A stack that is not thirds

A triad is two thirds stacked. That is a construction rule, and like every construction rule it can be run with a different input: stack two fourths instead, or two fifths, or two whole tones, and see what comes out.

Eleven such chords exist at three notes, one for each interval from a semitone to a major seventh, and the site has the machinery to score all of them without adding anything. Roughness is the summed Plomp–Levelt curve; root strength is the subharmonic matcher the previous rung used.

Eleven stacks, two measurements #

The number that ought to be surprising is the first one. Eight of eleven constant-interval stacks are smoother than the major triad, and the triad is not close to the top of the ranking: it scores 0.288, and stacked fifths score 0.140, stacked tritones 0.144, stacked fourths 0.218.

That is not a criticism of the triad and it is not news to this collection. Three is the largest agreeable number established that the triad is the unique three-note chord that is good on both counts — smooth and expressible in small whole numbers — and that scoring smoothness alone selects the wrong chords. What the stacks show is which wrong chords.

The fourth stack, in detail #

Take the stack of fourths on its own, because it is the one with a repertoire.

Its roughness is 0.218 against the triad’s 0.288 — measurably smoother. Its subharmonic fit is 6 : 8 : 11, with three empty slots in the series and a worst error of 28 cents, which is a poor fit by every criterion the matcher has. And its interval content is the thing that explains both.

The interval vector of a stack of fourths is 0 : 1 : 0 : 0 : 2 : 0 — one whole tone, two perfect fourths, and no thirds. The major triad’s is 0 : 0 : 1 : 1 : 1 : 0, one of each of the thirds and one fifth.

Two consequences fall straight out.

No third means no quality. Major and minor are names for which third a chord has, so a chord with neither is neither. That is not an evasion; it is arithmetic about the set.

And the absence of thirds is what makes the smoothness. The thirds are the rough intervals at any register a chord is likely to be voiced in, and they get worse as the chord goes down. A chord made of fourths and a whole tone has avoided the two intervals that generate most of the roughness, which is why it is smooth, and it has avoided the two intervals that define chord quality, which is why it has none. One fact, read twice.

The roughness curve for a string spectrum puts wells at the fourth, the fifth and the octave, so a chord built entirely of fourths lands on wells at every one of its intervals — which is exactly why it scores as it does and exactly why the score does not settle anything.

Four notes, and the absence becomes complete #

Extend the stacks to four notes and the pattern sharpens.

At three notes the fourth stack had a poor fit. At four notes it has none: no fundamental up to the sixteenth harmonic accounts for its four notes within thirty cents, at any tolerance the site’s other essays use. Widening the tolerance does not rescue it either — nothing appears until sixty cents, which is past the point where a “fit” is a fit, and what turns up there is 5 : 7 : 9 : 12 at fifty-seven cents out.

Compare the dominant seventh, which is the four-note chord tonal harmony is built on. It is rougher by a third, and it has the most specific root and the most specific continuation of any chord in the system — it is the one chord that names its key, because it contains the diatonic set’s only tritone.

So the two chords sit at opposite corners of the same pair of measurements. One is rough and points somewhere; the other is smooth and points nowhere.

Which is what it was adopted for #

That is not a coincidence and it is not an interpretation. It is what the composers who took it up said they were doing.

Quartal harmony arrives in European music at the point where functional harmony is being deliberately set aside — the opening of Schoenberg’s Chamber Symphony, Op. 9, in 1906, which announces itself with a stack of fourths; Debussy and Scriabin at about the same time; Bartók; and Hindemith, who wrote the theory of it in the 1930s and organised his whole Craft of Musical Composition around the ranking of chords by their intervals rather than by their thirds.

What a composer wanted from it is exactly the two properties measured above. A chord with no third has no quality to be resolved; a chord with no fundamental has no root to define a key. It sounds like a harmony and behaves like a colour.

The jazz history is the same argument arriving separately. The voicings associated with McCoy Tyner and with the modal repertoire of the late 1950s are quartal because a modal passage is one in which the harmony is meant to stay put; a mode is a scale with a tonic and no dominant to speak of, so a chord that implies a dominant is precisely the wrong tool.

