The notes name the ghost bass before it has a cycle

A ghost bass an octave lower needs notes twice as long put a clock on the bass that sounds under a passage in just thirds. The difference tones of a C major scale harmonised in thirds spell a diatonic line — C, A, C, F, G, F, G, C, as the bass line under a passage in thirds drew it — and a low pitch is not heard until it has run a few of its own cycles. With four cycles, the bass under a third from C4 arrives 61 milliseconds into the note, the lowest one 73, and a passage whose notes are shorter than that has no bass at all. An octave down, every number doubles.

The essay’s own list of limits named the assumption most likely to be wrong. The played notes already name the ghost bass had found that a pitch mechanism fitting the two played notes and their products together as one object names the implied fundamental from the notes’ partials alone. Those partials are two to four octaves above the bass and complete their cycles eight to twenty times as fast. If the mechanism can read the fundamental off them, it need not wait for the bass.

It need not. And the one third it cannot read that way turns out to be heard with a different bass altogether.

Every component on its own clock #

The rule is the earlier essay’s, applied to everything rather than to the bass alone. A component takes part in the pitch computation once it has run a stated number of its own cycles — four, as before, and swept below — so a partial at 1,300 hertz is available after three milliseconds and a difference tone at 65 hertz after sixty-one. At each moment a new component becomes available, the lowest six available components are fitted with the same harmonic template the earlier essays used: every assignment of harmonic numbers is tried, and the one leaving fewest empty slots and fitting best is taken.

For C4 against E4 the first partial to qualify is E4’s sixth, at 1,962 hertz, two milliseconds in. For the next three milliseconds there are too few components for any fit to mean much, and the fit wanders — 103, 82, 86, 82 hertz — as each new high partial arrives. At 5.1 milliseconds C4’s fourth partial joins and the six components present are the 8th, 10th, 12th, 15th, 16th and 20th harmonics of 65.4 hertz. From then on the fit names 65.4 hertz at every step, and the arrival of the lower partials, the cubic tone and finally the difference tone itself only confirms it.

The whole object names the bass twelve times as fast as the bass could name itself. That is the answer the earlier essay could not compute: its register limit was a consequence of listening to the bass alone.

Note by note, and the note that is different #

The passage in thirds changes interval at every step, and the same computation run on each third gives each note its own arrival.

Seven of the eight thirds behave like the first. C–E and E–G, both over a C2 bass, are named at 5.1 milliseconds; F–A and A–C at 3.8, G–B and B–D at 3.4, and the top third, C5–E5 over a C3 bass, at 2.5. Every one is about a twelfth of the time its own four cycles take, because in every one the partials that settle the fit are around the fourth to sixth harmonics of the notes, and those stand in the same ratio to the bass whatever the step.

The eighth is D–F, and it is never named. The one third whose bass the earlier clock found slowest is the one the whole object cannot supply at all.

What the object hears under D–F #

In the five-limit just scale D is 9 : 8 above C and F is 4 : 3, which makes D–F a minor third of 32 : 27 — the Pythagorean minor third, 294 cents, rather than the 6 : 5 of 316 that every other minor third in the scale has. Its difference tone is 54.5 hertz, the A1 of the bass line.

For most of the first fourteen milliseconds the fit names 58.4 hertz — not the difference tone but a pitch about a hundred and twenty cents above it, nearer B♭1 than A1. The reason is the tolerance. A harmonic template accepts a partial within thirty cents of where it predicts one, and 32 : 27 is twenty-two cents narrower than 6 : 5; so the notes’ partials fit a 6 : 5 template well enough, and a 6 : 5 over these two notes has its fundamental between D ÷ 5 and F ÷ 6, at 58.4. The fit is reading the Pythagorean third as the just minor third it nearly is.

It cannot hold that reading. At 16.7 milliseconds the cubic tone, 2D − F at 240 hertz, has run its four cycles and joins, and it sits where a 6 : 5 would not put it: the fit fails and names nothing. At 73 milliseconds the difference tone arrives and the only template consistent with everything present has its fundamental at 26.8 hertz, with nineteen empty slots — within thirty cents of an octave below the difference tone, but at the bottom edge of the range in which anything is heard as a pitch at all, and fitted only by leaving most of its harmonics empty.

