A moving tenor is heard because it is struck again

A bass chord low enough to balance has already hidden its tenor took the arrivals of I–vi–IV–V–I, voiced for four parts and moved into four registers, and set each chord at the level that makes it as loud as the written register’s chords. The balancing level is higher the lower the chord, and at that level every partial of each voice was tested against the masking the other voices’ partials cast. Two octaves down the tenor kept none of its eight partials, and an octave down it kept one. The chord was balanced by burying the voice an arranger most often wants heard.

That essay held every chord still, and ended on the variable it had fixed. A tenor that moves while the bass holds has onsets the bass does not, and the forward-masking course of a sound hides what came before it says how long after each change the old note goes on masking the new one. The question was whether a line an arranger writes in an inner voice is heard as a line by its motion alone, and whether the advice to keep inner voices moving in the low register is a response to exactly this.

It is, but not for the reason the question assumed. Motion is worth nothing to a masking count. What rescues a moving tenor is that on a piano, a harp or a guitar the new note is struck while the notes around it have been decaying, and on an organ or a string section, where nothing decays, the moving tenor is as buried as the held one.

The note the tenor leaves #

The first thing that changes when a tenor moves is that its old note stops. For some time afterwards it goes on masking, and its partials are close to the new note’s — a step away, the nearest any note can be. So the first question is how much that forward masking costs the new note.

It costs almost nothing. The forward-masking threshold falls away from the level of the masker’s own simultaneous masking along the published logarithmic course, and it is below the masking the three held voices are already casting within ten milliseconds of the move in every register. An octave up, where the held voices mask least, the old note costs the new fundamental 2.9 decibels for the first five milliseconds; everywhere else it costs a few tenths or nothing, and after ten milliseconds the new note sits exactly where a tenor that had never moved would sit. The note a tenor leaves is not what buries it. What buries it is the chord that is still sounding, and the chord is exactly as loud whether the tenor moves or not.

Motion on a sustained instrument #

On an organ, a string section or a choir, the held voices do not decay. A chord a second old is as loud as a chord just begun, and a moving tenor’s new note meets the same masking as a held tenor’s.

On a sustained instrument the moving tenor keeps precisely what the held one keeps: no partials two octaves down, one an octave down, three as written, three an octave up. That is the earlier essay’s result with a time axis added and nothing changed. Motion alone, measured as audibility against simultaneous masking, buys an inner voice nothing.

The same figure shows the other half. On a struck instrument, a second into the chord, the held tenor has decayed with the others and keeps what the held tenor on a sustained instrument keeps or a little more — it gains where its own partials happen to decay slower than the skirts above them. The re-struck tenor keeps six of eight partials two octaves down and all eight in every other register. Motion is worth nothing to a sustained tenor and everything to a struck one, and the difference between the two is not in the tenor.

What decay does for a new note #

A struck note’s partials decay from the moment of the strike, the higher ones faster. By the time a tenor moves half a second or a second into a chord, the bass, alto and soprano around it have lost several decibels each, and their upper partials — the ones whose skirts reach up over the tenor — have lost more. The tenor’s new note is struck at full level.

The result is steep. An octave down, where the balanced chord left the held tenor one partial, a tenor re-struck a quarter of a second into the chord keeps six, and from half a second on all eight. Two octaves down, the register where the held tenor keeps nothing at any time, the re-struck one keeps one partial a quarter of a second in, three at half a second, six at a second and all eight at two. As written and an octave up, a re-struck tenor keeps all eight at every time drawn.

The held tenor’s own curves go the other way. It decays with the chord, and in the higher registers its upper partials, which decay fastest, fall under the hearing threshold before the masking falls away: as written it keeps five partials a second in, two at two seconds and one at four. So on a struck instrument an inner voice that is held is heard less and less as the chord rings, and one that moves is heard more and more.

Why the skirt falls faster than the bass #

The size of the gain is out of proportion to how much the bass itself has faded, and the reason is in which of its partials do the masking. A second into the chord, the bass’s fundamental at C2 has lost five decibels. Its eighth partial, at 523 hertz, has lost more than twenty, because on a struck string each partial’s decay rate rises with its frequency. And it is the upper partials of the bass, not its fundamental, whose masking reaches up over the tenor: one sound hides another drew the upward skirt that a low partial casts, and the partials that cast it across the tenor’s register are the bass’s upper ones.

So the masking over the tenor falls far faster than the bass’s loudness does. A listener hears the chord as a little quieter a second after it was struck; the tenor’s region of the spectrum has been cleared by thirteen to twenty decibels. That is the fact the partials that do not die together found about a single struck note — its partials part company as it rings — seen from underneath: a struck chord goes dull as it rings, and a dull chord is one that has stopped masking the notes inside it.

How long a tenor has to wait #

In a written texture the time between the chord and the tenor’s move is a note value, so the figure reads directly as advice. At a moderate crotchet of about sixty beats a minute a tenor that moves on the second beat of a held chord moves one second in, and an octave down it has all eight partials back; at a quick hundred and twenty, half a second in, and still all eight. Two octaves down it needs longer — six partials at a second, all eight at two — so a very low inner voice is only fully rescued by moving on the third or fourth beat of a slow bar. A tenor that moves at the same instant as the chord, on the downbeat, gets none of this, because nothing around it has had time to decay: it is struck with the chord and buried with it.

Partial by partial #

The count of partials hides how large the margins are.

A second into the chord an octave down, the held tenor’s fundamental is 1.7 decibels above its masking and every other partial is below it, by as much as 5.8. The re-struck tenor’s partials are all above, by 4.7 to 9.9 decibels — enough that the margin is not a knife-edge but a clear gap. Dragging the time of the move earlier shows how quickly the gap opens: at a quarter of a second two of the re-struck tenor’s partials are still under, and by half a second none is.

