The dynamic chooses which D the scale has

The notes name the ghost bass before it has a cycle fitted a harmonic template to every component of a just third as each one completed its own cycles, and found the bass under seven of the eight thirds of a C major scale named within five milliseconds. The eighth, D–F, is the scale’s Pythagorean minor third, 32 : 27, and it was never named with its own difference tone. For the first fourteen milliseconds the fit read it as the 6 : 5 it nearly is and named 58.4 hertz, the bass that D–F would have if the D were a syntonic comma lower; then the products arrived and broke the reading.

That essay assumed every product was there. The products are what break the B♭, and a combination-tone bass needs a forte had already found that a third’s difference tone is not audible at all below the upper sixties of decibels. So its closing paragraph predicted that a quiet D–F would keep the B♭ for the whole note, and a loud one would lose it.

It does, and the boundary is sharper than the prediction. It also turns the older essay’s conclusion round.

Which products a dynamic lets in #

The fit is the earlier essay’s exactly: each note’s first six partials, each joining once it has run four of its own cycles; the lowest six joined components fitted with harmonic numbers up to thirty-two; a bass named when the fit names it and keeps naming it. The one change is at the door. A product joins only if, at the level the primaries are played, it clears both the threshold of hearing and the masking the primaries themselves put over it — the level law and the limits the earlier essay on the forte used, unchanged.

For D–F there are two candidates. The cubic tone, 2D − F at 240 hertz, sits a little more than a major third below D and inside the primaries’ masking at every level; it is never admitted. The difference tone, F − D at 54.5 hertz, grows two decibels for every decibel of the primaries and clears its limit from 76 dB.

So the passage has a staircase of dynamics at which its difference tones switch on, from 66 dB for the top third to 73.5 for the two thirds over C2, and D–F’s is the last of all. That ordering is not an accident of the level law; the difference tone of a narrower interval is lower in frequency and further below its primaries, and the Pythagorean third is the narrowest minor third in the scale.

Below a forte, B♭ for the whole note #

Below 76 dB nothing arrives that could contradict the 6 : 5 reading. The fit settles on 58.4 hertz at four and a half milliseconds and names it at every step to the end of the note, however long the note lasts.

That is the result the prediction was after, and its content is not small. The pitch that D–F is heard with is the bass of a different third: the D–F that a just scale would have if its D were 10 : 9 above C rather than 9 : 8. The two Ds are a syntonic comma apart, 21.5 cents, and every choir and string quartet that tunes a C major scale in just intervals has to choose between them — 9 : 8 gives a pure fifth to G and a Pythagorean third to F, 10 : 9 a pure third to F and a narrow fifth to G. The progression that never comes home is about what that choice costs a progression; a comma is a polyrhythm that never closes is about why it cannot be avoided.

The fit makes the choice on the players’ behalf. Whichever D they play, a listener hearing the pair as one object hears the bass of the 10 : 9 D. Below a forte, the syntonic comma in this one interval is resolved by the pitch mechanism rather than by the tuning.

Above a forte, the B♭ is taken away #

From 76 dB the difference tone is admitted, and it needs four of its own cycles: 73 milliseconds at 54.5 hertz. In any note shorter than that it never joins the fit, and the B♭ holds to the end of the note however loud it is. In any note longer, the difference tone joins at 73 milliseconds and the 6 : 5 template can no longer explain the set.

What the fit names then is about 27 hertz — within ten cents of an octave below the difference tone, an A0 against the difference tone’s A1. The template reaches it by leaving nineteen harmonic slots empty, and 27 hertz is at the lower edge of the range in which anything is heard as a pitch at all, so the honest reading is that the object loses its bass rather than gaining a new one. What the forte does is subtractive. It does not supply the difference tone’s A as the passage’s bass; it takes the B♭ away and leaves a hole the right pitch class would fill only an octave too low.

