An onset can only hand over what the chord leaves

A moving tenor is heard because it is struck again asked whether an inner voice that moves while the chord around it holds survives the masking that a bass chord low enough to balance found burying it. On a struck instrument it does, because the held voices have decayed by the time the tenor’s new note is struck. On a sustained one — an organ, a string section — the moving tenor keeps exactly the partials a held tenor keeps, and a masking count can see no difference at all.

Its closing section named the one mechanism left. A partial that starts some tens of milliseconds before or after its neighbours is heard out of the mixture, and what makes two partials one note measured how much asynchrony that takes: about thirty milliseconds. A moving tenor on an organ has exactly that asynchrony against the held chord, at every move. The question was whether it is enough to make the tenor heard.

It is enough to make the tenor’s surviving partials a new object. How much of a tenor that object is depends on how much the chord has left it, and in the low register that is very little.

Masking decides what is there; the onset decides what it belongs to #

It helps to separate two questions that the phrase “is the tenor heard” runs together, much as one sound hides another and a sound hides what came before it separated masking by what sounds with a note from masking by what sounded before it.

The first is whether the tenor’s components reach the ear at all — whether each partial stands above the threshold that the other voices’ partials put over it. That is the masking count the earlier essays made, and on a sustained instrument it is the same for a moving tenor and a held one: the old note stops masking the new one within ten milliseconds, and nothing else changes.

The second is what the components that do reach the ear are heard as. A chord sounding on an organ is a single stack of partials, and the partials of its tenor that survive the masking are in that stack alongside everything else. Nothing marks them as the tenor’s: they share an onset with the chord, they are near-harmonic with it, and they persist with it. They are heard as part of the chord’s colour. That is the fate of a held tenor’s surviving partials, and it is why a held inner voice on an organ is not heard as a line even when some of it is audible.

A moving tenor’s surviving partials begin when the tenor moves, while everything else continues. The auditory system has a well-established way of handling exactly that situation, which Bregman called the old-plus-new heuristic: when a sound continues and something new starts, the continuing part is attributed to the old sound and the change to the new one. The threshold for it, for a single partial against a harmonic complex, is the thirty milliseconds of onset lead the earlier essay measured. A tenor moving a second after the chord has thirty times that.

So the onset does something real and something bounded. It takes the partials the masking has left the tenor and makes them a separate object — a new note, with a pitch — and it can do nothing whatever for the partials the masking has taken.

What the chord leaves, register by register #

The hero figure is therefore a map of what an onset has to work with, and its shape follows the earlier essays’ register rule.

Two octaves below the written chord, the balanced bass buries every partial of the tenor, whatever the spectrum. There is nothing for an onset to hand over, and a moving tenor on an organ in that register is not heard as a line by any mechanism considered here.

An octave down — the register in which the earlier essay found a tenor keeping one partial of eight when held — the moving tenor keeps the same one, on a string spectrum, and the onset makes that one partial a new object: a pure tone at the tenor’s fundamental. On an organ spectrum, whose energy is spread less steeply, it keeps two or three. On a clarinet’s, whose even partials are weak and whose odd ones are strong, three or four. In the written register it keeps three to four on every spectrum, and above it about as many.

The curves are not smooth, and the unevenness is information rather than noise: each transposition changes which of the tenor’s partials fall in the gaps between the other voices’ partials, so a semitone either way can open or close a slot. What does not change is the ordering of spectra in the low register, where it matters: a spectrum with more energy above its fundamental gives the onset more to hand over.

Partial by partial, an octave down #

The register the question is really about is the one where the tenor has almost nothing, and there the difference between spectra is the difference between a line and a tone.

On a string spectrum an octave down, only the tenor’s fundamental clears the masking, by two decibels. Its second partial falls short by 2.8, its fourth by 0.9, the rest by more. The onset makes the fundamental a new object, and a new object that is one partial is a sine tone: it has the tenor’s pitch and none of its timbre. A listener would hear a line in the tenor register, but not a line in the instrument’s own colour.

