The scale's ii chord borrows the dominant's bass

The dynamic chooses which D the scale has settled a question about one interval. A just C major scale has one third that is neither 5 : 4 nor 6 : 5: D–F, where the scale’s D, a pure fifth below A and 9 : 8 above C, makes a Pythagorean 32 : 27 with F. A pitch mechanism that fits the pair as one object — both notes’ partials, and the difference and cubic tones the ear adds when they are loud enough — hears it as the just 6 : 5 it nearly is, with the bass that interval would have beneath it: B♭, the note a 10 : 9 D implies. Below a forte nothing contradicts that reading. So a scale in just thirds, played quietly, carries under its D–F the ghost bass of the other D, and the syntonic comma is resolved by the listener rather than by the players.

That essay ended on its own limitation. A scale in thirds is a line, not a progression, and in a progression the D arrives with a job. As the root of a D minor chord it wants to be 10 : 9, so that its fifth to A and its third to F are pure. As the fifth of a G major chord it wants to be 9 : 8, a pure fifth above G. The progression that never comes home is the account of what that double demand costs players who tune every chord justly: the progression I–vi–ii–V–I, held on its common tones, drifts down a comma. The question left was whether a listener hears the D each chord needs, whichever the players play.

The answer, for the D minor chord, is stranger than either possibility the lead named. The chord the scale makes on its second degree does not keep the B♭ its D–F had. Quietly, the fit hears it over G — the dominant’s bass — and the chord it hears is the upper half of a G ninth.

From an interval to a chord #

The fit is the one the ghost-bass essays built and the notes name the ghost bass before it has a cycle timed. Every note brings its first six partials. Every pair of notes brings its difference tone and its cubic tone, provided each is audible at the dynamic played — louder than the threshold of hearing at its own frequency and than the masking the two notes cast on it — which is the level law a combination-tone bass needs a forte fitted. Components within fifteen cents of each other count as one. The lowest six components are then fitted as harmonics of a single fundamental: every assignment of harmonic numbers up to 32 in which no component lies more than thirty cents from its harmonic is a candidate, and the candidate with the fewest empty harmonic slots between its lowest and highest members wins, the closer fit breaking ties. That fundamental is the bass the object implies.

With two notes this is the earlier essay’s fit exactly. With three it is the same fit with three notes’ partials in it and three pairs’ products. Nothing is timed here: the figures read a note long enough for every component to have joined, which is the whole of the note at any length above a few tens of milliseconds at a quiet dynamic, and above about seventy-five at a forte.

The three chords differ only in D. The scale’s ii is D–F–A with D at 9 : 8, F at 4 : 3 and A at 5 : 3, which makes it 27 : 32 : 40 — a Pythagorean minor third under a just major third, with a fifth a comma narrow. A just ii lowers the D by that comma, to 10 : 9, and becomes 10 : 12 : 15, a pure minor triad over B♭. The tempered ii is equal temperament’s.

Two readings, nearly tied #

The dyad’s reading is still there. Fitted as 10 : 12 : 15, the scale’s ii has its F and A within five cents of their harmonics and its D sixteen cents off — the D the chord would need is a comma lower than the one it has, and the fit notices. The worst error is 16.9 cents, comfortably inside the template’s thirty.

But a second reading has appeared, and it is one the dyad could not have. The three notes are also close to 6 : 7 : 9, a chord built on the sixth, seventh and ninth harmonics of G. Its D is almost exact, because the scale’s D is a pure fifth above G, and its A lies nineteen cents below the ninth harmonic. Its F is the seventh harmonic, and the seventh harmonic lies thirty-one cents below a tempered F; the scale’s F is 29.6 cents from it, which is inside the template’s tolerance by four tenths of a cent. And 6 : 7 : 9 leaves fewer harmonic slots empty than 10 : 12 : 15 does. Between a reading that fits every note to within seventeen cents but uses high harmonics and a reading that fits one note only just but uses low ones, a template that prefers low harmonics chooses the second.

The reading it chooses is a familiar chord. D, F and A over G are the fifth, seventh and ninth of a G dominant ninth: G–B–D–F–A with its root and third missing. Harmony textbooks have long described ii as a partial dominant — the same notes, a dominant ninth without its lower half — and the fit hears exactly that relationship, under the one tuning of the D that makes D a pure fifth above G.

