An inversion lasts as long as its outer sixth

A fifth on a piano is not a fifth a second later put two struck notes an interval apart and watched consonance split in two. The coincidence that makes a fifth a 3:2 — the lower note’s third partial on the upper note’s second — survives the strike for as long as both partials stay above the threshold of hearing, and across the octave those survival times came out in the order the theory of consonance ranks its intervals. The roughness that scores the same pair did something else: it reordered itself within a third of a second, and the fifth was overtaken by four intervals every treatise calls harsher.

Harmony is not built of dyads. The object it is built of is a triad, and a triad is three intervals sounding at once, each with its own clock. An inversion is a choice of which three. So the question the dyads leave is whether an inversion has a lifetime — whether the evidence that makes a six-four a six-four outlasts the evidence that makes a sixth chord a sixth chord, on the same instrument, from the same strike.

One bass, six voicings #

Every voicing here is drawn over the same bass note, which is the convention the inversion that cannot end a phrase used when it set roughness against the six-four. Holding the bass still is what keeps a comparison of inversions from being a comparison of registers: a six-four built upwards from the root of the same chord would have its bass a fifth higher and would score as smoother for that reason alone.

So over C3, a major triad in root position is C–E–G, its sixth chord is C–E♭–A♭, and its six-four is C–F–A. The minor triad’s three positions over the same C are C–E♭–G, C–E–A and C–F–A♭. The pitch classes differ from row to row; the intervals above the bass are what an inversion is, and they are what the drawing compares.

Each voicing contains three intervals — the lower pair, the upper pair and the outer pair — and each interval’s strongest coincidence has a survival time computed exactly as for a dyad: both notes decaying from the strike with every partial losing a rate that rises with its number, and the coincidence alive while partial p of its lower note and partial q of its upper note are both audible. The rule for the chord is the weakest link. A struck chord keeps the evidence of being that chord for as long as the shortest-lived of its three intervals keeps its coincidence, because a chord whose outer interval has lost the partials that identify it is no longer evidenced as that chord, whatever its other two intervals still say.

The dyads already set the order #

The six results come straight out of the dyad table, and it is worth seeing where.

A major sixth is 5:3 and a major third is 5:4, and the coincidence of both needs the upper note’s fifth partial — so the two intervals last almost exactly as long, 1.26 seconds. A minor sixth is 8:5 and needs the lower note’s eighth partial, which a struck note loses early: 0.74 seconds. A minor third, 6:5, needs the sixth: 1.04. The fourth and the fifth use partials four and three and outlast everything else in a triad.

Put those into the voicings. Both root positions contain a minor third and a major third under a fifth, so both are held to the minor third’s 1.03 or 1.04 seconds. The major sixth chord, C–E♭–A♭, has a minor third, a fourth and a minor sixth on the outside, and lasts 0.74. The major six-four, C–F–A, has a fourth, a major third and a major sixth on the outside, and lasts 1.26. In minor the two inversions trade their outer intervals: the sixth chord C–E–A has the major sixth and lasts 1.26, and the six-four C–F–A♭ has the minor sixth and lasts 0.74.

An inversion’s lifetime is its outer sixth’s. An outer major sixth lasts exactly as long as the major third beside it and holds the chord to 1.26 seconds; an outer minor sixth outlasts nothing else in the chord and holds it to 0.74.

The same two voicings, in every register #

A dyad’s coincidence life depends on its register, because the threshold of hearing is not flat and a note’s partials move across it as the note moves. The ordering of the voicings could have depended on the register as well.

It does not. Over C2 the long voicings keep their intervals for 1.16 seconds and the short ones for 0.72; over C5 the long ones for 1.41 and the short ones for 0.83. Every lifetime grows with the bass, because a higher note’s partials sit where the threshold of hearing is lower and a decaying partial takes longer to cross it. The gap between the long pair and the short pair is roughly half a second everywhere, and no register exchanges a long voicing for a short one.

That is the expected result, and it is worth saying why before moving on. A coincidence’s partial numbers are a property of the interval’s ratio, and a ratio has no register. What register changes is how fast each partial’s level falls against a threshold that differs by frequency, and that moves every lifetime up or down together. An ordering set by partial numbers survives transposition in a way an ordering set by hertz does not.