So the map in the hero figure has three regions, and they are set by two numbers the arithmetic already had. Below 76 dB, B♭. At or above it, B♭ for the first 73 milliseconds of every note and a broken fit after. A semiquaver at 200 beats a minute is 75 milliseconds, so a fast passage in thirds keeps its B♭ at any dynamic.

The line a dynamic draws #

The same question asked of the whole passage gives the line itself, and the dial plays it at each dynamic.

At 60 dB not one difference tone in the passage is audible, and the object names a bass under every third anyway: C2, B♭1, C2, F2, G2, F2, G2, C3. At 70 dB five of the difference tones are audible and the line is the same. At 80 dB all eight are audible, seven of them confirm what the object had already named, and the eighth removes its own note.

This is where the older essay’s conclusion turns round. A combination-tone bass needs a forte found that the difference tones are inaudible below the upper sixties of decibels and concluded that the ghost line is a forte phenomenon. On the account in which the notes’ own partials name the bass, the line is there at every dynamic, because it never depended on the products: the played notes already name the ghost bass said as much for a single third held still. The forte is not what gives the passage its bass. It is what takes one note of it away — and the note it takes is the one that was not the difference tone in the first place.

A tempered scale is heard with the just scale’s bass #

The same door, opened on an equal-tempered passage, produces the strangest result in this account.

The bass line under a passage in thirds found that tempering the thirds wrecks their difference-tone line: equal temperament’s major thirds are 13.7 cents wide, the third sound magnifies cents, not hertz showed that a difference tone moves several times as many cents as the interval that made it, and the tempered line comes out as C♯, A, B, F♯, A♭, E, F♯, C♯ — a chromatic scatter a quarter-tone and more from anything diatonic. A tempered third cannot be one sound with its own ghost then found that with its products present, no tempered third fits a single template at all.

At 60 dB none of those products is audible, and the fit has only the notes. Each tempered third is within the template’s tolerance of the just interval it approximates, so each is classified as that interval and given that interval’s bass: C2, B♭1, C2, F2, G2, F2, G2, C3 — the just scale’s object line, B♭ and all, to within ten cents at every note. The tempered difference tones, which are what the ear’s nonlinearity actually produces, are nowhere in it.

At 90 dB every one of those difference tones is audible, each contradicts the classification, and the fit names nothing under any third of the passage. So a keyboard playing thirds quietly carries, on this account, the bass line of a just scale it is not tuned to; played loudly, it carries none.

That inverts what equal temperament is usually said to cost a passage in thirds. The usual account — this collection’s included — has tempered thirds beating and their ghost line out of tune. The account here says the ghost line is not out of tune but absent at a forte and in tune at a piano, because what a quiet listener hears is the template the notes are nearest, not the products they make.

How much error the template must allow #

The whole result rests on one number the model sets, and it is worth seeing how much room there is around it.

The tempered major third is heard with the just bass once the template allows ten cents of error per partial; the Pythagorean minor third needs fifteen. Both thresholds are about half the interval’s departure from just — 13.7 cents and 21.5 — because a least-squares fit spreads the error across the two notes’ partials, so each partial carries roughly half of it. Below the threshold the fit names nothing: a template that will not accept the interval as a simpler one has no simpler one to offer, and the interval is not heard as one object at all. Above it, up to forty cents, the answer does not change.

So the result does not depend on the collection’s thirty cents; it depends on the ear’s tolerance being more than about half a comma. That is a much weaker assumption. Experiments that mistune one harmonic of a complex tone find it still pulling the pitch of the whole until it is about three per cent out — some fifty cents — which suggests the requirement is met with room to spare.

Why a narrower third is heard as a wider one #

The fit reads 32 : 27 as 6 : 5 because the template allows each partial thirty cents of error, and the Pythagorean minor third is 21.5 cents narrower than the just one: close enough that the notes’ partials are within the tolerance of a 6 : 5 template’s slots. The tempered minor third, 300 cents, is 15.6 cents narrower than 6 : 5 and is read the same way. In both cases the fitted fundamental lands between D ÷ 5 and F ÷ 6, which is within a few cents of the just third’s own bass.

