A room keeps a pizzicato from giving its note away

A doubled pizzicato gives its note away early set a plucked violin against a held flute on the same G and asked when the composite stops sounding like the pluck. The answer was 70 milliseconds into a one-second pluck, while the pluck was still five decibels the louder, because a string loses its upper partials first and the upper partials are what a pizzicato is recognised by. A pluck whose partials all faded together would have kept the note eight times as long.

Every number there was computed in a room with no walls. A pizzicato is heard in a hall, and a hall does to a plucked note the one thing the string does not: it keeps each partial sounding after the string has let it go, for as long as the room’s own decay time at that partial’s frequency. The essay ended on the obvious question. Does that reverberant tail return the note’s colour to the pluck after the direct sound has handed it over, and for how long?

The answer is that the room gives nothing back, because by the time the question arises the note has not left.

What a room adds to a pluck #

The hall is the one a room does not decay evenly built from its surfaces: 18,700 cubic metres of plaster, panelling, a wooden floor and an audience, ringing for 3.07 seconds at 125 hertz and 1.53 at 4 kilohertz. The seat is ten metres from the players, which is about twice the hall’s mid-band critical distance of 5.3 metres — the distance at which, as how far away the room takes over put it, the direct sound and the reverberant sound are equally loud. Most of the audience sits further back than that.

Each partial of the pluck now reaches the seat twice. Directly, it decays as the string makes it decay. And through the room: the partial feeds the reverberant field while it sounds, and the field decays at the room’s own rate in that partial’s band. A held note has been sounding long enough to fill the room, so it arrives as its direct sound plus a steady reverberant field of the size the seat’s distance sets.

The tail overtakes the direct sound almost at once, and it overtakes it soonest for the partials the string loses fastest: the eighth partial at 20 milliseconds, the fourth at 30, the fundamental at 40. By a quarter of a second the eighth partial’s direct sound is 120 decibels below where it started and its tail is 16. At a seat in the hall, the upper partials that dry pluck lost first are carried almost entirely by the room. Their tail decays at the hall’s rate near three kilohertz, about 1.7 seconds, rather than the string’s rate for its eighth partial, an eighth of a second.

The note stays for half a second #

That is enough to reverse the reason the note left. Dry, the composite’s top belonged to the flute within 70 milliseconds, because nothing was left of the pluck’s top. In the hall the pluck’s top arrives from the room for the better part of a second, and the composite stays nearer the pluck for 506 milliseconds.

The two distances in the first figure show how differently it happens. Dry, the distance from the pluck climbs steeply as its spectrum collapses and crosses the flute’s at 70 milliseconds. In the hall both distances move slowly, because both players are now arriving mostly as reverberation, and they cross once, at 506. The room does not return the note after the pluck has lost it. It keeps the pluck from losing it, for seven times as long.

The room also changes where the two players start. At the seat the flute arrives with its reverberant field already built: ten metres into the hall is nearly twice the critical distance in the middle of the spectrum, so the field there is nearly four times the direct sound and the flute arrives more than six decibels above its direct sound alone, while the pluck’s tail has not yet begun to build. At the instant of the pluck the composite is therefore about three decibels from the pluck’s colour rather than the dry 1.8. The pluck starts with less of the note and holds what it has for far longer.

Half a second is a musical length rather than an acoustic one. At a hundred and twenty crotchets a minute it is a whole beat. Dry, the pizzicato’s colour lasted 70 milliseconds, the front of a note; in the hall it lasts about as long as the note.

The seat matters for the first few metres #

The seat decides how much of the room a listener gets, and the handover follows it up and then levels off. A metre from the players in the hall the room adds almost nothing: 86 milliseconds against the dry 70. At five metres, near the critical distance, the flute’s handover is 445; at ten, 506; at thirty, 532. For the flute, a seat further back than about twice the critical distance changes the answer by a few per cent, because both players are already arriving mostly as reverberation; the oboe and the clarinet go on gaining a little further back.

