An open triad lasts as long as its tenth

An inversion lasts as long as its outer sixth struck three-note chords over a fixed bass and asked how long each keeps the partial coincidences of all three of its intervals. The answer was the weakest link, and for every voicing drawn the weakest link was a third or a sixth, because those are what a close triad is made of. The major six-four and the minor sixth chord came out the longest-lived; the major sixth chord and the minor six-four the shortest.

Every voicing in that essay was the closest one over its bass, and a close voicing is how a keyboard reduction writes a chord rather than how anyone spaces one. The essay’s last paragraph named the obvious next step. Raise the middle note an octave and the triad’s intervals change: a root position acquires a tenth, a six-four an eleventh, a minor chord a minor tenth. Whether spreading a chord makes it more durable or less, and whether the six-four’s advantage survives, is the same arithmetic on wider voicings over the same fixed bass.

It answers both questions, and neither answer is the same for major and minor.

Four compound intervals, four clocks #

A compound interval is an ordinary interval plus an octave, and on a harmonic series that is not a small change. The coincidence that names an interval is the lowest pair of partials that meet — partial p of the lower note on partial q of the upper — and adding an octave to the upper note halves every one of its partials’ numbers against the lower note’s. So a compound interval can need either fewer partials than its simple form or more.

Over C3, struck at 80 decibels and decaying with every partial losing sixty decibels in six seconds divided by its number, the four compound intervals an open triad can contain come out as follows.

The major tenth, 5:2, needs the lower note’s fifth partial and the upper note’s second. The major third it replaces, 5:4, needed the fifth and the fourth. The limiting partial is the same fifth, so the tenth lives exactly as long: 1.26 seconds.

The twelfth, 3:1, needs only the third partial of the lower note, and lives 2.14 seconds — as long as the fifth it contains, and longer than anything else a triad holds.

The eleventh, 8:3, needs the lower note’s eighth partial, as the minor sixth does, and lives 0.74 seconds, the minor sixth’s life.

The minor tenth, 12:5, is the one that matters most. Its simple form, the minor third, is 6:5 and needs the sixth partial; the octave above it needs the twelfth, which a struck note loses long before its sixth. It lives 0.47 seconds, less than half the minor third’s 1.04 and shorter than any interval a close triad contains.

Major chords open well and minor chords do not #

Put those clocks into the six voicings.

A major root position opened is C, G and the E above — a fifth, a major sixth between G and E, and a major tenth on the outside. Its close form was held by its minor third, 1.03 seconds; opened, it contains no third at all, and its weakest link is the tenth, at 1.26. Opening gains it almost a quarter of a second and makes it the joint longest-lived voicing a struck triad has.

A minor root position opened is C, G and the E♭ above — a fifth, a minor sixth, and a minor tenth outside. Closed, it lasted as long as its minor third, 1.04 seconds. Opened, it lasts as long as its minor tenth, 0.47. The same spacing that lengthens a major chord cuts a minor one to less than half.

The inversions sort the same way. The major six-four, C–F–A, opened is C–A–F — a major sixth, a minor sixth and an eleventh — and falls from 1.26 seconds to 0.74, held by the eleventh. The major sixth chord, C–E♭–A♭, opened is C–A♭–E♭, with a minor sixth, a fifth and a minor tenth outside, and falls from 0.74 to 0.47. The minor sixth chord, C–E–A, opened is C–A–E, a major sixth, a fifth and a major tenth, and keeps its 1.26. The minor six-four keeps its minor sixth and its 0.74.

The rule is the outer interval again, one octave further out. A close triad lasted as long as its outer sixth; an open one lasts as long as its outer tenth or eleventh, and the kind of tenth decides almost everything. A major tenth on the outside gives 1.26 seconds whatever the chord; a minor tenth gives 0.47.

The same order in every register #

The register does to the open voicings what it did to the close ones: it moves every lifetime together, because a higher note’s partials sit where the threshold of hearing is lower, and it moves no voicing past another. Over every bass from C2 to C5 the two voicings with a major tenth outside are the longest-lived, and the two with a minor tenth outside are the shortest.

