A ghost bass an octave lower needs notes twice as long

The bass line under a passage in thirds drew a C major scale harmonised in just thirds and the line its difference tones trace beneath it: C, A, C, F, G, F, G, C, a diatonic bass the ear supplies from two upper voices. The essays since have asked when that bass is heard and what it is heard as. The played notes already name the ghost bass found that a pitch mechanism fitting the played notes and their products together names the implied fundamental, and a tempered third cannot be one sound with its own ghost found that equal temperament’s thirds fall outside the window in which the notes and their products share one fundamental at all.

Every one of those essays held its interval still. Two notes sound, the products form, a template is fitted to what is there. The last of them ended on the variable they all fixed. A scale in thirds at a walking tempo gives each of its intervals a few tenths of a second, the bass under it moves by a whole tone or a fourth at every step, and a low pitch is not heard the instant its components are present. The question was what fraction of each note a listener has a bass under, and at what tempo the answer reaches zero.

The answer depends almost entirely on one thing, and it is not the tempo. It is the register. A low pitch needs a number of its own cycles before it is there, a bass an octave lower has cycles twice as long, and so every tempo limit halves with every octave the passage descends.

A pitch needs time, and a low one needs more #

A combination tone forms as soon as both primaries are sounding: it is a product of the ear’s nonlinearity, and it is present in the cochlea within a cycle or two of the upper note’s onset. What takes time is the pitch. A periodic sound does not have a clear pitch in its first cycle; pitch strength grows with duration over several cycles and settles only after some of them. No single count is where it starts, so the model here states one and sweeps it: a bass of f hertz is heard once it has run a stated number of its own cycles, four unless a figure says otherwise, and it arrives that many periods after the note begins.

Four cycles of a bass at C2, 65 hertz, is 61 milliseconds. Four cycles of the A1 under D and F is 73. Four cycles of the G2 under G and B is 41. These are short against a note of a third of a second and long against a semiquaver.

The three curves are one curve moved along the axis. From C4, a passage at three notes a second has its bass for 83 per cent of every note; at ten notes a second, 50 per cent; below 73 milliseconds a note, none. From C5 everything happens at half the note length: 92 per cent at three notes a second and none only below 37 milliseconds. From C3 everything happens at twice it: 67 per cent at three notes a second, 22 per cent at eight, and none below 147 milliseconds. The figure checks the doubling exactly, because it is arithmetic — an octave down halves every bass frequency and so doubles every arrival time — and the doubling is the whole of the register effect.

Note by note #

The average hides a line that is not uniform, because the eight thirds of a just major scale do not all have the same bass.

The line is the one the earlier essay drew, and it is the one the clock has to run on: a bass that moves at every note, by a third, a fourth or a whole tone, so that there is never a held bass to carry a pitch from one note into the next. Its lowest point is A1, under D and F, and its highest the C3 under the final third. In equal temperament the same passage would have a line that wanders by more than three quarters of a semitone around the just one, and a tempered third cannot be one sound with its own ghost found that its products would not fuse with the notes at all; the clock is therefore run on the just passage, the one in which a ghost bass is there to be timed.

The bass under C and E is C2 and arrives at 61 milliseconds. Under D and F it is A1, at 54.5 hertz, the lowest note of the line, and it arrives last, at 73. Under F and A and under A and C it is F2, under G and B and under B and D it is G2, all arriving between 41 and 46 milliseconds. The final C and E, an octave up, have C3 underneath at 31 milliseconds.

The latecomer is not an accident of the scale’s shape. D to F in a just major scale is not the 6 : 5 minor third of the others: D is 9 : 8 above C and F is 4 : 3, so D to F is 32 : 27, the Pythagorean minor third, a syntonic comma narrower than a pure one. Its difference tone is correspondingly lower — five twenty-fourths of the tonic, below every other bass in the passage — and it is the one bass that takes four cycles of more than eighteen milliseconds each to establish. The scale’s one impure third is the note that sets the tempo limit — the syntonic comma that somebody has to pay the comma is about, turning up here as the last twelve milliseconds of the wait.

Dragging the note length shorter shows the order in which the bass goes. At 150 milliseconds every note still has some bass and the passage has it two thirds of the time. At 100, the D–F note has it for only a quarter of its length. At 60, the D–F note and both notes over C2 have none, and only the five notes over F2, G2 and C3 keep some of theirs.

How many cycles, and why it hardly matters #

The four cycles are a stated number, not a measured one, so the result should be read against the number of cycles rather than at one value.

The note length below which the passage has no bass at all is the number of cycles times one period of its lowest bass, the A under D and F. From C4 that is 18 milliseconds a cycle: 37 for two, 73 for four, 147 for eight. The figure checks that the relation is exact, and the relation is why the uncertainty about the number matters less than it looks. Whatever the true count is, the register sets its scale: a listener who needs eight cycles puts the C4 passage’s silent limit where a listener who needs four puts the C3 passage’s. A factor of two in the ear’s requirement is worth exactly one octave of register, and every conclusion below survives being read an octave up or down.

The line at 125 milliseconds, a semiquaver at 120 beats a minute, crosses the C3 passage’s curve at three and a half cycles: a listener who needs more than that hears no bass under the D–F note of a semiquaver passage in thirds from C3 at that tempo. It crosses the C4 passage’s curve at about seven cycles and the C5 passage’s at about fourteen.

At the note values of a tempo #

Note lengths are easier to read as note values.