The church, 7,000 cubic metres of stone ringing for 6.32 seconds at 125 hertz and 2.37 at 4 kilohertz, has a critical distance of only 2.3 metres, so every seat more than a couple of metres from the players is in its reverberant field. A metre from the players the flute’s handover is already 440 milliseconds, and thirty metres back it is 824.

The partner matters as much as the room. The oboe’s handover rises from 52 milliseconds to 254 in the hall and to 326 in the church; the clarinet’s from 44 to only 118 and 94. A section has a loudest member found ownership among these instruments ordered the same way whatever the context, and the room keeps the order: the partners that took a doubled pizzicato fastest dry still take it fastest in a hall, and a clarinet takes it within about an eighth of a second at any seat drawn. In none of the sixty seats and partners drawn does the note return to the pluck once it has gone.

The room’s uneven decay takes part of it back #

A hall rings longer in the bass than in the treble, and that tilt runs in the same direction as the string’s, only far more slowly. So a room’s reverberation keeps a pluck’s upper partials alive, and a room’s tilt shortens that reprieve. The size of each effect can be read off by giving a room one decay time at every frequency — its own mid-band value — and comparing.

For the flute, the tilt takes back 30 per cent of what an even room would give in the hall and 40 per cent in the church; for the oboe, 24 and 32 per cent. The room’s reverberation is most of the effect and its tilt is a large minority of it. The church, whose treble decays two and a half times faster than its bass, loses more of its reprieve to the tilt than the hall does.

The clarinet is the exception in both rooms. With the church’s decay made even, the clarinet would take a doubled pizzicato at 35 milliseconds, sooner than with no room at all; the church’s actual tilt lets the pluck keep the note to 80. The tilt that shortens the pluck’s hold against a flute lengthens it against a clarinet. The drawings do not say why, and a partner whose colour differs from the violin’s mostly in its upper partials is where a room’s handling of the upper partials would be expected to cut both ways.

The loss law matters a tenth as much #

The dry result had one number in it that nobody has measured: the exponent relating how fast a string partial dies to its partial number. Across the plausible range it moved the dry handover by a factor of fifty, and the collapse belongs to the bass and counted in the decay, or not at all had each found the same unmeasured number deciding a different question.

In the room that dependence nearly disappears. Dry, the handover falls from 543 milliseconds at an exponent of nought to 11 at two; ten metres into the hall it falls from 931 to 184, and in the church from 1,439 to 254. The factor of fifty becomes a factor of five. The exponent still matters, but it decides whether a pizzicato keeps its note for a fifth of a second or most of one, not whether it keeps it for a hundredth or half.

The reason is the tail. The exponent sets how fast the string’s direct sound loses its top, and at a seat in the reverberant field the pluck’s top is arriving mostly from the room, whose loss depends on the room’s absorption at that frequency and not on the string’s partial number. The measurement nobody has made still decides the dry answer; in a hall it decides much less of the answer anybody hears.

The room is the clock #

Dry, the handover was a fixed share of the pluck’s decay time — seven per cent of it for a flute — because every loss in the model was proportional to time over that decay time. In a room that proportionality is gone. A pluck of a quarter of a second, the short dry pizzicato that dry loses its note in 18 milliseconds, keeps it for 372 in the hall; a pluck of four seconds, closer to a harp, keeps it for 654. Sixteen times the decay time buys less than twice the time.

In a room the pizzicato’s own decay stops setting the clock and the room’s decay starts. A short pluck and a long one both put their upper partials into the same hall, and the hall keeps them for as long as the hall keeps anything at those frequencies. What distinguishes the two plucks is how much energy they put in, not how long they would have lasted on their own.

What it means for a doubling in a hall #

Pizzicato strings under a sustained woodwind line are taught as a way of giving the line a defined start, and the dry computation found that description exact: the pluck owns the first twentieth of its own decay and the held instrument owns the rest. In a hall the description is too modest. Ten metres back, a pizzicato doubled by a flute or an oboe keeps its colour for between a fifth and four fifths of a second, depending on the partner and the room, which at ordinary tempi is the length of a written note rather than of its onset.

