A singer needs a quarter of the breaths the page offers

Ask a singer where a phrase ends and the answer is often where the breath goes. The ceiling everybody names is the loose one priced that breath and found it two or three times looser than the listener’s own limit on what can be held as one thing, which removed the breath as the explanation for why phrases are the length they are. It left something else open. A sung performance contains an event no boundary detector reads: a silence with an intake of air in it, placed by a person who has just decided something about the music.

Adding that event to a detector looks easy and is awkward, for the reason that essay gave. A singer breathes at a boundary the detector is supposed to find independently, so a detector that used breaths would be scoring its own answer sheet. The way out it proposed was a case where the two disagree, which should occur where the written phrase is longer than the air: slow, high and loud.

That is a claim that can be run in full before anyone goes near a microphone. The air is arithmetic, the tunes are written down, and the placement of breaths through a tune is a small exact optimisation. The corner the claim predicts turns out to be there, and to be much smaller than the argument suggests. The more useful finding is what the rest of the map says about the corner’s complement: nearly everywhere else, the air does not decide.

The air as a budget #

Every note spends air at a rate set by its pitch and its level, which is the same arithmetic as the earlier essay’s: a flow of 140 millilitres a second at middle C and 70 decibels, rising 1.2 per cent a semitone and 3.5 per cent a decibel, drawn from 2.8 litres of usable volume for a trained singer and 1.8 for an untrained one. So each note uses a known share of a full breath. A plan is a set of gaps in the tune at which the singer breathes, and it is possible if no stretch between two breaths spends more than one full lung.

Two questions then have exact answers by simple search. What is the fewest number of breaths a tune can be sung on, wherever they fall? Walk the tune spending air and breathe only when the next note will not fit, which is provably as few as any plan can use. What is the cheapest plan that keeps the breaths on the phrase ends? Dynamic programming over the gaps, with a breath at a notated phrase end costing nothing and a breath anywhere else costing however weak the detector’s boundary is there. A breath that lands inside a phrase in the second plan is one the air has forced: no plan with a free phrase end could have avoided it.

The tunes are the three the detector has been tried on since where a phrase ends: Ode to Joy, Twinkle and Frère Jacques. Their phrases are not long, and that is the first thing the method makes visible. Ode to Joy’s and Twinkle’s phrases run to about eight beats; Frère Jacques ends a phrase every four. At crotchet 60, eight beats is eight seconds, about half of what a singer can do at a comfortable level.

The corner is real and very small #

Put that on a grid of seven tempos and seven levels, for all three tunes, from D4, and read off how often the air forces a breath off a phrase end.

A trained singer in this register is forced inside a phrase on 8 settings of 147. Across the whole grid there are 332 breaths in the plans, and 301 of them, 91 per cent, are at phrase ends. The boundary of the corner is easy to state exactly for Ode to Joy. A phrase at crotchet 40 lasts twelve seconds. At D4 and 85 decibels the air lasts 11.7 seconds, and at 80 decibels it lasts 14. The crossing is that single inequality, and everything else in the grid is on the safe side of it.

So the essay that predicted this corner predicted it correctly: slow, high and loud is where the disagreement between breath and phrase is to be found. Taking the same tunes up an octave to D5 raises the number of forced settings from 8 to 14, because the flow’s rise with pitch over that octave is worth about four decibels. But the corner is not where music usually is. Crotchet 40 at 85 decibels, unaccompanied, on tunes of this length, is a place for a demonstration rather than a reading. A collection of disagreements gathered from recordings of these tunes would have to be gathered from there, and it might be empty.

The larger fact is the map’s other side.

What the air actually asks for #

If the forced corner is small, then outside it the singer has room. How much is the question the second number answers: the fewest breaths possible, set beside the phrase ends the page offers.

At a comfortable level a singer needs about a quarter of the phrase ends a tune offers, 28 per cent across the grid at 70 decibels. At 60 decibels it is 15 per cent, at 80 it is 43, and even at 90 it is 67. Ode to Joy at crotchet 60 and 70 decibels has three phrase ends and needs one breath. Frère Jacques at the same setting has seven and needs one. Twinkle at crotchet 120 has five and needs one, and Ode to Joy and Frère Jacques at crotchet 120 could be sung without breathing at all.

Now read what that means for the event a listener hears. A singer who breathes at every phrase end at that setting is taking three breaths where one would do, or seven where one would do. All those extra breaths are decisions. They could be left out without consequence for the air, and when they are taken, the reason is musical: a new thought, a clearer shape, or a taught habit of how the phrase should sound. In information terms, an event that has to happen tells a listener nothing about why it happened, and an event that need not happen tells the listener something.

That is the answer to the question as it was left. The earlier essay asked whether a breath is a cue a listener uses or a cost they discount. It is not settled here, since nothing in this essay has a listener in it. What the arithmetic settles is which of the two the breath could be, and where. Almost everywhere a breath can serve as a cue, because it is almost everywhere optional. In the forced corner it cannot, since a listener who took it as evidence of a boundary would be reading the singer’s lungs.

It also says why the detector’s answer-sheet problem is not quite what it seemed. A breath is a performer’s vote on the question the detector is asking, and since the air rarely forces that vote, it is usually an informed vote. A detector reading breaths would be reading another listener’s reading rather than a neutral acoustic cue. That would make it a better model of what an audience actually hears and a worse test of what a score contains, and those were always different questions, as a family of readings has already concluded about its own window.

Who runs out, and what that is worth #

The corner grows quickly with less air, and the air’s arithmetic gives this a compact form.

