The seventh has two places to be

A seventh chord cannot be spaced to last like a triad censused every inversion and spacing of five seventh chords and found each of them bounded by its own pitch classes. A chord keeps its consonance while the partials at which its notes coincide are still above threshold — the mechanism a string does everything at once laid out and a fifth on a piano is not a fifth a second later turned into a clock — and it lasts as long as its worst pair; a dominant seventh’s worst pair is the tone between its root and its seventh, which coincides at the ninth partial and lasts 0.64 seconds, so no spacing of the chord reaches two thirds of a second where a triad reaches 1.26.

That census took its ratios from the five-limit table used throughout these essays, in which a minor seventh is 9 : 5. Its own list of omissions put the alternative first:

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.

Run with the seventh moved, the census says so, and it says four other things that the prediction did not contain.

The retuning is a change of pitch, not of table #

It is worth being exact about what was altered, because the difference decides everything below.

The computation does not consult a ratio table when it asks how long a pair lasts. It takes the interval as played, looks for partials of the two notes that fall within twenty cents of each other, and asks how long both survive. So “tuning the seventh as 7 : 4” is not the substitution of one entry for another. It is the seventh played 31.2 cents flat of where a keyboard puts it, and the coincidence that follows is a consequence, not an instruction: a note that far down finds the seventh partial of the root and loses the ninth.

That matters because the change is continuous and the coincidences are not. A pair of partials is either inside the tolerance or outside it, so what a sweep of the seventh’s pitch produces is a series of plateaux, and the plateaux need not touch.

They do not. As tempered, the root and seventh are a tempered tone short of an octave apart and the 9 : 5 coincidence sits 17.6 cents away, just inside the tolerance, and the chord lasts 0.66 seconds. Move the seventh down by six cents and that coincidence is 23.6 cents away, outside it, and gone; 7 : 4 is still 25 cents away, and not yet found. The chord lands on 0.46 seconds, which is worse than either destination and worse than the tempered chord it started from.

That is the valley, and its width is the tolerance’s. It runs from about four cents flat to about thirteen, and a performer who narrows the seventh a little because it sounds sharp has walked into it.

Which pairs move #

The gain is not where the prediction put it either, and the six-pair table is where to look.

The root-to-third and root-to-fifth pairs are a major third and a fifth whatever is done to the seventh, and they last 1.26 and 2.14 seconds either way. The third-to-fifth pair is a minor third at 1.26. None of those three is touched.

Of the three pairs that contain the seventh, the root’s is the one the prediction was about: a tone distance, 0.64 seconds as tempered, and 0.87 retuned. The third-to-seventh pair is a tritone at 0.87 both ways, because a tritone’s best presentation already coincides at a seventh partial and moving the seventh thirty cents does not change which partials meet.

The fifth-to-seventh pair is the one nobody was watching. As tempered it is a minor third — 1.26 seconds, one of the chord’s longest-lived pairs. Retuned it is a minor third thirty-one cents narrow, which loses 6 : 5 and finds 7 : 6, and it falls to 0.87.

So all three arrive at the same number, and the number is what a seventh-partial coincidence is worth. The retuning does not improve the chord’s pairs; it levels them. One pair rises by a third of a second, one falls by a third of a second, and the chord’s life — which is its worst pair’s — rises because the worst pair is now as good as any of the three the seventh is in.

That is a different kind of result from the one the prediction imagined, and it is the reason the ceiling lands where it does. The septimal seventh cannot lift the chord past 0.87 however it is spaced, because every pair it belongs to is stuck at 0.87 together.

The seventh partial, which these essays keep arriving at #

There is a pattern in the last three sections worth pulling out before the ceiling, because the same number keeps appearing and it is not a coincidence in the loose sense.

Every pair that the retuning touches ends at a coincidence involving a seventh partial: the root and seventh at 7 : 4, the third and seventh at 7 : 5, the fifth and seventh at 7 : 6. Three different intervals, three different pairs of partials, one number — 0.87 seconds — because what decides a coincidence’s life is how high up the series the two meeting partials sit, and a seventh partial is a seventh partial whichever note it belongs to.

That is the same fact an inversion lasts as long as its outer sixth and an open triad lasts as long as its tenth each found from one side: a chord’s life is read off a partial number, not off an interval name. The interval names in the figures above — tone, minor third, tritone — are three ways of arriving at the seventh harmonic, and once they have arrived they are interchangeable.