The spread between them is wider than it was in close position. Close, the longest and shortest voicings differed by about half a second at every bass. Open, they differ by 0.69 seconds over C2, 0.79 over C3, 0.72 over C4 and 0.97 over C5. Spacing widens the difference that the choice of chord makes, because it moves the weakest link of the best voicings up to a tenth and the weakest link of the worst down to a minor tenth, and the second is a much longer way down.

The one register where the order among the middle voicings moves is the highest, and it moves for a reason worth isolating.

Over C5 the open major six-four and the open minor six-four stop lasting alike, and the reason is not the interval that separates them on paper. Both contain the same eleventh, C5 to F6, and both are held over C5 not by it but by a minor sixth, 8:5 — which needs the eighth partial of its lower note. What differs is which note that is. In the open minor six-four, C5, A♭5 and F6, the minor sixth is C5 to A♭5 and its lower note is the bass, whose eighth partial is at 4.2 kilohertz; it lasts 0.83 seconds. In the open major six-four, C5, A5 and F6, the minor sixth is the upper pair, A5 to F6, and its lower note is A5, whose eighth partial is at seven kilohertz; it lasts 0.66.

An octave lower every one of those partials sits in the flat middle of the threshold curve, and the two minor sixths last the same 0.74 seconds whichever note they are built on. Over C5 the higher of the two eighth partials has climbed to where the threshold of hearing is rising again, and a partial that starts nearer the threshold crosses it sooner. The partial numbers set the order among intervals, and the register breaks only the ties between the same interval on different notes — this is the one such tie the four basses contain.

Every open voicing is smoother, and the order does not hold #

The lifetimes are one of the two measurements the dyad essays split consonance into. The other is roughness, and it behaves the way the spacing rules of thumb say it should.

At the strike, every open voicing is smoother than its close form, at every bass from C2 to C5 — all twenty-four pairs of an open inversion and its close form were checked. Over C3 an open major root position is about a third less rough than a close one at the strike, and a second later less than a third as rough. That is the effect a rough arrival is rough because of its spacing measured for four-part voicings: moving notes out of each other’s critical bands takes pairs of partials out of the roughness sum.

What spacing does not preserve is the order among the inversions.

In close position the major six-four was the smoothest inversion at the strike and at every moment after it. Opened, over middle C, it starts as the roughest of the three and the root position as the smoothest. At 0.72 seconds the order turns over, and from then on the open six-four is the smoothest and the open root position the roughest.

The minor inversions keep their smoothest voicing — the sixth chord, as in close position — and swap the other two at 0.53 seconds. Close, they did not move at all.

The summary over four basses is the point. Close voicings keep their smoothest inversion through the decay in all eight cells; open voicings keep it in four. The major chord opened over C3 and C4 starts smoothest in root position and ends smoothest as a six-four; the minor chord opened over C2 starts smoothest as a sixth chord and ends as a six-four, and over C5 ends in root position.

The reason is the one a fifth on a piano is not a fifth a second later found for dyads. Roughness is decided by pairs of partials within a critical band, and opening a voicing moves its closest pair up into the part of the spectrum that the decay removes first. What separated the close inversions was their lowest pair, sitting on the bass note’s durable low partials; what separates the open ones is a pair an octave higher, and a decay tilts it. Opening trades a ranking decided at the bottom of the spectrum, which a decay cannot overturn, for one decided in the middle, which it can.

Two measurements of the open six-four #

The six-four has now been measured by the two clocks at both spacings, and they disagree about it in opposite directions.

By coincidence lifetime, opening the major six-four is a loss: it falls from the longest-lived major voicing, 1.26 seconds, to 0.74, because its outer interval becomes an eleventh with the minor sixth’s eighth partial in it. By roughness, the open six-four is the smoothest major inversion after the first seven tenths of a second, which close position had it at from the start.

That split is the same split the dyads showed between the coincidence that names an interval and the roughness that scores it, now measured inside a chord. The inversion that cannot end a phrase found every measurement it tried favouring the close major six-four over the practice that restricts it. The open six-four is the first voicing of it that one of the site’s measurements does not favour — and it is the coincidence measure, the one the traditional ranking of consonances agrees with, that turns against it.

The spacing the manuals give #

Orchestration and keyboard-harmony manuals give one spacing rule more often than any other: spread the low notes of a chord, and put the third high. Their reason is roughness — a third is rougher in the bass and a low chord stops being a chord when its partials crowd — and the roughness figures above bear it out, for every voicing.