The comparison rows say what the 58.4 was. Lower the D by a syntonic comma, to 10 : 9 above C, and D–F becomes a true 6 : 5, whose difference tone is 58.1 hertz — and the whole object names it at 5.7 milliseconds and holds it to the end. So for the first fourteen milliseconds of every D–F, the object hears the bass the scale would have if its D were the other D: the one the comma separates, the one a comma is a polyrhythm that never closes and the pump in the progression that never comes home are both about. The ghost line becomes C, B♭, C, F, G, F, G, C for as long as the products have not yet contradicted it.

That is a strange line for a C major scale to carry. B♭ is not in the key; a bass that steps from C down to B♭ and back is the bass of a passage in F, or of a C major passage borrowing the flattened seventh. The difference tones of the just scale spell a line made entirely of the scale’s own notes, which is what made the bass line under a passage in thirds remarkable; the whole object, fitted in the first milliseconds, spells one with a note from outside it, and the note it brings in is exactly the one the syntonic comma puts there. The pitch mechanism is not making a mistake in any local sense. It is choosing the simplest template the partials allow, and the simplest template for a Pythagorean minor third is a just one.

The tempered D–F, 300 cents, fits the same 6 : 5 template for the same reason and is named 58.4 as well; like every tempered third in a tempered third cannot be one sound with its own ghost, it is never named with its own difference tone.

What the passage keeps #

With every bass but one arriving in a few milliseconds, the share of the passage that has its bass is almost independent of how fast it moves.

From C4 at eighty milliseconds a note — semiquavers at about 190 beats a minute — the bass-alone clock gives the passage its bass for 38 per cent of the time. The whole object gives it 83. At forty milliseconds the figures are 3 per cent and 79; at twenty, none and 71. From C3, at a hundred milliseconds a note, the bass-alone clock gives 12 per cent and the object 81.

The object’s curves flatten at seven notes in eight because D–F contributes nothing at any length. The bass-alone curves climb past them only at half a second a note and longer, and they climb past them because the bass-alone clock does give D–F its A1 once four of its cycles have run. So there is a crossover, and it has a meaning: at slow tempi the bass-alone clock supplies more of the line than the object does, because it grants the difference tone to the one third whose partials, fitted together, say something else. Which of the two a listener does is a question about whether a pitch mechanism can be overruled by a difference tone once it has settled on a fit, and the model has no term for that.

The two clocks can be set side by side note by note at the same speed, and the contrast is the whole of the change.

At eighty milliseconds the bass-alone clock hands the bass over late in almost every note: the C2 thirds keep it for the last quarter, the A1 under D–F for the last twelfth, and a listener hearing the passage would hear a bass line that arrives, flickers and goes. The object’s version is a line that is simply there from the first few milliseconds of seven notes in eight and absent from the eighth. Those are different musical objects: one is a bass that fails with speed and the other a bass that fails on one degree of the scale at every speed.

Four partials, six or eight #

The number of partials each note is given is a choice, and it is the choice most likely to move an arrival time, since the fastest components are the highest partials. With four partials a note, the latest named bass in the passage from C4 is 5.1 milliseconds; with six, 5.1; with eight, 5.1 again, because it is the C–E third’s fit that settles last and it settles on the same six components however many higher ones are available. The quickest thirds move a little — the top third from 2.5 milliseconds to 2.0 with eight partials — and D–F is never named with any of the three. The cycle count scales everything together, as the next figure shows, and at two cycles the latest arrival is 2.5 milliseconds and at eight 10.2.

The register rule survives, rescaled #

The earlier essay’s headline was that an octave down every limit doubles. That is still true, because every frequency in the object halves together and every entry time doubles with it.

The latest named arrival in the passage from C4 is 5.1 milliseconds at four cycles; from C3, 10.2; from C5, 2.5. At one cycle each is a quarter of that, and at eight, twice. The scaling with the cycle count is exact, and the figure computes it both ways — at one cycle and at four, from scratch — and checks one against the other, because every entry time is the count over the frequency and changing the count cannot reorder what enters when.