Why the seventh harmonic is within reach #

The G reading depends on one coincidence, and it is worth seeing that it is a coincidence of the scale rather than of the chord. Measured from G, the scale’s F is a minor seventh of 16 : 9, 996 cents. The seventh harmonic is 7 : 4, 969 cents, the flat “natural” seventh that four notes cannot get below the seventh partial found a barbershop dominant seventh reaching for. The two are 27 cents apart; the least-squares bass the fit places under all three notes stretches that to 29.6, since it moves the fundamental to suit D and A as well. A thirty-cent template lets the seventh harmonic stand in for the scale’s F, and a twenty-five-cent one would not.

The other two notes are what make the stand-in worth taking. The scale’s D is exactly the third harmonic of G, twice over: 9 : 8 above C is 3 : 2 above G. And the scale’s A is 5 : 3 above C, which from G is a major ninth of 20 : 9, nineteen cents below the ninth harmonic’s 9 : 4. So of the three notes one lands on its harmonic, one is within twenty cents, and one is just inside the limit. The 10 : 12 : 15 reading, by contrast, has two notes nearly exact and the third — D — sixteen cents out, because it needs the D a comma lower. The scale’s own tuning, in other words, has built most of a G ninth into its second degree, and the only thing standing between the ii and the dominant is a thirty-cent allowance on its F.

That is also why the vi of the fixed-scale progression has no such trouble. A minor in the scale is A–C–E at 5 : 3, 1 and 5 : 4, which is exactly 10 : 12 : 15 over F. In the voicing the progression uses, A below C and E, the fit names F; its inversions are read over D and B♭, with errors of 27 and 28 cents, which is the same template accident appearing in an exact chord once its voicing puts awkward partials lowest. The just ii’s second inversion, read over F with a 29-cent error, is the same accident again. So the voicing figure’s lesson is about the scale’s ii and not only about it: close to the template’s limit, what an inverted chord is heard over is decided by which partials fall lowest, and a chord in root position is where the comparison is cleanest. The bass line under a passage in thirds found the same thing for the scale’s thirds taken one at a time: every one but D–F is exact, and D–F is where the scale’s tuning shows.

A preference, not a fact #

That G owes as much to the template as to the chord is plain from the sweep. The scale’s ii names nothing at all while the template forgives less than 17 cents, because neither reading fits that well. From 17 to 29 it names B♭, the dyad’s answer. From 30 it names G. The earlier essays settled on thirty cents and never needed to revisit it, and the scale’s ii lands exactly on the boundary.

Two comparisons keep the finding honest. The pair D–F, on its own, names B♭ at every tolerance from 13 cents up: the G reading needs the A, since two notes cannot be read as 6 : 7 : 9. And the just ii names B♭ at every tolerance up to 34 cents, because its 10 : 12 : 15 reading is exact; only a template that forgives more than a third of a semitone prefers G for it too. So the G is not the scale’s ii’s inevitable bass. It is what a template that prefers simplicity to accuracy hears in a chord whose accurate reading needs a D the chord does not have.

That makes the result a statement about the listener, and a precise one. Whether a quiet ii from a fixed just scale is heard as its own chord or as part of the dominant depends on whether the ear, choosing between a simple fit and a close one, weighs a thirty-cent error as heavily as the template does. The root an ear supplies found a tempered minor triad fitting two series a major sixth apart, with nothing to choose between them. The scale’s ii is the just version of that ambiguity, and the choice here is between roots a sixth apart too — B♭ and G.

Each inversion hears something else #

The hero’s grid adds the observation that decides between the two readings in practice. A chord’s bass should not depend on which note is on top, and for the just ii it barely does: B♭ in root position and first inversion at both dynamics, and in second inversion at a forte. Only the quiet second inversion, A–D–F, is read otherwise, over F.

The scale’s ii has a different bass in every cell. Quietly, G in root position, B in first inversion, A in second — three different notes, none of them the chord’s root or the B♭ the dyad implied. At a forte, nothing in root position and second inversion, and F in first. Rearranging the notes changes which partials fall lowest and which products are audible, and a chord whose notes fit no series well is at the mercy of those accidents. The tempered ii behaves the same way, G in two voicings and A in the third quietly, D in all three at a forte.

So the D a ii needs is the one the fit can build a chord on. Only the 10 : 9 D gives the ii a bass that survives being voiced. With the scale’s D the chord has no stable bass at all: what a listener hears under it is decided by the voicing, not by the harmony.

A forte leaves it with nothing #

The dynamic moves the answer in the way the earlier essays’ level law says it should. At 40 to 60 dB the products are below their masked thresholds and the fit sees partials only; the scale’s ii is heard over G. At 70 dB the difference tones begin to join, and the one between F and A, 87 hertz, is the third harmonic of a B♭ — so the fit briefly swings back to B♭. From 76 dB the difference tone between D and F joins too, at 54.5 hertz, and it belongs to no series the chord’s other components share: the fit names nothing. Loud, the scale’s ii has no ghost bass.