The other stacks, briefly #

The census names the rest and they are worth a sentence each, because two of them are famous for the same reason.

Stacked major thirds — the augmented triad — is smoother than the major triad, has no fundamental worth the name, and is transpositionally symmetric: three transpositions of it exhaust the twelve. Its use is nineteenth-century and it is used exactly where a composer wants the tonal centre suspended.

Stacked minor thirds at four notes — the diminished seventh — is likewise symmetric, likewise rootless by the matcher, and was the century’s other favourite modulating chord for that reason. Any of its four notes can be treated as the root, which is another way of saying it does not have one.

Stacked tritones collapses: two tritones is an octave, so the chord has two pitch classes rather than three. It is the smoothest thing in the census and it is not a chord.

Stacked fifths is the interesting near-miss. It is the smoothest genuine three-note stack, at 0.140, and its fit — 4 : 6 : 9, with a worst error of three cents — is the best fit in the whole census. So a stack of fifths does have a root and is not neutral, which is why it reads as an open, incomplete tonic rather than as a colour, and why the power chord of rock guitar is that interval and not the fourth.

On the lattice of fifths and thirds a stack of fifths and a stack of fourths occupy a straight line rather than a triangle, which is the same fact as having no third: a triangle needs a step in both directions and these chords take steps in one.

The most diatonic chord there is #

Here is a fact about the stack of fourths that reads as a contradiction until it is looked at properly.

A stack of fourths is three adjacent links of the chain of fifths. So is a stack of fifths, read the other way. Which means that the chord with the least tonal implication of any in the census is built from the most consecutive possible segment of the very chain that generates the diatonic set — the chain that produces the pentatonic at five links and the diatonic set at seven.

A major triad is not three adjacent links. C, E and G are at positions 0, 1 and 4 of the chain, so the triad reaches across five of its links and skips three. That gap is where the third comes from, and the third is where the quality comes from, and the quality is where the function comes from.

So the two constructions are not opposed and are not even far apart. They are the same generator read with and without a gap, and everything tonal harmony has — quality, root, function, resolution — lives in the gap.

The quartal chord’s own register problem #

The previous rung showed that where a chord is put dominates what its notes are, and the stack of fourths is a good test of that finding because its interval content is so different.

That is a real and usable property. A quartal voicing survives being put in the bass in a way a close triad does not, which is part of why the twentieth-century orchestral repertoire that uses it uses it low, and why the same chords work on a guitar’s bottom strings where a close triad is unplayable.

The corresponding cost is at the top. High up, the roughness of everything collapses and the triad’s disadvantage disappears, so a quartal voicing buys nothing in the treble — which is where the practice puts triads and the practice is right.

Two absences are not one absence #

There is a distinction inside all of this that is easy to lose and worth keeping.

Having no third is a property of the pitch-class set. It is exact, it is a fact about the interval vector, and it survives any voicing, any register and any temperament.

Having no fundamental is a property of a model of hearing, and running the matcher’s own knobs shows it is more model-dependent than this paragraph originally claimed — and dependent on different things.

It does not change with the tuning. Tuning the three-note stack as pure fourths, 9 : 12 : 16, and asking again returns the same answer the tempered version does: 6 : 8 : 11, at 29.0 cents. The exact 9 : 12 : 16 reading is in the candidate list, at zero error, and the matcher ranks it second — because it sorts on empty slots before goodness of fit, and 6 : 8 : 11 leaves three gaps in the series where 9 : 12 : 16 leaves five. In equal temperament the same thing happens with 9 : 12 : 16 sitting at 2.7 cents. So both tunings offer a near-exact fit high in the series, and the matcher declines both for the same reason: it is a parsimony rule, and what it is reporting is not that no fundamental fits but that no fundamental fits low down and without gaps.

That is the right rule for the question this ladder asks — a root a listener supplies has to come from the bottom of a series — and it should be named as the rule rather than read as an absence.