At a moderate hundred beats a minute, a passage in thirds from C5 keeps its bass for three quarters of the time even in sextuplets, and for 83 per cent in semiquavers. From C4 it keeps two thirds in semiquavers and half in sextuplets. From C3, where a cello or the left hand of a keyboard would play thirds, it keeps a third of the time in semiquavers and an eighth in sextuplets: the bass is there for the slow notes and for a sliver at the end of the fast ones.

One number summarises each register. The passage has its bass for half its length when each note lasts twice as long as the wait for its slowest bass, near enough: from C4 that is at about a hundred milliseconds a note, which is semiquavers at 150 beats a minute; from C3 it is two hundred milliseconds, semiquavers at 75; from C5, fifty milliseconds, semiquavers at 300, which nobody plays. Put as a rule a player could use, the tempo at which a passage in thirds keeps its ghost bass half the time doubles with every octave the passage rises, and in the treble it lies above any tempo in the repertoire.

The same rule applies to any chord whose products spell a bass, not only to thirds. A major triad’s combination tones are its own notes, so a triad’s products land on pitches the chord already has and need no new pitch to be established; a passage of thirds is the case where every product is a new note, which is why it is the case where the clock bites.

That makes the answer to the question the earlier essay posed a register-dependent one. It asked whether the Tartini bass is a laboratory object that exists only under held intervals, or the thing eighteenth-century players said they listened for. In the treble, where violinists play thirds as double stops, the bass is there for most of every note at any tempo those passages are played at. In the tenor register a moving passage of thirds has a bass only when it moves slowly.

Repeating a third before it moves #

A passage need not change interval at every stroke. Many figures in thirds repeat each third before moving to the next, and a bass that continues across the repeat does not have to be established again.

Only the first stroke of each group waits for the bass, so repeating a third spreads the wait over more time. From C3 at semiquavers, a passage that changes third every stroke has its bass 22 per cent of the time; one that strikes each third twice has it 60 per cent of the time, and three times, 73. The model’s assumption here is that a bass heard at the end of one stroke is still heard at the start of the next identical one, which is right if the notes are joined and less right if there is a gap between them.

So the figures of the eighteenth-century violin repertoire that repeat a double stop, or hold one while the other voice moves, are the ones that give the ghost bass the most time. A player tuning a third by its third sound, as Tartini recommended, holds the third, and on this arithmetic that is the one situation in which the bass is present from almost the start of the sound in every register.

The model the clock runs on #

The passage is the C major scale on C3, C4 or C5 in the five-limit just tuning, each degree paired with the degree a third above it, eight thirds in all; the bass under each third is its difference tone, the upper note’s frequency minus the lower’s, as the essay that drew the line computed it. The earlier essays’ fitting showed that for just thirds the pitch mechanism’s object fit names that same fundamental, so the difference tone is taken as the bass to be heard. A bass is heard once it has run a stated number of its own periods from the note’s start, and a note whose bass has the same pitch as the note before keeps it; the share of a note with its bass is what is left of the note after the bass arrives. Nothing here depends on level: the question of whether the difference tone is loud enough to be heard at all is a combination-tone bass needs a forte’s, and the clock here applies only where the answer to that is yes.

Where the clock is too simple #

The count of cycles is a stated number. Pitch emerges over several cycles of a periodic sound and grows stronger for longer; there is no single cycle at which it switches on. The sweep over the count is there because of that, and it shows the conclusions scale with the count without changing shape.

A combination tone may take longer than its primaries. The products are formed by the ear’s nonlinearity from the primaries’ waveforms, and they build up as those waveforms do. A note whose attack takes forty milliseconds to reach full level gives its products a slower start than the clock here, which begins counting at the note’s onset.

The residue may not wait for its own cycles. The earlier essays found that a pitch mechanism fits the played notes and the products as one object. The played notes’ partials are at much higher frequencies and establish their periodicity sooner, and a mechanism that infers the fundamental from them could have the bass before the bass’s own cycles have run. That would make every number here pessimistic, by an amount this essay cannot compute.

Legato and articulation. A passage played legato joins its notes; a detached one separates them, and a bass carried across repeated strokes needs the join. The repetition figure assumes it.

Whose thirds #

The passages in question are the thirds of the string and keyboard repertoire, played at the levels the ghost bass drops a twelfth at a forte found change which bass the products name: double stops on the violin, which are what Tartini wrote about and are played in the treble, and thirds in the right hand of a keyboard, which in the classical repertoire run in semiquavers. The third sound was known to eighteenth-century violinists as a check of intonation, and the third sound magnifies cents, not hertz showed why it is so sensitive a check. The clock here adds that it is available to them for most of the time in the register in which they play thirds, and to a cellist playing the same figures two octaves down only when the figures move slowly.

Still open: the object that does not wait #

The pessimistic assumption above is the one worth testing. A pitch mechanism that fits the played notes and their products together, as the earlier essays’ fitting does, has the played notes’ partials to work from, and those are two or three octaves above the bass and complete their cycles eight times as fast. If the mechanism can name the implied fundamental from the upper partials’ periodicity alone, it does not have to wait for the bass’s own cycles, and the ghost bass would arrive in the time the notes themselves take to establish a pitch.

The computation that would settle it gives the fitted template a time course: each component contributes to the fit only once it has run its own stated number of cycles, so that early in a note only the fast upper partials are in the fit and the slow products join later. It would say whether the ghost bass is named within the first tens of milliseconds, from the notes, and then merely confirmed by the difference tone — in which case the register limit above is an artefact of listening to the bass alone — or whether it waits for the bass, as the clock here assumes.