The just ii names B♭ at every dynamic, because every product it makes is a harmonic of B♭ as well: 10 : 12 : 15 has difference tones at 2, 3 and 5, the whole-number differences of whole numbers. That is the old argument for just intonation seen from the listener’s side, and it holds here without qualification. The tempered ii, whose products fall between the harmonics of anything, trades its quiet G for a loud D — the chord’s own root, which its tempered difference tones, landing close to D’s low harmonics, make the simplest reading once they join.

The progression, three ways #

Put the chords in order and the lead’s question gets its answer. Played quietly from a fixed just scale, I–vi–ii–V–I is heard over C, F, G, G, C. The vi is heard over F — A minor as the top of an F major seventh, the same reading one step down — and the ii over G. The ii’s bass is the V’s; the two chords are heard as one harmony with a changing top, and the fixed scale’s D, a pure fifth above G, is the D the dominant needs. The progression comes home exactly, because nothing in it moved.

Retuned chord by chord, as players holding the common tones would tune it, the same progression is heard over C, F, B♭, G, C — the ii over its own B♭, as a just ii should be, and then the drift the progression that never comes home describes, carried into the bass: the G twenty cents low and the final C twenty-two. The listener hears every chord as its own, and hears the progression come home a comma flat.

Tempered, it is heard over C, D, G, G, C, a few cents sharp throughout: the vi, in this voicing, reads as a chord over D, and the ii again over G.

So the listener’s D, in the ii chord, is the one the progression after it needs. The scale’s fixed D does not become the 10 : 9 D in the listener’s ear, as the dyad D–F did. It makes the ii sound like the dominant arriving early. A comma is a polyrhythm that never closes explained why the two Ds cannot be reconciled on paper; here a quiet listener reconciles them by a different route, not choosing the ii’s D but absorbing the ii into the V.

What the fit assumes #

The template’s rule. Everything above turns on the rule that fewer empty harmonics beats a closer fit. That is how the residue fit in these essays has always ranked its candidates, empty slots first and closeness second, and in every case they drew the ranking was between candidates that fitted about equally well. It was never tested on a choice between a simple poor fit and a complicated close one, which is exactly the choice the scale’s ii presents. A listener who weighs accuracy more would hear B♭ where the fit hears G.

Steady state. The chords are read with every component present. The earlier essays found that products need several of their own cycles before they join, so in a short note at a forte the fit would see the partials alone for the first tens of milliseconds, and the quiet reading, G, would hold for that long.

Six partials of equal weight. Each note contributes its first six partials whatever its register or timbre, and the lowest six components of the whole chord are fitted. An instrument with weak upper partials, or a voicing wide enough to push the products below the partials, would change which six are fitted.

Pure tones for the products. The difference and cubic tones are computed from the fundamentals alone. Products between higher partials exist and are weaker, and the fit leaves them out.

Whose progressions #

The claim concerns chords held long enough to be heard as objects, in close position in the middle of the keyboard, played by an ensemble or an instrument that can tune them justly — a string quartet, a choir, an organ in just intonation — and the progression is the textbook I–vi–ii–V–I in C. It says that such an ensemble’s choice of D decides not only the tuning of the ii chord but what a quiet listener hears beneath it: the 10 : 9 D gives the chord its own bass, B♭, in every voicing and at every dynamic; the scale’s 9 : 8 D gives it the dominant’s bass quietly, a different bass in each inversion, and no bass loud. It does not say that a listener hears ii as V. It says that under the one assumption the ghost-bass essays have made throughout, the pitch mechanism would, and that the third sound magnifies cents, not hertz — the tuning of these chords is exactly what the products are most sensitive to.

Still open: whether the ear prefers the simple fit #

The scale’s ii sits on the boundary between two readings, and which one a listener takes is decided by a rule the model imposes rather than measures: that a fit with fewer empty harmonics beats a closer one. The experiment that would settle it is small. Play the scale’s ii and the just ii, quietly and in close position, each followed by a low G and a low B♭, and ask which bass completes each chord. If listeners choose G for the scale’s ii and B♭ for the just one, the fit’s preference is the ear’s, and a ii chord from a fixed scale is heard as the dominant arriving early; if they choose B♭ for both, the ear forgives the D its comma, as it did for D–F alone, and the template’s rule has to be revised to weigh accuracy against simplicity. Either answer would say how the ghost-bass essays’ template should score every chord that sits, like this one, between a simple series and a close one.