Four notes cannot get below the seventh partial

The seventh has two places to be moved one note of a dominant seventh thirty-one cents flat, onto the seventh partial of the root, and the chord’s longest-lived spacing rose from 0.66 seconds to 0.87 — the diminished seventh’s figure, and the highest any four-note chord in the census of seventh chords had reached. It ended by pointing out that one note is not what a singer moves. A barbershop quartet tunes the whole chord to itself, 4 : 5 : 6 : 7 over a fundamental two octaves below the root, and that moves the third fourteen cents flat and the fifth two cents sharp as well.

The question it left had two possible answers and they mean different things. If the whole chord tuned to the series reached past 0.87, then 0.87 belonged to a half-measure and the ceiling was a fact about keyboards. If it did not, then 0.87 might be a ceiling for any four notes however they are tuned, and the reason seventh chords do not last would be arithmetic about the number four.

It does not reach past it — not by a millisecond. And once one chord has been asked, every chord of four notes a harmonic series holds can be asked too, which turns a statement about the dominant seventh into a statement about four notes.

What the third and the fifth were already doing #

Start with the dominant seventh itself, because the result there is exact and its exactness is the first clue.

The one-note retuning put the chord’s best voicing at 0–16–19–22 — root in the bass, the third, fifth and seventh an octave and more above it — lasting 0.868 seconds. Tune the third and fifth to the series as well and the best voicing is 0–16–19–22, lasting 0.868 seconds. The two figures agree to the last digit printed, and the reason is visible pair by pair.

The tempered major third is 13.7 cents wider than 5 : 4 and the tempered fifth 2.0 cents narrower than 3 : 2. The model a pair’s life rests on — two partials that fall within twenty cents of each other count as meeting, and the pair lasts as long as both are audible — does not ask how close the meeting is, only whether it happens. A tempered third is inside the tolerance of 5 : 4 and a tempered fifth is nowhere near the edge of 3 : 2, so both already coincide at the partials they would coincide at if tuned exactly. Tuning them exactly moves them from inside the window to its centre, and the life of a coincidence does not depend on where inside the window the two partials fall.

So root to third stays at 1.26 seconds, root to fifth at 2.14, third to fifth at 1.26. None of the three moves.

The three pairs with the seventh in them behave exactly as they did when the seventh alone was moved. Root to seventh rises from 0.64 to 0.87, third to seventh stays at 0.87, fifth to seventh falls from 1.26 to 0.87, and the chord’s bound — the worst of its six pairs at their best — is 0.87. The earlier essay called that a levelling rather than an improvement, and nothing about tuning the other two notes disturbs it, because the other two notes were never the pairs that limited the chord.

The whole-chord tuning is the one-note tuning plus two corrections the model had already granted. That is why the ceiling does not move. It is also why this result is not the one a barbershop singer would predict: to the ear, a third that has been lowered to 5 : 4 is a different sound from a tempered one, because the beating between its fifth partial and the root’s fourth — beats are arithmetic counts them at about ten a second at C4 — stops. That is a real difference and it is not a difference in how long the coincidence lasts. The lifetime measured here and the locking a quartet listens for are two properties of the same tuning, and only one of them is on the figure.

Tuning the rest of the chord costs seven voicings #

If the whole-chord tuning buys nothing at the top, it is worth asking what it does lower down, where the one-note census had found the median voicing unmoved.

Twenty-two of the thirty-two voicings are unchanged. Three gain, and each by about a thousandth of a second, which is register rather than tuning. Seven lose, and not trivially. The largest loss is the voicing 0–4–7–22 — a close root-position triad with the seventh an octave and a half above it — which falls from 0.54 seconds to 0.43. The median voicing falls with the seven, from 0.47 seconds to 0.43.

The mechanism is the same one that made the valley in the earlier essay, and it deserves its own sentence because it runs against intuition: a tempered interval can sit inside the tolerance of a coincidence its just version misses. In 0–4–7–22 the limiting pair is the third against the seventh, seventeen and two-thirds semitones apart. With a tempered third and a septimal seventh that distance is 1,769 cents, which is 17.7 cents from 11 : 4 — inside the window, meeting at the eleventh partial of the third. Lower the third by 13.7 cents to 5 : 4 and the distance becomes 1,783 cents, which is 31.7 cents from 11 : 4 and outside, and the nearest coincidence left is 14 : 5, at the fourteenth partial. A fourteenth partial dies sooner than an eleventh, so the pair dies sooner, so the chord does.

That is the sense in which a tempered chord is not simply a just chord made worse. Its intervals are each a few cents from a simple ratio, and in a compound spacing those few cents can land a pair near a different simple-ish ratio, a coincidence the pure tuning cannot reach. The loss is real only inside this model, since it rests on a tolerance chosen once and held; but it is a warning that “tuned to the series” and “longer-lived in every spacing” are not the same claim, and the second one is false.

Every chord a series holds #

The dominant seventh as 4 : 5 : 6 : 7 is one chord a harmonic series contains. It is the first one, in the sense that its parts are the lowest possible — but whether it is the best is a separate question, and the series contains a great many others.