And the four-note result is a fact about the ceiling. “No fundamental up to the sixteenth harmonic” is exactly true and exactly conditional: the four-note stack of pure fourths is 27 : 36 : 48 : 64, whose top note needs a harmonic beyond the ceiling. Raise the ceiling and 9 : 12 : 16 : 21 fits within 14.2 cents — with nine empty slots, which is a worse parsimony score than anything in the census and is why the sixteen was set where it was. The absence is real at the resolution the model works at and it is not the flat impossibility the sentence above it reads as.

The two agree here, and they need not have. A chord could have no third and a strong root — a stack of fifths does — or a third and no root, which is what the minor triad approaches in equal temperament. The stack of fourths is a chord where both are missing, and it is the coincidence of the two that made it useful.

The test, and the thing it would take to fail it #

The claim is that the fourth stack’s usefulness is the coincidence of two computable absences. That has an observable consequence and a failure condition.

The consequence: a chord with no root should not be able to be a goal. And it is not — quartal harmony does not cadence. Where a piece using it ends, it ends by some other means: on a unison, on an octave, on a fifth, on a held note, or by simply stopping. Five separate signals make an ending in tonal music and the harmonic one is unavailable here, so what is left is the metrical, the durational and the dynamic.

The failure condition: if a repertoire were found using bare quartal stacks functionally — as goals, with a consistent expectation of what follows one — then the absence measured here would not be doing the work claimed. Nothing in the twentieth-century repertoire does that, and it is worth noticing that the composers who used the chord most said so themselves, in print, before anyone measured anything.

What the picture cannot show #

Roughness is one model and it has been used four ways in this ladder. It is summed Plomp–Levelt on an idealised string spectrum, with no masking, no level dependence and no duration. The ordering of these eleven stacks is robust across the spectra in the site’s table; the numbers are not.

The matcher’s silence is not a listener’s, and the section above establishes that it is not even a silence. “No fundamental fits” means no candidate wins a parsimony ordering over a stated range of harmonics at a stated tolerance, and both the range and the ordering do real work in the answer. It is a strong statement about a model and a weak one about what anybody hears, and nobody has been asked whether a stack of fourths has a root.

And a real quartal chord is not a bare stack, which is worth scoring rather than conceding. The voicings actually used add a note: the So What voicing is three fourths and a major third on top. Running it through both measures gives a chord that has recovered one of the two absences and not the other.

The added third puts a quality back — there is now a major third in the interval vector, so the chord has one — and it costs 0.152 in roughness, taking the voicing from 0.381 to 0.534, which is within four per cent of the dominant seventh’s 0.556. The claim that the third is placed on top because that is where it costs least is confirmed: the same third put at the bottom of the same five notes scores 0.574, seven per cent worse.

What it does not recover is the root. The matcher returns nothing for the So What voicing at the same ceiling and tolerance that returns nothing for the bare stack. So the voicing practice actually uses has bought a quality at the price of most of the smoothness, and has left the rootlessness exactly where it was — which, if the argument of this essay is right, is the half that was doing the work.

Whose harmony, and when #

Chords built on fourths are not a twentieth-century invention; what is new is using them as the basic unit. Fourths are the primary consonance of medieval organum, and the prohibition on the fourth against the bass arrived only when the bass became the structural reference. A quartal harmony is, from that angle, a return: it is what harmony looks like when nothing is measured from the bottom.

Outside Europe the construction is ordinary rather than exotic. Fourth-based tunings and fourth-based simultaneities are common wherever the string tuning is in fourths, which is a great many places, and no theory in those traditions has ever had to explain the absence of a third because no theory in them required one.

Which is the shape of result this collection keeps arriving at. The triad is not the general case of a chord. It is the chord that survives two criteria at once, in a repertoire that needed both — and the moment one of the two criteria is dropped, the census has eight better answers waiting.

The ladder from here #

Seven rungs, and every one of them has treated a chord as a set of notes sounding together. What is left is the thing a chord does that a set cannot: the next rungs of this ladder go to what a chord implies about what comes next, which is the progression ladder’s subject, and to what a chord is in a texture of independent lines, which is voice leading’s.