The partner still decides most of it. A clarinet takes a doubled pizzicato within about an eighth of a second in either room at every seat drawn, so a pizzicato under a clarinet is heard as the clarinet’s attack in a hall as it is dry. A flute leaves the pluck its colour for half a second in the hall and more than three quarters of a second in the church. An orchestrator who wants the pluck heard as a colour, and not only as an onset, is choosing the partner and relying on the room, and the choice of partner is the part of that decision that survives a change of hall.

The loss law makes the same point from the other side. Dry, whether a doubled pizzicato was heard as itself for a hundredth of a second or for half a second depended on a property of the string that nobody has measured. In a hall it depends mostly on the hall, whose decay is a measurement anyone can make band by band.

The arithmetic #

The pluck and the held note are the radiators the dry release used, at 392 hertz, with the pluck twelve decibels up and its partial n losing sixty decibels over the decay time divided by n to the loss exponent. The rooms are the two whose surfaces the earlier essay on uneven decay tabulated, with Sabine’s decay time in each octave band, air absorption included above a kilohertz, and a partial’s decay time read by interpolating between bands in log frequency. The critical distance in each band is the site’s formula with a directivity factor of one, and a seat r metres away hears a steady source’s reverberant field at (r over the critical distance) squared times its direct power.

A partial whose direct power decays at a rate g feeds a field decaying at the room’s rate k, and its tail at the seat is its direct power at the pluck, times that ratio, times k(e^(−gt) − e^(−kt))/(k − g). That closed form is checked against the field integrated step by step. The held note arrives as its direct power times one plus the ratio. Ownership is the dry release’s rule: log-spectral distance over the first sixteen partials, each spectrum normalised to its own power, against the pluck as it now arrives, direct and tail together. With no room the computation reproduces the dry handovers to the millisecond. The handover is the end of the pluck’s last stretch of ownership in four seconds.

What the room model assumes #

A diffuse, exponentially decaying field. Sabine’s field is the same everywhere past the critical distance and decays at one rate per band. A real hall has early reflections that arrive in the first eighty milliseconds from particular surfaces and colour the sound before the diffuse tail does; the flute’s and the oboe’s handovers in both rooms fall after that, and the clarinet’s at the nearer seats do not.

That the held note has filled the room. A flute that began with the pluck would have its own reverberant field still growing, which would slow the flute’s arrival at the seat as the room slows the pluck’s departure.

That the direct sound is not beamed. A directivity factor of one is right for a violin’s lower partials and wrong for its upper ones, which an instrument points found beaming above the frequency at which the radiating surface is about a wavelength across. A listener on the instrument’s axis gets more direct top than this model gives, which shortens the reprieve at a near seat and changes little at a far one.

What the model cannot show #

What a listener calls the owner. The drawings say when the composite’s spectrum stops being nearer the pluck’s. A listener who heard the pluck’s attack may hear a pizzicato for longer than that, and one attending to the flute may hear a flute sooner. Two players on one note set up the rule; nothing here has tested it against anyone.

The orchestrator’s hall. The effect is a property of the seat and the room, and a composer writing a doubling does not know either. What the arithmetic says is that the dry answer, a twentieth of the pluck’s life, is the answer for the front of a dry room, and that in the halls this repertoire was written for the pizzicato keeps its colour inside a doubling for about half a second.

Still open: whether the room lets a pluck have a note back #

Every doubling here starts with the pluck and a held note together. The case the room ought to change most is the reverse: a pizzicato entering on a note a sustained instrument already holds, with the held note’s reverberant field already full and the pluck’s tail still to build. The blend arrives before the note does found an attack turning a balance by a bounded amount; in a room the pluck’s attack arrives against a held note that is mostly reverberation, and its own reverberation arrives tens of milliseconds late. The computation is this one with the held field steady and the pluck’s tail starting at zero — which is what it already is — scored instead by whether the pluck ever owns the note at all, seat by seat. It would say whether a pizzicato entering under a held line can be heard as a colour in a hall, or only as an attack.