By roughness, the inversions keep their places #

The dyads’ second finding was that roughness does not keep its order through a decay, and that the fifth — the smoothest interval at the strike — was the smoothest for only a sixth of a second at C3. A triad’s inversions are the obvious place to look for the same reversal.

Over middle C the major six-four is the smoothest of the three inversions at the strike and at every moment after it. The root position and the sixth chord exchange places once, early — the sixth chord starts roughest and has become smoother than root position by 0.15 seconds — and after that nothing moves, although every curve falls by two orders of magnitude.

The minor inversions do not reorder at all. The sixth chord is smoothest, the six-four next and root position roughest, from the strike to the moment the notes are down to their last few partials.

Across four basses and both qualities that is the whole of the movement: one exchange between the two non-smoothest major voicings over C4, and nothing else. A dyad’s roughness order is overturned by the decay and a triad’s inversions keep theirs, and the smoothest inversion is the same one — the major six-four, the minor sixth chord — at every register and every moment drawn.

The register effect runs the way it does elsewhere. Over C2 the three major inversions are within four per cent of one another at the strike, because the low notes of a chord are rough whatever they are, and by C5 the smoothest and roughest differ by a quarter. What changes with register is how much the choice of inversion matters, not which inversion wins.

The difference is a matter of what separates the candidates. The eleven dyads differ in their ratio, and a fifth’s smoothness was made of coincidences high in the spectrum that the decay removes first. The three inversions over a fixed bass differ in which intervals they stack, but every inversion contains a third and a fourth or fifth and a sixth or fifth, and what separates them is mostly the lowest pair, which sits on the bass note’s low partials and loses least. The tilt a decay applies falls on all three inversions nearly alike, and a ranking that was decided near the bottom of the spectrum is not a ranking a tilt can overturn.

And the six-four’s lead widens as the note dies #

Keeping its place is not the whole of what the smoothest inversion does. Over middle C the major six-four’s roughness at the strike is 14 per cent below root position’s and 17 per cent below the sixth chord’s. One second later the notes have lost most of their partials, every roughness has fallen by a factor of twenty or more, and the six-four is 45 per cent below root position and 37 per cent below the sixth chord. Over C5 the gap grows further: at the strike the six-four is a tenth smoother than root position and a fifth smoother than the sixth chord, and a second later it is less than half as rough as either.

That is the same direction a fifth on a piano found for the whole octave of dyads — the spread between smooth and rough widens as a struck note decays, rather than closing — arrived at here among three voicings instead of eleven intervals. The difference is that among the dyads the widening came with a reordering, and among the inversions it does not. The voicing that starts smoothest becomes more decisively smoothest, and the same chord is harsher when it is louder already established the complementary fact that the ranking survives a change of level. A triad’s inversions keep their roughness order under both of the operations a piano applies to a chord, turning it down and letting it die.

Whether the definition decides the result #

The lifetimes above rest on the weakest-link rule, and the rule is a choice. The obvious alternative is that a listener needs two of a chord’s three intervals rather than all three, so that a chord stays evidenced until its second interval goes.

Under that rule the numbers move and the short pair does not. Both root positions last until their major third goes, at 1.26 and 1.27 seconds; the major six-four until its major third, at 1.27; the minor sixth chord until its major sixth, at 1.26. Those four now tie. The major sixth chord lasts until its minor third goes, at 1.04, and the minor six-four until its minor third, at 1.03. The two voicings with an outer minor sixth are the least durable under either rule, and what the looser rule dissolves is only the advantage of the long pair over the root positions.

So the robust claim is narrower than the headline and still definite. An outer minor sixth makes an inversion fragile whatever a listener needs; an outer major sixth makes one durable if a listener needs every interval, and merely ordinary if a listener needs two. That is also the order two notes and a ratio would predict from the partial numbers alone — the minor sixth’s 8:5 asks for the eighth partial, and no rule about how many intervals are needed can rescue a coincidence that has gone.

The six-four, measured a fourth time #

The six-four has been set against three measurements already. The inversion that cannot end a phrase found it the smoothest of the three positions by roughness at every register a cadence occurs in, the best explained by a virtual-pitch model, and not separated from the sixth chord by support from the bass’s harmonics. A major triad’s combination tones are its own notes added that the major six-four, 3:4:5, is one of the two voicings whose products land on its own notes, and that the sixth chord is not.

