A seventh chord cannot be spaced to last like a triad

A fifth on a piano is not a fifth a second later found, after a string does everything at once laid out the partials a struck note is made of, that the thing that makes a struck interval consonant — its two notes’ partials coinciding — has a lifetime. Every partial decays, the higher ones fastest, and an interval keeps its consonance only while the partials where its notes coincide are still above threshold. An octave coincides at the lower note’s second partial and lasts; a major third coincides at the fifth and goes sooner. A chord is several intervals at once, and it lasts as long as its shortest-lived pair.

A minor triad can be spaced to last enumerated every spacing of both triads over one bass and found the same ceiling for each: 1.26 seconds over C3 — after an inversion lasts as long as its outer sixth and an open triad lasts as long as its tenth had each found one spacing’s weakest link — reached by at least one spacing of each quality, because a triad always contains a third or a sixth and the best any presentation of one does is 1.26. It ended on the obvious extension. A dominant seventh has four notes and six pairs, so more chances to contain something short-lived — and it contains a minor seventh, 9 : 5, which coincides at the ninth partial. If every spacing of a seventh chord has to include something that bad, the lifetime measure would say something quite strong about why seventh chords are the chords that resolve.

The census has been run over four notes, for five kinds of seventh chord, and it says that. It also says something the question did not expect.

What a pair of pitch classes can be #

The earlier essays’ single most useful observation is about menus. Two pitch classes a given distance apart can be presented as only a few intervals. A distance of three semitones is a minor third, a major sixth, a minor tenth or a major thirteenth; a distance of two is a major second, a minor seventh, a major ninth or a minor fourteenth. The distances run from one to six, since seven is five the other way round.

Each presentation has a just ratio and so a coincident partial, and so a lifetime. The best entry on a distance’s menu is the longest that pair of pitch classes can possibly last, however the chord is spaced.

Read as menus, the table gives six numbers. A semitone distance — minor second, major seventh, minor ninth — coincides at the fifteenth partial in every presentation, and the best of them lasts 0.38 seconds. A tone distance — major second 9 : 8, minor seventh 9 : 5, major ninth 9 : 4 — coincides at the ninth, and the best lasts 0.64. A minor third distance is best as a major sixth, 5 : 3, at 1.26. A major third distance is best as a major third or tenth, also 1.26. A fourth distance is best as a fifth or a twelfth, 2.14. A tritone has a 7 : 5 presentation that lasts 0.87.

The minor seventh’s menu is the one to notice. A minor seventh, a major second and a major ninth are three very different intervals to a musician and they all coincide at the ninth partial of their lower note, so no choice among them can buy a dominant seventh a longer life. The seventh the earlier essay named was never the problem on its own: the problem is the tone distance, and the chord cannot be spaced away from it.

The bound, before any spacing is chosen #

That gives a ceiling for any chord that is computed from its pitch classes alone. A chord can last no longer than the best presentation of its worst distance, give or take the few hundredths register adds.

A major triad’s distances are a minor third, a major third and a fourth: the worst menu is 1.26, which is the ceiling the earlier census found. A dominant seventh, C E G B♭, has distances of a major third (C–E), a fourth (C–G), a tone (C–B♭), a minor third (E–G), a tritone (E–B♭) and a minor third (G–B♭). The worst menu is the tone’s, 0.64. The major seventh C E G B has a semitone distance between B and C, and its bound is 0.38. The minor and half-diminished sevenths both contain the tone distance between the seventh and the root, bound 0.64. The diminished seventh, C E♭ G♭ A, contains only minor thirds and tritones, and its worst menu is the tritone’s: 0.87.

The census checks the bound against every inversion and every spacing. Over C3 the best spacings land at 0.66, 0.42, 0.69, 0.69 and 0.86 against bounds of 0.64, 0.38, 0.64, 0.64 and 0.87. The small excesses are register, the same few hundredths the earlier essay traced: the bound takes every pair at the bass, and a pair whose lower note sits higher up has its partials where the threshold of hearing is lower and lives slightly longer. For every chord in the figure, the bound is reached to within register — some spacing presents its worst distance in its best form without spoiling any other pair. That is not guaranteed, and the next section is the chord for which it fails.