On an organ spectrum the fundamental clears by 1.6 decibels, the second partial sits exactly at its threshold, and the fourth clears by six. Fitted as a harmonic template, those three partials name 110 hertz — the tenor’s A2 — with no ambiguity, because the first, second and fourth harmonics of anything leave only one candidate. The onset gives the listener a thin but genuinely pitched note: the organ tenor is heard, as a quieter and duller version of itself.

That is a finding about registration as much as about masking. The organ stop the arithmetic uses is a generic flue spectrum; a reed stop, or a stop with more upper partials, would give the onset more to work with. What an organist’s moving tenor sounds like in the low register is decided by the stop, because the stop decides how many partials survive for the onset to hand over.

What survives is the tenor’s octaves #

In the written register the tenor has more to hand over, and which partials they are turns out to follow a rule.

On a string spectrum in the written register the tenor’s fundamental clears the masking by 7.1 decibels, its second partial by 2.8 and its fourth by 3.5. The third falls short by 4.3, the fifth and seventh narrowly, the sixth by 5.2. On an organ spectrum the pattern is the same and starker: the first, second and fourth clear by six to twelve decibels, and the third, fifth and sixth fall short by more than ten. An octave down, the organ’s survivors were the same three.

The reason is the chord. In a close four-part voicing the other three voices sit a third, a fifth and an octave or so from the tenor, and their partials crowd the frequencies of the tenor’s third, fifth and sixth harmonics — the harmonics that make those intervals in the first place. The tenor’s octave harmonics fall between them. So the partials the chord leaves a moving tenor are its octaves, and what the onset hands over is a hollow tone made of octaves: unmistakable in pitch, since the first, second and fourth harmonics of a note have only one fundamental, and nothing like the instrument’s colour, which lives in the partials the chord has taken.

That is a sharper form of the earlier point about spectra. A stop, a bowing or a timbre that puts more energy into a note’s octave partials gives the onset more to hand over, and a stop rich in thirds and fifths — the mixtures and mutations of an organ — gives it less, because those are exactly the partials the other voices already fill. Whether an inner line is heard is decided less by how loud the tenor is than by how its energy is distributed across the harmonics the chord leaves free.

A tenor that moves with the chord has no onset #

The asynchrony is the whole mechanism, so it matters when the tenor moves.

The count is a step. A tenor that begins its new note at the same moment as the other voices’ new notes — a tenor moving at a change of chord, which is most tenor motion in a block-chord texture — has no onset lead at all and gets nothing from the heuristic. Its partials start with everyone else’s, and they belong to the new chord exactly as the held tenor’s belonged to the old one. A tenor moving thirty milliseconds or more after the others gets all of its surviving partials heard out.

So the motion that carries an inner voice on a sustained instrument is the motion between the harmony’s changes — a passing note, the resolution of a suspension, an anticipation, a neighbour note: any figure in which one voice moves while the others hold. A tenor that moves only when the chord does is, to this mechanism, a voice that never moves at all. The advice to keep inner voices moving is, on this reading, advice to give them their own rhythm.

A semiquaver at 120 beats a minute is 125 milliseconds, four times the threshold, so even a quick passing note carries a clear onset. The threshold matters only at the very edge — spread chords, arpeggiated attacks, the unintended asynchronies of an ensemble — where a tenor that is a few milliseconds late relative to its neighbours may or may not come out.

What the earlier count was, seen from here #

The earlier essay’s comparison has a new reading. On a struck instrument, a re-struck tenor gains partials — the masking falls because the held voices have decayed, and more of the tenor reaches the ear. On a sustained instrument the tenor gains no partials, but the ones it has are heard as a note rather than as colour. The piano’s advantage is quantity and the organ’s is attribution, and the struck instrument has both, since its re-struck tenor also has an onset.