A chord tuned to a series is a choice of partials, and since an octave changes a partial’s number without changing its pitch class, it is really a choice of odd parts: 1 for the fundamental and its octaves, 3 for the twelfth and its octaves, 5, 7, 9, and so on. Four pitch classes need four odd parts. Up to fifteen there are eight odd numbers and seventy ways of choosing four, and reading each choice down the series instead of up — as 1/1, 1/3, 1/5, 1/7 rather than 1, 3, 5, 7 — gives its mirror image. With the handful that collapse two parts onto one tempered pitch class removed, that is 140 chords, each tuned exactly and each put through every spacing within two octaves.

The hero figure above is the whole of the answer. Grouped by the largest odd part the chord needs, the best chord in each group lasts 0.87 seconds at seven, 0.73 at nine, 0.57 at eleven, 0.54 at thirteen and 0.69 at fifteen. The top two positions are held by one choice of parts, 1, 3, 5 and 7, read both ways, and nothing else comes within a tenth of a second of them.

The group at fifteen outdoes the groups at eleven and thirteen, which looks like a crack in the rule and is not. Fifteen is three times five, so a chord like 3 : 5 : 9 : 15 has pairs whose ratios reduce: 15 against 9 is 5 : 3, 15 against 5 is 3 : 1. The largest part a chord uses is not the partial at which its worst pair meets. What decides a chord’s life is the ratios between its notes, and 1 : 3 : 5 : 7 is the only choice of four parts in which every one of the six ratios has both of its terms at seven or below.

The number four #

Which suggests the argument that does not need a series at all.

A chord lasts as long as its worst pair, and a pair can only be presented as one of a few intervals — its interval class, placed in some octave. So the ceiling for four notes is a question about which interval classes four notes must contain, and how long the best-tuned member of each class can possibly last.

The sweep here is deliberately generous. Each class is allowed any tuning within a quarter-tone either side of its tempered size — past that it would be a different class — and any presentation up to two octaves, and the figure records the longest life found anywhere in that region.

The three classes a triad is made of come out well. A fourth or fifth can meet at 3 : 2 and last 2.14 seconds; a major third or minor sixth at 5 : 4, and a minor third or major sixth at 5 : 3, each 1.26. The other three are capped. A tone tuned as 7 : 4 below the octave, and a tritone tuned as 7 : 5, each reach 0.87 seconds and no tuning within a quarter-tone does better. A semitone reaches only 0.52, and even that needs the interval pushed to the very edge of its class, where it is a major seventh nearly a quarter-tone flat meeting at 11 : 6.

Then the counting. There are 165 sets of four pitch classes containing a given note. A set whose six pairs all fell in the three good classes — every pair a third, a sixth, a fourth or a fifth — would be a four-note chord with no semitone, no tone and no tritone anywhere in it. There is no such set. All 165 contain at least one of the three capped classes; 130 contain a semitone, 130 a tone, 85 a tritone. The sixteen that avoid both the semitone and the tritone all contain a tone, and exactly one set avoids both the semitone and the tone: 0, 3, 6, 9, the diminished seventh, whose only capped pair is its tritone.

So every four-note chord has a pair that cannot last past 0.87 seconds, and the ceiling is proved rather than observed: four notes, tuned any way within a quarter-tone of the twelve, cannot keep their coincidences longer than a seventh partial lets them. Three notes can, because a major or minor triad is the one arrangement of three pitch classes made entirely of the good classes — the reason three notes at once found two of the four ways of stacking thirds so much simpler than the other two, and why the four-note stack of fourths in a stack that is not thirds, smooth as it is, carries a tone between its outer notes. A fourth pitch class has nowhere to go that is a third, sixth, fourth or fifth from all three of the others.

That is the arithmetic the earlier essay’s last sentence was reaching for. The first six partials of a series hold three pitch classes, which is why the series is not a chord found a major triad and nothing else in them; the fourth pitch class the series offers is the seventh partial, and no fourth pitch class from anywhere does better.

The chord the one-note retuning ruined #

The mirror-image chord at the top of the hero figure is worth separate attention, because it is a chord the earlier essay condemned.

Read downwards, 1/1 : 1/3 : 1/5 : 1/7 over C is C, F, A♭ and D — a half-diminished seventh on D. Put D at the bottom and its notes stand in the ratios 1/7 : 1/6 : 1/5 : 1/4, which places 7 : 6 between root and third, 7 : 5 between root and fifth and 7 : 4 between root and seventh. Against the tempered chord that lowers the third 33.1 cents, the fifth 17.5 and the seventh 31.2.

In the seventh has two places to be, moving only the half-diminished seventh’s seventh down to 7 : 4 cost the chord a third of its life, from 0.69 seconds to 0.42, because it wrecked the minor third between the fifth and the seventh. Tuned as a whole to the mirror series, the same chord reaches 0.87 — the ceiling — with its upper three notes, a minor triad, left exactly as they were.