Where the bound is not reached #

The bound is a ceiling and not a promise, and one chord shows exactly why. The augmented triad, C E G♯, has three distances and all three are major thirds, whose best presentation lasts 1.26 seconds. Its bound is therefore a triad’s. Its census, over the same twelve spacings any triad has, reaches only 0.74.

The reason is the same kind of constraint that decides which chord a guitar tuning gives up. Three major thirds stacked add up to an octave, so the three pairs cannot be presented independently. Put E a major third or a tenth above C, and G♯ a major third or a tenth above E, and C to G♯ is then a minor sixth or a compound minor sixth — which is the same distance, four semitones the other way round, presented in its bad form, 8 : 5, at 0.74 seconds, or 16 : 5 at 0.36. Every arrangement of the three notes has to present at least one of its identical distances the long way round the octave, and the best it can do is 0.74. The menus are right about each pair and wrong about the chord, because the pairs of a chord are not free.

The seventh chords escape that problem because their worst distance occurs only once or is flanked by distances with room to spare, and so some spacing presents it well without forcing any other pair into a bad form. The diminished seventh’s four minor thirds would have the augmented triad’s problem if a minor third’s best form were what limited it — four minor thirds round an octave cannot all be major sixths — but the tritone ends the chord first, at 0.87, and a minor third presented badly as 6 : 5 still lasts 1.04.

The register residue can also be larger than a few hundredths. A suspended fourth, C F G, has a tone distance between F and G, bound 0.64, and its best spacing reaches 0.74 by putting F and G two octaves above the bass, where a major second’s ninth partial sits in the ear’s most sensitive region and outlasts the same interval at the bass by a tenth of a second.

Every voicing of a dominant seventh #

The census itself shows how the ceiling is reached and how far below it most voicings sit.

The best-lived voicing, 0–3–6–8 semitones over the bass, is the chord in first inversion packed close: the third in the bass, then the fifth, the seventh and the root a tone above it. It lasts 0.66 seconds and it ends on that tone — the root and seventh presented as a major second, which is no worse than presenting them as a minor seventh would have been, since both coincide at the ninth partial.

Most of the rest end sooner, because they present some distance in a worse form than they had to — a compound minor third, which coincides at the twelfth partial, or a compound tritone — and the shortest-lived voicing that lasts at all, 0–15–18–20 semitones, keeps its coincidences for 0.36 seconds. Four voicings end at once, because they place the seventh a minor fourteenth above the root, twenty-two semitones, and that interval has no just presentation within twenty cents of a coincidence at all: its partials beat from the strike.

So a dominant seventh can be spaced badly in many ways and well in only a few, and even the well-spaced ones lose their consonance in about half the time a triad keeps it.

The diminished seventh, which lasts longest #

The surprise is at the other end of the figure at the head of the essay.

The diminished seventh is the chord harmony textbooks treat as the most unstable of the seventh chords — ambiguous, restless, resolving in several directions — and on this measure it is the most durable of them by a clear margin. Its close root position lasts 0.86 seconds, two tenths longer than any dominant seventh and more than twice as long as any major seventh. It is ended by its tritones, whose best presentation, 7 : 5, coincides at the seventh partial.

The reason is the menu. The diminished seventh is four minor thirds stacked round the octave, so its distances are four minor thirds and two tritones. It has no tone and no semitone distance, and those are the two short menus. Every other seventh chord has one of them, because a seventh chord that contains a root and a seventh a tone or a semitone apart as pitch classes cannot avoid the ninth or fifteenth partial however it is spaced.

That is a reminder of what the measure is. Coincidence lifetime is about how long a struck chord’s intervals keep their partials aligned, not about tonal function. The diminished seventh’s instability in harmony is a matter of its symmetry — every note can be heard as a leading note — and of the tritones that pull toward resolution, and neither of those is a question about partials. What the census says is only that, struck and left to ring, the diminished seventh stays in consonance longer than the dominant seventh does.

The order does not change with register #

The triads’ ceilings drifted by a few hundredths across the compass. The seventh chords were checked the same way.