It is also why the seven-limit is a boundary rather than a step. A stack that is not thirds censused every chord built from one repeated interval and found the good ones clustered at small numbers; three notes at once found the same of the four ways of stacking two thirds. Going from five-limit to seven-limit buys a chord one more family of coincidences and it buys it at the cost of thirty-one cents, which is more than a third of a semitone and audible as a change of note. Going to eleven would buy another at a cost nobody would call a tuning adjustment.

The ceiling it reaches, and whose it is #

0.87 seconds is not an arbitrary figure. It is the diminished seventh’s.

The earlier census found the diminished seventh the most durable of the seventh chords, at 0.86 seconds, limited by its tritones — whose best presentation, 7 : 5, coincides at the seventh partial. A dominant seventh with a septimal seventh is limited by three pairs that also coincide at seventh partials. They are the same ceiling because they are the same partial.

So the septimal seventh does not make the dominant seventh a better chord than the diminished seventh. It makes it exactly as good, and by the same mechanism. Any four-note chord in this census whose worst pair coincides at the seventh partial lands at about 0.87 over C3, and nothing here reaches past it, because a chord of four notes has six pairs and the arithmetic of the twelve gives at least one of them a distance whose best presentation is septimal or worse.

The register check is the same one that census ran and it says the same thing. Every value moves by a few hundredths between C2 and C4, and no comparison moves at all. The retuned chord and the diminished seventh coincide to within four hundredths of a second at every bass.

The spacing changes completely #

A finding that the census was in a position to produce and nobody asked for: the best voicing is not the same voicing.

As tempered, the longest-lived arrangement is 0–3–6–8: the third in the bass with the fifth, seventh and root packed above it. Retuned, it is 0–16–19–22: root position, spread over nearly two octaves, with the seventh a compound harmonic seventh above the bass. The pair that ends it is that outer one, coinciding at the seventh partial of the bass against the second of the seventh — which is a very strong coincidence and needs the width to be reached.

The rest of the ranking barely moves. The median voicing lasts 0.47 seconds at either tuning, and the two curves are indistinguishable through their middles — which is the shape of a gain that belongs to one arrangement rather than to the chord.

Why the wide voicing wins is worth a sentence, because it is not the width. A pair’s coincidence lasts longer when the meeting partials are low, and the outer pair of 0–16–19–22 meets at the seventh partial of the bass against the second of the seventh. Spread over two octaves, the seventh is high enough that its second partial is where the bass’s seventh is; packed close, the same two notes meet at the seventh of the lower against the fourth of the upper, which is two partials further up and dies sooner. The width is not buying separation. It is buying a lower partial number on one side of one pair. What the retuning buys is a better best rather than a better chord, and a player who tunes the seventh down and keeps the close voicing gets nothing; the two decisions have to be made together.

That is a practical statement of an unusual shape for this collection. Voicing and intonation are ordinarily separate subjects — one belongs to the arranger and one to the performer — and here the arithmetic says they are one decision, because the presentation a pitch class can best be given depends on how the pitch class is tuned.

What the same move does to four other chords #

The prediction’s last clause was that if the retuning failed it would be because the chord’s other tone distances ended it first. It does not fail for the dominant seventh. It fails, badly, for two of its neighbours.

The major seventh is unmoved — 0.42 to 0.41 — because its worst distance is the semitone between its seventh and its root, which coincides at the fifteenth partial and is not what was retuned. Moving a major seventh down by thirty cents makes it a worse semitone rather than a better anything.

The minor and half-diminished sevenths lose a third of their life, from 0.69 to 0.42. Their seventh stands a minor third above their fifth rather than a major third, and a minor third narrowed by thirty-one cents is nowhere: it has left 6 : 5 and it is not near 7 : 6, which sits 15.6 cents the other side. Their root-to-seventh pair gains exactly as the dominant seventh’s does, and it is not the pair that ends them.

So the septimal seventh is not a general improvement to seventh chords, and it is not even a property of the interval. It is a property of the chord the interval sits in, and specifically of what stands a third below the seventh. Where that is a major third the retuning helps; where it is a minor third the same retuning costs more than it buys.

That sorts the repertoire in a way a reader can check against practice. Barbershop’s seventh chord is a dominant seventh and the tuning is the one this figure rewards. A jazz minor seventh sung the same way would be the second case, and the arithmetic says it should not be — which is a prediction rather than an observation, and a falsifiable one.

Whose sevenths #

Three practices tune a dominant seventh’s seventh deliberately flat and the arithmetic treats them differently.

