather than a measurement: the tuned chords have no slow beating whatever. The same comparison made on roughness, the quantity these essays have used for how harsh an interval sounds, puts them nowhere in particular. Tuning a dominant seventh removes every beat from it and leaves its roughness within a few per cent of where it was.

Two measures of a chord that does not coincide #

A pair of partials from different notes can do one of three things. If their frequencies are equal, they coincide, and that is the event the ceiling counts. If they differ by a few hertz, the ear hears one tone waxing and waning at the difference, which is beating: beats are arithmetic, and the rate is the difference in hertz. If they differ by a sizeable fraction of a critical band, the fluctuation is too fast to follow and is heard as roughness, the grainy harshness the Plomp–Levelt curve describes, which peaks about a quarter of a critical band apart and fades both toward unison and toward wide separations.

These are different quantities and the essays have mostly measured the third. Roughness can be computed with the Plomp–Levelt kernel summed over every pair of partials, and it gives slow beats very little weight, because a difference of a few hertz is near the kernel’s zero at unison. So two measures are computed here and kept apart. Slow beating is the sum, over partial pairs from different notes whose frequencies differ by between a tenth of a hertz and 15 Hz, of the quieter partial’s pressure. Roughness is the Plomp–Levelt sum. Both are integrated over the first three seconds of each chord’s decay. The strings are those of the ceiling essay: sixteen partials falling as one over their number at 80 decibels, each decaying in a sixty-decibel time of six seconds divided by its number, counted only while above the threshold of hearing.

Why a tuned chord cannot beat #

The theorem is one line, and the figure below shows it at work. If four notes are tuned exactly to 4 : 5 : 6 : 7 over some fundamental, every partial of every note is a whole multiple of that fundamental: the chord is a harmonic series with some members missing. So the difference between any two of its partials is also a whole multiple of the fundamental. Either it is zero, and the two coincide, or it is at least the fundamental itself.

At the ceiling spacing over C3 the tuned dominant seventh sounds C3, E4, G4 and B♭4 — parts 2, 5, 6 and 7 of a series on C2 — so its fundamental is 65.4 hertz and no two of its partials differ by less than 65.4 Hz without coinciding exactly. That is a fast fluctuation, well into the roughness range and nowhere near beating. The half-diminished seventh tuned to its mirror has, among the partials audible here, no difference smaller than 21.8 Hz — for a different reason, taken up below. Neither has a single pair in the slow range.

The tempered diminished seventh is the opposite case. Its minor thirds are 15.6 cents narrower than 6 : 5 and its tritones 17.5 cents wider than 7 : 5, so the partials that nearly coincide miss by a few hertz, and at the strike 125 pairs of its partials beat at under fifteen per second. The tempered dominant and half-diminished sevenths have 131 and 115.

So the ceiling’s three chords divide cleanly: one beats and two do not. That is the answer the earlier essay’s lead predicted, and it is stronger than predicted, since it is exact rather than approximate: a chord tuned to a series has no slow beating, not merely less.

The mirror chord is a harmonic series only on paper #

The theorem holds for the dominant seventh and needs a qualification for the half-diminished. Read downwards, the mirror chord is 1/7 : 1/6 : 1/5 : 1/4, which over C3 in its longest-lived spacing is C3, a septimal minor third above, a septimal tritone above that, and a septimal seventh: frequencies in the ratio 60 : 70 : 84 : 105. That is a harmonic series too, but its fundamental is C3 divided by sixty, 2.18 hertz, far below hearing and far below the slow range. Two of its partials can differ by 2.18 Hz, or by 4.36, or by any small multiple, and each of those would be a slow beat.

None does, among the partials audible here. The closest any two come without coinciding is 21.8 Hz, ten times the series fundamental, so the mirror chord has no slow beating in this model — but as a fact about which multiples its sixteen partials happen to land on, not as a consequence of its tuning. A timbre with more partials, or a spacing that brought the upper notes’ high partials together, could put a pair within 15 Hz. The dominant seventh cannot be made to beat by any choice of partials, because its series fundamental is an audible frequency set by the bass. The half-diminished seventh escapes beating in this model by arithmetic luck.

The asymmetry has a musical reading. The root an ear supplies found listeners hearing a chord’s root where its partials imply one. A 4 : 5 : 6 : 7 chord implies a root at a pitch the ear can use, two octaves below its lowest note. The mirror chord implies one at 2.18 hertz, which is not a pitch at all. The two chords at the ceiling that do not beat are therefore not alike in the one respect that would make them sound fused: only the dominant seventh has a periodicity a listener could hear.

How long the beating lasts #

The tempered chords’ beating is concentrated at the strike and gone within half a second. By four-tenths of a second the diminished seventh’s has fallen to 2 per cent of its initial value, and by the ceiling at 0.86 seconds to nothing measurable. The reason is that the pairs that beat are mostly high partials: two partials that are near multiples of each other’s notes are high in both series, and in a struck string the high partials die first.

That means the beating and the ceiling happen at different times. The ceiling is a statement about the chord’s last low coincidence, near a second into the note. The beating is almost all in the attack. The tempered diminished seventh therefore begins with a flutter the tuned chords do not have, and by the time the three reach their shared limit the flutter has gone. On a struck instrument the tuned chord’s advantage is in its first half-second, which is when a listener hears most of it.

Roughness does not follow #

The obvious expectation is that removing the beats makes a chord smoother, and in roughness it hardly does. The first figure’s roughness column seems to say otherwise: the tuned dominant seventh is 0.39 of the diminished seventh’s. But the two are at different spacings. The dominant seventh’s longest-lived spacing is open — root, then the third, fifth and seventh an octave and more above — and the diminished seventh’s is close, four notes a minor third apart.