The dominant seventh and the half-diminished seventh are the same four odd parts, one read up and one read down, and the tie between them is exact because a pair’s menu does not care which of its notes is on top: a minor triad can be spaced to last found that symmetry for triads and it holds for any pair. What differs is where the seventh partials sit. In the dominant seventh every pair containing the seventh meets at a seventh partial; in its mirror every pair containing the root does. One chord has a note that is the seventh of everything, and the other a note that everything is the seventh of.

That second reading has a history. Nineteenth-century harmonic dualism held that a minor chord was a major chord hung downwards from its top note, and the root an ear supplies found the difficulty with that at once: no instrument produces an undertone series, so a listener’s pitch mechanism has nothing downward to latch onto. The census does not contradict that — it knows nothing about roots — but it does say that the dualist tuning of the half-diminished seventh is exactly as long-lived as the harmonic tuning of the dominant seventh, which is more than the dualists ever claimed for it.

The same ceiling at every bass #

A result about partial numbers ought to be independent of where the chord is played, and it is.

Over C2 the three four-note chords each last 0.83 seconds; over C3, 0.87, 0.87 and 0.86; over C4, 0.90. The triad lasts 1.16 seconds over C2 and 1.26 over the other two. Register moves every value a few hundredths — a higher chord puts its partials where the ear’s threshold is lower — and it moves no comparison at all.

The three chords agree because they are three instances of one limit, reached by three routes. The diminished seventh gets there by accident: it is the only four-note set with neither a semitone nor a tone, and its tempered tritone is 17.5 cents from 7 : 5, inside the tolerance, so it touches the seventh-partial ceiling without being retuned. The dominant seventh gets there by lowering one note thirty-one cents. The half-diminished seventh gets there by lowering three. Every other four-note chord is below them, and nothing that can be done to its tuning lifts it through.

How a pair’s life is measured here #

Each note is a struck string of sixteen partials falling as one over their number, sounded at 80 decibels, every partial decaying with a sixty-decibel time of six seconds divided by its number. Two notes coincide at p : q when partial p of the lower and partial q of the upper fall within twenty cents of each other; the pair’s life is how long its strongest such coincidence — the one with the lowest partials — stays above the threshold of hearing on both sides. A chord lasts as long as its worst pair, and its best spacing is found by trying every inversion and every placement of the upper notes within two octaves of the bass.

Tuning enters as an offset in cents for each pitch class, applied after the spacing is chosen, so which octave a note is in and how it is tuned are separate decisions. A chord tuned to a series takes its offsets from the exact ratios of its odd parts. The interval-class sweep tries every whole-cent tuning within 49 cents of each class, both of its forms and every octave displacement up to two octaves, and keeps the longest life found.

What the ceiling does not say #

That a four-note chord is less consonant than a triad. It says the partial coincidences that evidence a chord’s intervals die sooner, which is one account of why a struck seventh chord blurs. Roughness is a different quantity — eighty-one chords, one number found a dominant seventh’s roughness varying by a factor of five and a half across its spacings — and a chord whose coincidences are gone can still be smooth.

That the twenty-cent tolerance is right. The loss of seven voicings rests on it: a narrower window would cost the tempered chord its accidental elevenths, and a wider one would give more tempered pairs coincidences they have no business having. The proof that four notes cannot pass the seventh partial does not rest on it, because it is a statement about which ratios each interval class can be tuned to, and a quarter-tone of freedom is far wider than any tolerance.

What a real voice or a real piano does. The strings here are equally loud and decay by one law; a quartet’s voices do not decay at all while held, and what they hear is the beating stop, which is instantaneous rather than a lifetime. The ceiling is a property of struck chords in this model and a statement about the arithmetic of the twelve, and the arithmetic part is the only part that would survive a change of instrument.

Anything about five notes. A fifth pitch class cannot make things better, since any five pitch classes contain four and therefore a capped pair — but how much worse, and whether a ninth partial or an eleventh sets the next ceiling, is a separate count.

Still open: whether a chord the ceiling allows is one a listener prefers #

Three routes reach the same 0.87 seconds and they are not the same sound. The diminished seventh reaches it with every note on the keyboard’s pitch and four equal minor thirds; the dominant seventh by lowering its seventh a third of a semitone; the half-diminished seventh by lowering three notes, one of them by a third of a semitone and another by a sixth.

Within this model they are indistinguishable, and that is the point at which the model has said all it can. The quantity it would need to separate them is one it already has for pairs and has never totalled for a chord: the beating that remains among the partials that do not coincide, which is where the three chords differ most. The diminished seventh is a lattice of identical intervals and every one of its non-coincident partial pairs beats at a rate the others repeat; the dominant seventh tuned to the series has no beats among its low partials at all. Summing the beating a chord’s non-coincident partials produce across its life, at its best spacing, would say whether a ceiling shared in lifetime is also shared in how much a listener hears the chord move — and it would be the first comparison in these essays that could put a tuned seventh chord above the diminished seventh rather than beside it.