Over C2 the dominant seventh’s best spacing lasts 0.65 seconds, over C3 0.66 and over C4 0.74; the diminished seventh’s 0.83, 0.86 and 0.90; the major seventh’s 0.40, 0.42 and 0.41. The triads sit at 1.16 over C2 and 1.26 over C3 and C4. At every bass the triads lead, the diminished seventh comes next, the three sevenths with a tone distance tie near two thirds of a second, and the major seventh is last. The ranking is a property of the pitch classes and the register only scales it.

Which computation produced the numbers #

Each note is a struck string with partials falling as one over their number, sounded at 80 dB, every partial decaying with a sixty-decibel time of six seconds divided by its number. An interval’s life is the time its strongest coincident partial pair stays audible, using the five-limit ratio for each interval class and allowing twenty cents between the pair; a chord’s life is the least of its pairs’ lives, which the figures verify for every voicing. Voicings keep the chosen bass note fixed and place every other pitch class in any octave that keeps the chord within two octaves of the bass; inversions choose which pitch class is in the bass. The bound takes, for each of the chord’s pitch-class distances, the best life among that distance’s presentations within two octaves, with the lower note at the bass, and the least of those.

What the census leaves out #

The minor seventh has more than one just ratio. The census tunes it as 9 : 5. As 7 : 4 — the harmonic seventh, which barbershop singers and some brass players reach for — it coincides at the seventh partial instead of the ninth and would last noticeably longer, lifting the dominant seventh’s bound. That is a tuning decision, not a spacing one, and the census has held the tuning table fixed.

A consonance that has decayed is not a dissonance. A chord past its lifetime still sounds, and the top that falls while the note lasts showed how much of its series is still audible then; its partials simply no longer align above threshold, and what a listener hears as roughness depends on which partials beat and how fast, which eighty-one chords, one number priced separately for dominant sevenths and found varying by a factor of five and a half across spacings.

The strings are equally loud and equally damped. A real voicing weights its notes — a bass struck harder, a doubled root — and a piano’s decay times vary across the compass by more than the partial-number law allows, as does the stretch of its partials that the piano is tuned wrong on purpose to accommodate.

What a lifetime cannot say about a resolution #

Whether seventh chords resolve because their consonance is brief. The arithmetic makes the earlier essay’s proposal checkable and consistent: every seventh chord’s intervals lose their alignment in about half the time a triad’s do, at every spacing and in every register. It does not show that anyone hears a chord’s resolution as a response to that, and the diminished seventh is an immediate counter-example to the strongest reading, since it is both the most durable seventh chord here and one of the most urgently resolving in practice.

Whether the lifetime measure means anything past three notes. It gives clean, bounded, register-stable answers for four-note chords, which is some evidence that it does. But a four-note chord’s lifetime is always one pair’s, and a listener hears six pairs at once, several of them long gone and several still aligned.

Whose sevenths #

Seventh chords in Classical harmony are almost always prepared and resolved within a beat or two, and at those tempos a struck chord is released well within its half-second or so of coincidence; the finding bites on sustained chords — a fermata on a dominant seventh, an organ holding one, a piano with the pedal down. Jazz voicing practice, which leaves sevenths unresolved and often spaces them to separate the root from the seventh by a ninth or more, is choosing presentations of the tone distance, and on this arithmetic every one of them is 9-limited and none escapes the ceiling. Barbershop’s harmonic seventh escapes it by retuning rather than respacing.

Still open: the seventh tuned as a harmonic seventh #

The census held the minor seventh at 9 : 5 because the collection’s ratio table does. A dominant seventh with its seventh tuned as 7 : 4 changes the tone distance’s menu — the minor seventh then coincides at the seventh partial, while the major second and ninth above the seventh stay 9-limited unless retuned too — and so it changes which presentation is best and possibly which pair ends the chord.

The computation is the census above with one entry in the ratio table replaced, and it would answer a question with a practical edge. If the harmonic seventh lifts the dominant seventh’s ceiling to the diminished seventh’s or beyond, then the singers and players who tune it that way are buying consonance that lasts, and the chord’s reputation for needing to resolve is partly a fact about equal temperament’s 9-limited seventh rather than about seventh chords. If it does not, because the chord’s other tone-distance presentations still end it first, the ceiling belongs to the chord’s shape.