That ordering matches a familiar pair of instructions. Pianists are usually told that an inner voice comes out if it is struck a little more firmly, while organists, who cannot strike harder, are taught to articulate an inner line — to detach it, to give it its own attack — if it is to be heard. The arithmetic says the two instructions are about two different mechanisms, and that on the organ the articulation is the mechanism: a line that is not articulated has no onset of its own.

The earlier essay’s hero is the struck case drawn against time, and it is worth setting beside the sustained one, because it is the other half of that division of labour.

On a struck instrument the new note gains partials the longer the chord has been decaying, and from half a second on it has all eight back an octave down. None of that happens on a sustained instrument, and the most the onset can do there is attribute what the flat line allows.

How the numbers were made #

The chords are the earlier essays’: the arrivals of I–vi–IV–V–I in four parts, transposed, each balanced to the loudness of the written register’s chords. Each voice has a stated spectrum — the collection’s string spectrum of eight partials, its generic flue organ of six, or its clarinet, with strong odd partials — and every partial of the tenor is tested against the threshold that the other voices’ partials put over it, through the same spreading functions as every masking figure here. On a sustained instrument nothing decays, and the reading is taken a second after the chord.

A surviving partial is one above that threshold. The pitch the surviving partials name is the fundamental of the best harmonic template fitted to them, the same fit the combination-tone essays use; a single partial names its own frequency. The onset rule is a step at the thirty milliseconds what makes two partials one note used, which is a round number from the published range for hearing out a partial with an onset lead, not a measured constant of any listener.

What a grouping rule cannot settle #

Whether an attributed partial is heard as a note or as a change of colour. An entrance is a change of colour found that a part entering a texture is often heard as the texture changing rather than as something added to it. A single heard-out partial is the limiting case of that, and which way it goes may depend on whether the listener is already following the tenor.

Whether one partial is a line. A single sine component, heard out by its onset, has a pitch and nothing else. Whether a listener follows a succession of such tones as the tenor line, or hears them as a faint accompaniment that is not quite anyone’s, is a question about sequential grouping — the subject of a voice is a stream — rather than about any one note.

How long the attribution lasts. The old-plus-new heuristic is described for the moment of a change; whether a partial heard out at its onset stays heard out for the second the note lasts, or gradually rejoins the chord’s colour, is not settled by the step rule here. On a sustained instrument the answer would decide whether a long moving note is heard for its whole length or only at its start.

Registration and room. The organ spectrum is generic and the room is absent. A real organ in a reverberant church adds a tail to every note, which blurs exactly the onsets the mechanism depends on; how much reverberation it takes to erase a thirty-millisecond lead is the question the release is on the wrong side asked of endings, asked of beginnings.

Whose inner voices #

The practice this is about is four-part writing for organ, choir and strings — the chorale, the hymn, the string quartet’s inner parts — where the textbook advice to keep inner voices moving was formed. The arithmetic says that advice works on a sustained instrument only for motion that is rhythmically independent of the chord changes, and only in registers where the chord leaves the inner voice something to move: an octave below the written register, a string-like spectrum gives the listener a pitch without a colour, and two octaves below, nothing. It also says that an organist’s registration and articulation are doing the work that on a piano the decay does for free.

Still open: the moving tenor in a room #

The mechanism rests on a clean onset, and the instruments it matters for are played in rooms that smear onsets. A reverberant tail keeps each old note sounding while the new one begins, which does two things at once: it adds masking, which the counts here do not include, and it blurs the asynchrony, since the old note’s tail and the new note’s start overlap.

The computation would add the room’s decay to the held voices and to the old tenor note, as the essays on endings did, and ask two things of the new note’s partials at each reverberation time: whether they still clear the masking, and whether their onset still leads what surrounds them by the thirty milliseconds the heuristic needs, measured as the time for the new partial to rise above the decaying tail. It would say whether the organist’s articulated inner voice survives a church — and whether the reverberation time at which it stops surviving is anywhere near the times of the rooms organs are built for.