What changes is where the doubling starts. The bass-alone clock put the semiquaver at 120 between the C4 and C3 passages’ limits; the object puts every passage from C3 upward an order of magnitude inside it. Even at eight cycles — a conservative count for a pitch to settle — the object’s bass under thirds from C3 arrives by twenty milliseconds. For a just passage the register limit is gone at any tempo a string or keyboard player plays thirds. What remains is a limit of tuning, and it is the same at every register: the Pythagorean third has no bass of its own in any octave.

The model the fit runs on #

Each note is the first six partials of its fundamental, all present from the onset; the products are the difference tone and the cubic tone of the two fundamentals, which the earlier essays found audible at the dynamics at which thirds are played. Components within fifteen cents of each other are one, since a just third’s upper fourth partial and lower fifth are the same frequency. A component joins the fit when it has run the stated number of its cycles; the lowest six joined components are fitted with harmonic numbers up to thirty-two, the candidate with fewest empty harmonic slots and then the least error winning, every partial within thirty cents of its assigned harmonic. A bass is named from the moment every subsequent fit names the difference tone within thirty cents.

The frequencies are the notes’ own, not a ratio: the D–F third is fitted at 294.3 and 348.8 hertz, and the fit is free to find that no small-integer ratio describes them exactly.

What an arrival time cannot settle #

Whether a fit is a percept. The object fit is a model of how a pitch mechanism might group components, and the earlier essays found it consistent with what the products alone name at a forte. It is not a measurement of what a listener hears under a moving third; the claim here is conditional on the mechanism being of this kind.

The first few milliseconds. Before about four milliseconds the fit has three to five components and wanders among pitches that mean nothing. A real pitch mechanism has integration windows of its own and would not report those, and the figures do not count them as naming anything; but where exactly a mechanism begins to trust a fit is not something this arithmetic sets.

Onsets. Every partial here starts at full level at the note’s start. A bowed or blown attack takes tens of milliseconds to build — a blown note does not start late put a clarinet’s at 121 — and a component that is still growing may not count until it is near its full level. That would add the attack to every arrival here, and it would matter more to the object’s five milliseconds than to the bass’s sixty.

Whether a difference tone can overrule a settled fit. For D–F the model names a B♭ bass for ten milliseconds, loses it, and names nothing useful after the difference tone arrives. A listener might keep the B♭ — the pitch first heard — or switch to the A the difference tone supplies. The model has no memory from one fit to the next and cannot choose.

Whose passages #

Thirds in the treble of a keyboard and double stops on a violin are the repertoire this is about, and both are usually played in equal temperament or something close to it — in which, by the arithmetic here and in the essay on tempered thirds, the object never names any third’s own difference tone. The just passage is the case a string quartet, a barbershop quartet or a choir can approach, and in a just C major scale the D–F problem is the familiar one: a choir tuning D as a pure fifth above G sings a Pythagorean third against F, and a choir tuning D as 10 : 9 — a pure fifth below A — sings a narrow fifth against G. The arithmetic adds that the first choice costs the passage its bass under that third, and that the object, left to itself, quietly supplies the bass of the second.

Still open: whether the object resolves the comma it hears #

The fit hears D–F as the 6 : 5 it nearly is, for exactly as long as the products let it, and the products are what the earlier essays found appear only at a forte. So the prediction is that the bass under a Pythagorean third is B♭ in a quiet passage and fails in a loud one — the difference tone and the cubic tone are what break the fit, and they are below threshold at a piano.

The computation would run the object clock with the products’ levels from a combination-tone bass needs a forte — a product joins the fit only if it is audible at the passage’s dynamic as well as after its cycles — and ask, dynamic by dynamic, what the object names under D–F for the whole length of the note. If it is B♭ throughout at a piano, then a just scale played quietly carries the bass line of a scale with the other D, and the syntonic comma is resolved by the listener rather than by the players.