Take every spacing of each chord and the picture changes. Each chord’s roughness ranges over a factor of about two and a half, from its smoothest open spacing to its roughest close one, and the three curves lie almost on one another. The smoothest diminished seventh, the smoothest tuned dominant seventh and the smoothest tuned half-diminished seventh are within 20 per cent: 1.00, 0.83 and 0.86. In roughness, where the notes are matters much more than which chord they spell or how it is tuned. That is the result eighty-one chords, one number found for the dominant seventh’s spacings, extended to the whole ceiling.

It also clarifies the first figure. The diminished seventh reaches the ceiling only in its close spacing, which happens to be its roughest, and the tuned dominant seventh reaches it in an open spacing near its smoothest. Their difference in roughness at the ceiling is a difference between spacings.

A seventh on its way beats more #

The clearest way to see what tuning does to beating is to move one note continuously.

With the third and fifth already tuned, the seventh starts tempered, 31.2 cents sharp of the seventh partial. There its mismatches with the root’s partials are large — tens of hertz for the partials that matter — and they fall mostly in the roughness range. As it is lowered toward 7 : 4 those mismatches shrink and slide down into the slow range, and the chord’s slow beating rises: 0.29 of its maximum when tempered, 0.47 at twenty cents flat, 1.00 at thirty. At exactly 31.17 cents flat every mismatch reaches zero at once and the beating vanishes. A cent either side it is back near its maximum.

This is familiar to anyone who has tuned by ear: the beats slow down as the note approaches the target, which makes them easier to hear, and stop only when it arrives. The beat a tuner can actually use found that a tuner needs beats slow enough to count. The arithmetic here says that a seventh partly tuned is audibly worse, in beating, than one left tempered. It is a narrow target — the notch is well under a cent wide in this model — which is one reason a quartet’s septimal seventh is either locked or plainly beating, with little in between.

Roughness does the reverse, slightly. It rises by about 6 per cent across the sweep, and is 4 per cent higher at 7 : 4 than tempered. The partials the seventh brings into exact coincidence were near-coincidences, near the Plomp–Levelt kernel’s zero, and contributed little roughness; the ones it moves apart were in the kernel’s range and move further into it.

Across the bass #

The same comparison over nine basses from C2 to C4 shows how little the roughness moves. Tuned against tempered, the chord is 1.4 per cent rougher at C2, level at E3 and 4.4 per cent smoother at C4. The beating result, meanwhile, holds at every bass: the tuned chord never beats, and its smallest partial difference is the series’ fundamental — 32.7 Hz over C2, 130.8 over C4 — which even at the bottom of the range is faster than the slow beating range.

At a bass lower than drawn it would not be. A 2 : 5 : 6 : 7 chord over C1 has its series fundamental at 16.4 Hz, on the edge of the beating range, and the tuned chord’s own periodicity would then be heard as a flutter at the series fundamental. That is the sense in which a tuned chord buzzes. Its partials do not beat against one another; the whole chord fluctuates, periodically, at the one frequency they all share — the pitch the note that is not there would hear as the chord’s root.

What the three chords share and what they do not #

The earlier essay’s ceiling said the three chords’ coincidences last equally long. This essay adds that the tempered one beats in its attack and the tuned ones do not; that their roughness depends on spacing far more than on identity or tuning; and that the tuned chords replace beating with a single periodicity at the series’ fundamental. None of that changes the ceiling. It says what the ceiling was leaving out: in a struck chord’s first half-second, the diminished seventh sounds unsettled and the tuned sevenths sound fused, and the difference is audible as beating rather than as harshness.

That puts the tuned dominant seventh above the diminished seventh, as the lead hoped, on one quantity and not on the other. A listener who hears beating as the mark of a chord out of tune would rank them that way. A listener, or a model, that measures harshness would not rank them at all.

It also separates the two tuned chords from each other, which the ceiling could not do either. The dominant seventh tuned to the series has no slow beating by necessity and a periodicity at an audible frequency; the half-diminished seventh tuned to its mirror has no slow beating by the accident of where its sixteen partials fall, and a periodicity far below hearing. Of the three chords at the ceiling, then, one beats, one does not beat and implies a root, and one does not beat and implies nothing. Those are three different sounds reaching one lifetime, which is what the earlier essay suspected and could not show.

What the measures assume #

A fixed boundary between beating and roughness. The slow range is cut at 15 Hz, a round number near where fluctuation stops being heard as beats. Moving it changes the tempered chords’ totals but not the tuned chords’, which are zero for any boundary below the series fundamental.

Every beating pair counted alike. Slow beating is weighted by the quieter pressure, which is a proxy for modulation depth, and counts every pair once whether or not the ear can separate it from the pairs around it. Several slow beats at different rates in one critical band are heard as one irregular fluctuation, not as several beats.

Struck strings. On a sustained instrument the high partials do not die, the tempered chord’s beating does not fade in half a second, and the difference between the chords would last the whole note. Barbershop singing, the practice most associated with the tuned dominant seventh, is sustained, and there the whole of this essay’s contrast holds for as long as the chord is held.

Still open: whether the buzz is heard as a pitch #

A chord tuned to a series has one periodicity, its series’ fundamental, and at the spacings drawn here that frequency lies an octave or two below the bass. Whether a listener hears it — as a low pitch under the chord, as a buzz, or not at all — depends on how many of the series’ low members the chord supplies. That is the question which harmonics carry the pitch asked of single tones. Asked of the ceiling’s three chords, it would say whether tuning a seventh chord adds a note below it that the tempered chord does not have. The computation is the missing-fundamental estimate of the essays on residue pitch, run on each chord’s combined spectrum and compared with the tempered chord’s, which has no single periodicity at all.