So a pitch mechanism of this kind does not measure a minor third and report it; it classifies it, against a small set of templates, and reports the template’s fundamental. The difference tone is the one component that measures the third exactly — it is at F − D, whatever the tuning — and so it is the one component that can overrule the classification. Where it is inaudible, the classification stands; that is the whole mechanism of the result.

It also says which other intervals should behave this way. Any interval within the template’s tolerance of a simpler ratio, and whose own difference tone is quiet, will be heard with the simpler ratio’s bass. The tempered major third, 13.7 cents wide of 5 : 4, is the obvious candidate, and a tempered third cannot be one sound with its own ghost found its object fit failing at a forte for exactly the reason D–F’s does here.

What the model assumes and does not test #

The level law is the one the earlier essays used, with its published constants: a difference tone 50 dB below primaries of 60 dB, growing two decibels per decibel. The earlier essay swept those constants and found the onsets moving by a few decibels either way; the 76 dB boundary moves with them, and the order of the onsets does not. Played partials are admitted at every level, which is an idealisation for the sixth partial of a quiet note; a quiet note’s upper partials are the first to fall below threshold, and without them the fit settles later.

The template’s tolerance of thirty cents is the one the whole account has used, and it is doing real work here. The figure above puts the B♭'s threshold at fifteen cents, half the comma; at a much larger tolerance many more intervals would be classified as simpler ones than the scale’s thirds, and a template that accepted everything would hear nothing precisely. Where a listener’s pitch mechanism sits in that range is not something this arithmetic can settle.

Whose thirds are played quietly #

Thirds are one of the standard textures of keyboard and string writing, and they are often marked quiet. Chopin’s étude in thirds in G♯ minor opens pianissimo, sotto voce, in semiquavers — equal-tempered, fast and soft, the corner of the map in which every third of its scale is classified as just and the object’s line is the just scale’s. The same passage played forte would, on this account, have no ghost line under it at all. A violinist’s double stops in thirds are the other case: tuned by ear, usually closer to just, and played at every dynamic, so that the Pythagorean D–F problem is a live choice of finger position rather than a fixed tuning.

In both, the arithmetic here makes a specific prediction about practice: a player who adjusts a third by listening for its ghost bass, as eighteenth-century violinists were taught to, gets useful feedback only at a forte. Below it the ghost is the classification’s, not the interval’s, and it sounds in tune whether the third is or not.

What cannot be read off a fit #

Whether the B♭ is heard as a bass. A residue pitch of two notes a third apart is a faint thing, heard by some listeners and not others, and the object fit says which pitch would be heard if one is, not that one is. The claim here is that the pitch is B♭ rather than A below a forte, not that it is prominent.

Whether a performer’s D moves with it. Singers and string players adjust intonation by ear, and an ear hearing a B♭ under D–F is hearing a third in tune. Whether that removes the pressure to lower the D — which is how a just scale’s comma problem is usually described — is a question about performance practice, and no measurement of it is drawn on here.

Still open: whether the listener’s D is the one the progression needs #

A C major scale in thirds is a line, not a progression. In a progression the D arrives with a function — the fifth of a G chord, where it wants to be 9 : 8, or the root of a D minor chord, where it wants to be 10 : 9 against F and A — and the progression that never comes home is the account of what that ambiguity costs. The result here says that, heard quietly, the pair D–F always brings the 10 : 9 D’s bass with it, whichever D is played.

The computation would run the object fit over the chords of a progression rather than the thirds of a scale — each chord’s partials together, admitted by the same cycles and the same level law — and ask, chord by chord, which D the object’s bass implies. If the listener’s D is always 10 : 9 in a D minor chord and 9 : 8 in a G chord, the comma pump is heard as the players would tune it; if the object’s classification pulls both toward one D, a quiet progression may never come home to the listener either — or come home when the players do not.