The untrained singer’s volume, 1.8 litres against 2.8, is worth 12.8 decibels of flow. The two models are the same function with a scale change, so a trained singer at 90 decibels and an untrained singer at about 77 run out of air on the same notes of the same tune. Forced settings go from 8 of 147 to 34 from D4, and to 45 from D5. At 90 decibels the fewest breaths an untrained singer needs is 1.16 per phrase end from D4 and 1.31 from D5, so here the page’s phrases are longer than the air can reach and a singer has no way of breathing only at the phrase ends.

That has a simple practical reading for anyone who has sung in a congregation. Nobody in an untrained group reaches the end of a loud, slow hymn phrase on one breath; they breathe when they run out, at no particular place. Since this model places a forced breath at the cheapest point in the phrase it fell into, it overstates how well they do it. The difference between the congregation and the soloist is thirteen decibels in the model and a matter of practice in the room.

One qualification is worth stating as a number. Nobody sings down to the last of the 2.8 litres. If a singer uses only seventy per cent of their usable volume before they feel short, then that is worth about ten decibels on the same scale, and the corner of the first grid moves towards the middle by the same ten. It still does not reach ordinary singing at moderate tempos, but it stops being quite so remote.

Where a forced breath falls #

The last check is on the idea that makes this most tempting as a cue. A phrase longer than eight seconds is longer than the psychological present, so by the time the air runs out, the listener already holds the phrase in pieces. The level the tempo chooses gives a detector set to that present its own boundaries inside the phrase. It would be neat if a forced breath fell at those inner boundaries, so that the place the singer has to breathe is the place the listener was going to divide the phrase anyway.

It does not. Forced breaths land near the listener’s own boundaries about as often as a gap taken at random does: 71 per cent against 67, 65 against 66, 61 against 63, 63 against 62. That test is not sharp, because a window of one note either side covers about two thirds of all gaps in these tunes, so a real effect of modest size could hide inside it. What can be said is that nothing beyond chance appears in 396 forced breaths, and the cheap plan is given every chance of finding the inner boundary, since it is charged by exactly the detector’s strength.

The reason is mechanical. A forced breath’s place is set mostly by the budget: it has to fall late enough that the air lasts and early enough that the remainder fits, and in a tune of equal crotchets the budget leaves only a narrow window. The tune’s own structure has little say in that window. So when a breath is forced it is doubly uninformative: it says nothing about the singer’s reading, and it says little about the music.

What a tank of air leaves out #

The model is arithmetic on a stated flow. Each note spends air at the rate its pitch and level imply, nothing else consumes it, and a breath refills the lungs at no cost. That last assumption is generous and it matters: a real breath takes a few hundred milliseconds that must come from somewhere, and in a singer’s hands they come from the end of the note before. Charging that time would make every breath cost something audible, and would make phrase ends relatively cheaper still.

The flows are published ranges, reduced to a formula with two exponents. A real singer’s flow depends on vowel and on technique as well as pitch and level, and unvoiced consonants spend air far faster than any vowel. A long sung text will have a higher effective flow than a neutral vowel, by an amount not modelled.

The detector’s strength is a charge, not a measurement of harm. The forced breaths’ placement rests on it, and nothing says that a breath in a weak gap is actually worse than one in a strong gap by this amount. The 1-minus-strength cost is a convenience. Where the numbers depended on the placement, the conclusion was chance, which is the conclusion it could most easily have been made to miss.

Three tunes, none of them written for long breath. They were chosen by earlier essays for other reasons, they are short, and they repeat. A tune with a sixteen-beat phrase, a Handel melisma or a long Romantic line, would shift the corner a long way and possibly into ordinary performance. That is exactly the direction this method would be useful in, and the tunes here cannot show it.

Nothing here describes what singers do. Nobody has checked the plans against any recording, and it would be surprising if a performer’s breaths lay so neatly on the notated ends. There is also the phrase-end lengthening a singer adds, which a detector whose resolution the performance sets has shown to be large; it spends more air at the end of each phrase than these written durations do, which moves a few of these numbers against the singer.

Whose breath #

The numbers are about a solo line sung unaccompanied, and they do not transfer directly. A choir staggers its breathing so that no one breath is heard, which removes the event altogether: the section can sing longer than any of its members, and the cue is gone. One of these three tunes does that by construction. Frère Jacques is a round with entries two bars apart, so each voice’s phrase ends arrive eight beats after the voice ahead of it, and a breath taken at the same place in the tune is never taken at the same moment in the room. A round is a staggered-breathing machine, which nobody needs to have intended, and what it hides from a listener is exactly the event this essay has been pricing. A singer accompanied by an instrument can take time differently, and a wind player has a similar budget with different numbers. A tradition in which breath is deliberately audible, or deliberately placed against the phrase, is doing something the model has no term for. Western vocal pedagogy’s taught rule, which says to breathe at the punctuation of the text, is stricter than the air requires almost everywhere on this map, and that is itself the finding: the rule is enforcing something other than the lungs.

A long silence is a separate matter again. A silence long enough to be an ending puts the threshold at which a pause becomes an end at several seconds, and a rest is a diminuendo computes what a pause does to a listener’s running impression of level. A breath at normal speed is a fraction of a second, well under either. What it gives a listener is a mark in the sound, rather than time.

Still open: a breath the listener can hear coming #

Everything above treats a breath as an event at one moment. A listener following a line can also hear the air running out: a held note’s level and steadiness change towards the end of a breath, a singer shortens the note before it, and long-breathed lines are heard as long because a listener has some idea of what they should have cost. The computation would use the plans here to mark, note by note, how far through its budget each note sits, and it would ask whether the moment of a forced breath can be predicted from the budget alone by a listener who knows the singer’s capacity from the first phrase. It would say whether an audience’s sense of a phrase being too long is a sense of the music or a sense of the singer’s lungs, and whether the two can be separated in a performance at all.