The coincidence lifetimes add a second reason and a warning. For a major chord in root position the rule’s spacing is root, fifth, and the third a tenth above, and that is the longest-lived voicing a struck triad has: its weakest link is a major tenth, lasting as long as the best close voicing’s sixth. For a minor chord the same rule gives root, fifth and a minor tenth, and that is the shortest-lived voicing a struck triad has. The manuals’ spacing is the best a major chord can do and the worst a minor chord can do, by the measure that ranks intervals the way the theory of consonance does.

That is a result about partial numbers rather than about practice, and it is robust in the way two notes and a ratio says such results are: the 12:5 coincidence needs a twelfth partial in every register, on every struck instrument whose partials decay faster the higher they are, and no loss law that keeps that property gives it a long life.

The model under the lifetimes #

The model is the one the close voicings were drawn with, changed in nothing but the voicings. 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 first approximation the note that gets duller as it dies used. A partial is present while it is above the threshold of hearing at its own frequency. A coincidence is two partials of an equal-tempered pair within twenty cents, its strongest coincidence the one with the lowest partial numbers, and an interval’s life is the moment the first of that pair goes. A chord’s life is the shortest of its three intervals’ lives. An open voicing is the close one with its middle note raised an octave, so the bass never moves and the top note of the close voicing becomes the middle one.

Roughness is the Plomp–Levelt sum over every pair of partials drawn from different notes, the three pairs of a triad added, read at the strike and through two seconds of decay.

What the open voicings still leave out #

One kind of opening. Raising the middle note an octave is one spacing of several. Raising the top note instead leaves a close lower pair under a wide upper one; doubling the bass at the octave adds a fourth note and three more intervals. Each changes the intervals a chord contains, and the weakest-link arithmetic would have to be run on each.

The weakest link, still. A listener might need two of a chord’s three intervals rather than all three. For the open voicings that rule would lift the minor root position from its minor tenth’s 0.47 seconds to its minor sixth’s 0.75 — still much shorter than the same chord closed under the same rule, whose second-shortest interval is a major third. The lifetimes say which coincidence goes first; they do not say how many a listener needs.

Equal temperament. A tempered minor tenth’s twelfth partial and fifth partial are about sixteen cents from coinciding, inside the twenty-cent tolerance. A just minor tenth coincides exactly and lives exactly as long, because a life is set by which partials are involved and not by how close they are.

And one loss law. The seconds scale with the decay time and bend with the exponent; the ordering, set by partial numbers, is the part that survives.

What a lifetime cannot tell a player #

Whether the chord is heard as the same chord. A spread minor triad is still a minor triad, and a listener does not stop hearing it as one at 0.47 seconds. What ends is the partial coincidence of its outer interval, which is one kind of evidence among several a listener has — a major triad’s combination tones are its own notes is another, and it is not affected by spacing the same way.

Whether a composer should care. A chord struck and held on a piano is a short event in most music, and a minor chord in open position that keeps its full evidence for half a second is still sounding for as long as the pedal holds it.

Whose chords #

The practice is keyboard and ensemble writing in the common-practice tradition, where the spacing rules were formulated and where minor and major chords were spaced by the same rule. On an organ, or in a choir, the notes do not decay, every coincidence lives as long as the note is held, and nothing in the lifetimes above exists; the roughness at the strike is then the whole of it, and open voicings are smoother whatever their quality. The asymmetry between major and minor is a property of struck and plucked instruments — the piano, the harp, the guitar — and it is exactly on those instruments that a minor chord in wide spacing is a sound that thins quickly.

Still open: whether the asymmetry is the minor chord’s or the minor tenth’s #

A minor chord can be opened without a minor tenth. Raise the top note instead of the middle one and a minor root position becomes C, E♭ and the G an octave up — a minor third, a major tenth and a twelfth — whose weakest link is the minor third again. So the fragility found here may belong to one spacing of the minor chord rather than to the minor chord. Running every spacing of each triad within two octaves over the same bass — each note placed in either octave, the bass fixed — would say whether any spacing of a minor triad matches the best major one, and whether the manuals’ rule is the best spacing of a major chord or merely a good one.