On a struck instrument the major six-four is now also the longest-lived voicing of its triad and the smoothest at every moment of the decay. Four measurements, and every one of them favours the voicing the practice restricts. That essay’s conclusion — that the prohibition is a rule about a two-voice skeleton in which a fourth over the bass counts as a dissonance, not a claim about how the three notes sound — gains a fourth result it has to overrule.

The minor six-four is the one voicing on which the arithmetic and the practice agree. Its outer interval is a minor sixth, so it is the least durable voicing of its triad, and it is not the smoothest: the minor sixth chord is. So the six-four prohibition, which common practice applies to both qualities alike, is contradicted by the arithmetic in major and supported by it in minor, and the asymmetry comes from a single fact about the outer interval — which is the kind of fact a rule about fourths over the bass does not contain.

The model the lifetimes are drawn from #

Each note is struck at 80 decibels with a spectrum falling as one over the partial number, and each partial loses sixty decibels in six seconds divided by its number — the standard first approximation the note that gets duller as it dies used, under which a sixth partial dies six times as fast as the fundamental. A partial is present while it is above the threshold of hearing at its own frequency and absent afterwards, which is the convention the top that falls while the note lasts established. A coincidence is two partials of an equal-tempered pair within twenty cents of each other, and its life is the earlier of the two moments at which one of its partials goes. The roughness is the Plomp–Levelt sum over every pair of partials drawn from different notes, the three pairs of a triad added.

Every choice there was made for the dyads, and none was changed for the chords. The only new decision is the weakest link.

What the three clocks leave out #

The weakest link is a definition, not a measurement. A listener might identify a chord from two of its three intervals once the third has gone, in which case the chord’s effective lifetime would be the second-shortest rather than the shortest, and the ordering here would change: by that measure the root positions, held by a fifth and a major third after their minor third goes, would last longest. The drawing settles which intervals keep their coincidences and not how many a listener needs.

Close position only. A triad voiced with its third an octave higher has a tenth instead of a third, and a tenth’s coincidence uses different partials and has a different life. Every voicing drawn here is the closest one over its bass, which is how a keyboard reduction writes a chord and not how an orchestra spaces one.

Equal temperament. The coincidences are between tempered partials within twenty cents, which is how a piano is tuned, and a tempered major third’s coincidence is already fourteen cents from exact. A just chord’s coincidences would be exact and their lifetimes the same, since what decides a lifetime is which partials are involved and not how close they are; the roughness would differ, roughest in the bass where a tempered third’s near-miss sits inside a critical band.

No doubling, no pedal. A four-part chord doubles a note and adds three more pairs; a sustaining pedal lets other strings ring in sympathy. Both change the population of partials a coincidence lives among, and neither is in the model.

And one loss law. The exponent of one is the textbook’s and not a measured piano’s, and the lifetimes scale with the decay time and bend with the exponent. The ordering, which comes from partial numbers, is the part that survives a different law; the seconds are the part that does not.

Whose chords #

The practice is keyboard harmony on a struck instrument — the chorale-style textures of piano reductions and hymnals, and the cadential six-four that closes a phrase in every style from the late eighteenth century on — and the result is a statement about what the piano itself does to a chord held for a beat. On an organ or in a choir the notes do not decay and every voicing keeps every coincidence for as long as it is held, so none of the lifetimes here exists; the roughness order is then the order at the strike, which is the order the steady-tone essay measured.

Still open: what spacing does to the weakest link #

The weakest link of every close-position voicing here was a third or a sixth, because those are the intervals a close triad is made of. Open the voicing and the intervals change: a root-position triad with its third raised an octave contains a tenth, 5:2, whose coincidence needs only the upper note’s second partial and the lower note’s fifth, and a fifth, and a sixth above that. Whether spreading a chord makes it more durable or less, whether the spacing that orchestration manuals recommend for the bass register is the spacing that keeps a struck chord evidenced longest, and whether the six-four’s advantage survives opening, are the same arithmetic applied to wider voicings over the same fixed bass.