How fast the chord dies away #

The whole effect is a race between the held voices’ decay and the time the tenor waits before it moves, so it depends on how fast the instrument decays.

The piano’s low strings, the law every struck-string figure in this collection uses, take about twelve seconds to fall sixty decibels, and at that rate a second of decay is five decibels at C2 and more for every partial above it. A chord that dies in anything from three to twenty-four seconds leaves a re-struck tenor all eight partials a second in. Only when the chord rings for the better part of a minute does the gain start to shrink — six partials at forty-eight seconds, four at ninety-six — and a chord that never decays at all is the sustained case, where the gain is gone. The re-struck tenor is heard in proportion to how much the chord around it has died.

The tenor does not have to move #

One corollary follows at once and is worth stating on its own, because it is the cleanest test of the explanation. If the gain comes from the strike and not from the motion, a tenor that is struck again on the same note should gain exactly as much as one that moves a step. It does: re-struck on its own pitch, the tenor keeps three partials two octaves down half a second into the chord, six at a second and eight at two, and all eight an octave down from half a second on — the same counts, partial for partial, as the moving tenor. The forward masking of the old note was never what held the new one back, so it makes no difference whether the old note was the same pitch or a step away.

So the keyboard version of the advice is not really about lines at all. An inner voice on a struck instrument is heard if it is struck again after the chord, whether it moves or repeats, and the thing to avoid is an inner voice held under a bass that goes on sounding. Repeated-note accompaniment figures, which re-strike the same inner voices on every beat, are on this arithmetic among the most effective ways there are of keeping a low chord’s inner voices audible.

The advice, reconsidered #

The advice to keep inner voices moving in the low register is found in keyboard and orchestral writing alike, and the arithmetic here splits it in two. On a keyboard, a guitar or a harp, re-striking an inner voice is exactly what a low chord needs: the held notes have decayed, and a fresh note stands clear of their masking within half a second. On instruments that sustain, it does nothing a masking count can measure, and if the moving inner voice is heard it is heard for other reasons.

Those reasons exist, and none of them is level. A voice is a stream, and the ear decides which found that a sequence of notes is followed as one voice when its steps are small and its timing regular, which is a property of the line rather than of any one note. A blown note does not start late, it starts slowly found that onsets are where the ear separates sources, since partials that start together are heard as one. A moving tenor on a sustained instrument has onsets the held voices do not, and those onsets are a cue for hearing it as a separate voice even while its partials are below the masking — the auditory scene the masking count cannot see. The surprising part is that the two instruments need the same advice for opposite reasons: a pianist’s moving tenor is heard because it is louder than what surrounds it, an organist’s because it starts when nothing else does.

The count underneath the curves #

The chords are the first arrival of I–vi–IV–V–I in four voices, as the earlier essay voiced it, moved by zero, one or two octaves and set at the level that makes each register’s chord as loud as the written register’s. Each voice has eight partials of a string spectrum, and a partial’s masking is the power sum of the masking every other partial casts on it, with the spreading function the masking essays use, floored at the threshold of hearing. On a struck instrument every held partial has fallen by sixty decibels times the time since the chord, times its frequency’s ratio to C2 to the power 0.7, over twelve seconds; the re-struck tenor is a step from its old note and at full level. The forward masking of the old note is the published logarithmic decay from five to two hundred milliseconds, starting from the old partial’s simultaneous masking at the new partial’s frequency.

What a masking count cannot say about a tenor #

Whether a partial above its masking is heard as the tenor. A count says a partial is audible, not that the listener assigns it to a voice. Eight audible partials of a re-struck tenor are a strong case for hearing the note; they do not guarantee the line.

The attack. A struck note’s first milliseconds are a noisy transient, and a re-struck tenor’s attack is itself a masker and a cue. Neither is in the steady partial levels here.

Pedalling. A piano with its sustaining pedal down lets the held voices ring on undamped and adds the sympathetic resonance of every other string, which slows the chord’s decay toward the sustained case. On this arithmetic, pedalling a low chord takes away some of what re-striking its tenor buys.

One chord. The figures are the first arrival of one progression. Other voicings put the tenor nearer or further from the bass’s skirts, and the balancing level changes with them.

Whose inner voices #

The inner voices in question are those of four-part keyboard and ensemble writing: the tenor line of a hymn or chorale texture, the inner parts of a piano accompaniment, the second horn or viola in an orchestral chord. The balanced low register is where the earlier essays found them buried. On the evidence here, a composer writing for piano can rescue a low inner voice by re-striking it, and one writing for strings or organ cannot rescue it by level at all and must rely on the line. That is one more consideration to add to those why the exercise is in four parts weighs: every voice added to a low texture is another skirt over the voice beneath it.

Still open: the onset the count cannot see #

The one mechanism left that could make a moving tenor audible on a sustained instrument is its onset, and it has a quantity the masking essays already use for something else. A partial that starts some tens of milliseconds before or after its neighbours is heard out of the mixture more easily than one that starts with them, and what makes two partials one note measured how much asynchrony it takes. A moving tenor on an organ has exactly that asynchrony against the held chord, at every move.

The computation would give each partial of the new tenor an advantage for the first stretch after its onset — a release from masking that grows with the asynchrony and fades as the note settles — and ask whether it lifts a buried tenor above its masking for long enough to be followed. It would say whether the organist’s moving tenor is heard by a cue of the same size as the pianist’s decay, or whether on a sustained instrument the advice to keep inner voices moving is advice about writing a line, and not about hearing a note at all.