A barbershop quartet’s seventh is the case the figures are about, and it is sung in a close voicing in the middle of the compass — which is exactly the arrangement the census says gains least. The retuned chord’s long life needs the outer pair to be a compound harmonic seventh, and a quartet singing within an octave and a half cannot present it that way. What such a quartet buys is the levelling rather than the ceiling: its worst pair rises from 0.64 seconds to 0.87 even in a close spacing, which is the same gain as the census’s best voicing makes, but its best pair falls from 1.26 to 0.87 along with it.

A natural-trumpet or natural-horn player has no choice at all. The seventh harmonic is where the instrument puts it, and the register where the series becomes a scale is about what that costs a melody. What it gives a chord is the tuning these figures reward, arrived at by having no valve to press.

An orchestral wind section playing a dominant seventh in equal temperament is at the top of the figure’s left-hand plateau and cannot get off it by ear, because the first thing a small downward adjustment does is make the chord worse. The valley is a real obstacle for a player tuning by listening: the direction of improvement is not the direction of local improvement.

Which computation produced the numbers #

Each note is a struck string of 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. A pair’s life is the time its strongest coincident partial pair stays audible, where a coincidence is any two partials within twenty cents of each other; a chord’s life is the least of its pairs’. Voicings keep the bass fixed and place every other pitch class in any octave within two octaves of it, over every inversion.

The retuning is applied as a cent offset to one pitch class, after the spacing is chosen and before the frequencies are computed, so that which octave a note is in and how it is tuned are separate decisions. The bound is computed the same way as before, over the presentations the retuned distances have: a distance of 9.69 semitones can be presented as 9.69, 21.69 or 2.31, and the bound is the best of them.

Where the model stops #

Twenty cents of tolerance decides the plateaux. A wider tolerance would narrow the valley and might close it; a narrower one would widen it and would also cost the tempered chord its 9 : 5, since that coincidence is already 17.6 cents out. The valley is a robust prediction of the shape; its width is not a robust number.

Nothing here is about roughness. A chord whose coincidences have decayed still sounds — the top that falls while the note lasts measured how much of its series is still there — and the partials that no longer align still beat at rates beats are arithmetic computes; eighty-one chords, one number priced that separately and found it varying by a factor of five and a half across spacings of a dominant seventh. A voicing that lasts longer is not therefore smoother.

The strings are equally loud and equally damped, and a real piano’s decay times vary across its compass by more than the partial-number law allows.

Only one note is moved. A singer who tunes a chord by ear moves all of it, and what the arithmetic would say about a dominant seventh tuned 4 : 5 : 6 : 7 throughout is a different computation with four offsets in it rather than one.

What a lifetime cannot say about a practice #

Whether barbershop singers tune the seventh for this. They tune it because it locks, and locking is more plausibly about the beats stopping than about the coincidence lasting — and the tone on the root changes hands at the fifth shows a third thing a septimal interval does, which is put its combination tones on the chord’s own notes. The two are different quantities that a septimal seventh improves at the same time, and nothing here separates them.

Whether anybody hears the valley. The dip between the two tunings is two tenths of a second on a figure of two thirds, and whether a listener can tell a chord whose coincidences last 0.46 seconds from one whose last 0.66 is a question for an experiment.

Whether the spacing result survives a real instrument. The retuned chord’s best voicing puts its seventh nearly two octaves above the bass, where a piano’s decay times and its inharmonicity both depart from the model — and the piano is tuned wrong on purpose is about exactly that departure.

Still open: the whole chord tuned to the series #

One note was moved because one note was what the earlier census’s omission named, and moving one note is the smallest change that answers it. It is not the change a singer makes.

A dominant seventh sung in tune with itself is 4 : 5 : 6 : 7 over its root — the fourth, fifth, sixth and seventh harmonics of a fundamental two octaves below — which moves the third fourteen cents flat and the fifth two cents sharp as well as the seventh thirty-one flat. Every one of its six pairs is then a septimal or five-limit ratio exactly, and the census’s tolerance is not being asked to absorb anything at all.

That is four offsets instead of one in a function that already accepts them, and the question it answers is sharper than this one’s. If the whole chord tuned to the series reaches past 0.87, then the ceiling found here belongs to the one-note retuning rather than to four-note chords, and the earlier bound is a fact about a keyboard. If it does not — if the outer pair still ends the chord at a seventh partial’s life — then 0.87 is the ceiling for any four notes, however they are tuned, and the reason seventh chords do not last is arithmetic about the number four rather than anything about temperament.