Two mechanisms, and the seam between them

Ask anybody who has sung at all to slide slowly up from the bottom of their range and something happens on the way. The voice changes character, usually with a lurch, at a place that is roughly the same each time and that singers have named in every language that has a singing tradition. The English word for it is the break.

The standard account of the break is a piece of pedagogy: it is a defect, technique removes it, and a trained voice is one in which it cannot be heard. That account is not wrong about the training. It is wrong about the object, and the previous rung’s description of the larynx as a valve is enough to say why.

Two regimes, not one setting #

A valve that sustains an oscillation by feeding back on itself is a nonlinear system, and nonlinear systems characteristically have more than one stable regime available at the same parameter settings. The folds have two, and the difference between them is which part of the tissue is entrained.

In the first, the whole depth of the fold takes part. It closes completely once a cycle, and closes hard. In the second, only the mucosal edge moves; the folds are open a much larger fraction of the period and often never quite meet. The names are old and confusing — chest and falsetto, heavy and light, modal and loft — and the ones that carry no theory with them are Roubeau’s, from the electroglottographic work that established the scheme: M1 and M2.

Everything audible about the difference follows from those two shapes, and the transform of the pulse says how.

That single number, the level of the first partial above the second, is the standard acoustic correlate of the open quotient in phonetics, and it is the measurement that distinguishes the mechanisms without anybody having to look at the folds. It comes out of the model rather than being fitted to it.

Which makes the break a question about kinds #

Here is the distinction that decides what sort of object the break is, and it is worth stating in the abstract before measuring anything.

A threshold is a value of some parameter at which a system changes state. Water boils at a hundred degrees. Approach the value from below and the state changes there; approach it from above and it changes back there. The crossing is in the same place both ways, because there is only ever one state available at each temperature.

A bifurcation into two coexisting regimes is different. Over some range of the parameter, both states are stable, and which one the system is in depends on where it has been. Coming up, it holds the low state until that state stops existing; going down, it holds the high one until the same happens. The two crossings are therefore in different places, and the gap between them is hysteresis.

Hysteresis is not a subtlety of the theory. It is a measurement, and it can be made by any singer with a piano.

So the break is a bifurcation, and the eight semitones of overlap are not a defect to be trained away. They are the region where two stable regimes coexist, and the point of technique is to choose between them deliberately rather than to be handed whichever one the voice was already in.

Why the ranges are offset at all #

The two mechanisms do not cover the same notes, and the reason is available from the same reasoning that makes a bass string thicker than a treble one.

Frequency in any tensioned system goes as the square root of the tension divided by the mass in motion. The tension the larynx can apply is the same either way — the cricothyroid muscle does not know which mechanism is in use — but the mass in motion is not: M2 entrains a fraction of the fold’s depth and M1 entrains all of it. Less mass at the same tension is a higher frequency, so M2’s whole range is offset upward from M1’s, by roughly the square root of the mass ratio.

This is the same argument as the one that decides how a piano’s strings are wound and the same one that puts a bass viol’s strings where they are: for a given tension and length, mass sets the pitch. The unusual part is that here the mass is a variable of the performance rather than of the instrument.

That is also why the ranges overlap rather than meeting. Nothing constrains the tension to be the same in both, so the top of one mechanism’s range and the bottom of the other’s are set by different limits and there is no reason for them to coincide. They do not, and the measurement says by how much: 220 to 349 hertz, 799 cents, which is the eight semitones this essay has been calling the overlap.

The square-root argument is stated above as though it settled the offset, and it does not. The entrained depths in the model are 1.00 and 0.45, so the square root of the mass ratio is 691 cents — and the measured ranges are offset by 1,200 cents at the top and 1,709 at the bottom. The prediction is out by roughly a factor of two at one end and two and a half at the other, and worse than that, the two ends disagree with each other by 509 cents. A mass ratio predicts a translation, and what the measurements show is not a translation.

The same arithmetic says what the model is missing. Stress in the string picture runs over a hundredfold, so both ranges ought to be 1200·log₂√100 — 3,986 cents, and the same width for both, since the mass cancels out of a ratio. The measured spans are 2,507 and 1,999 cents, both about half of that, and differing from each other by 508. So the tension available is not the same in the two mechanisms and does not run over the same range, which is the assumption the offset argument quietly makes. The reasoning is right about the sign — less mass at a given tension is a higher pitch — and it is not a quantitative account of anything.

One more thing falls out of putting the hysteresis inside the overlap rather than beside it, and it is the asymmetry a picture of two crossings hides. The upward crossing at 330 hertz sits 702 cents into the 799-cent shared band, with 97 cents of M1’s range left; the downward crossing at 294 sits 502 cents into it, with 502 cents of M2 abandoned below it. Going up, the voice runs the low mechanism out and switches because there is nowhere left; coming down, it leaves the high mechanism with more than a third of the shared band unused. Those are two different reasons for a transition wearing the same name, and only the first is the bifurcation running out of a branch.

What the ear does with a seam #

The mechanisms differ in spectrum rather than in pitch, so the question of whether a break is heard is a question about how tolerant the ear is of a sudden change in spectrum with no change in frequency. The site has measured this from the other side.

That is the seam in the time domain rather than the spectral one. The two mechanisms differ in how long the folds stay shut, and everything about the two spectra follows from it: a longer closed phase makes a sharper corner, and a sharper corner makes more upper partials.

What makes two partials one note is common onset, common fate and a harmonic relation, and all three survive a change of mechanism: the partials keep their frequencies, they keep starting and stopping together, and they stay harmonic. So the two productions fuse into a single stream — nobody hears the break as two voices — and the ear assigns them to one source without difficulty.

What it does hear is a discontinuity in timbre inside that stream, and it hears it well, because spectral change is precisely what identifies an instrument. The break is audible for the same reason a clarinet is distinguishable from an oboe, and inaudible when a singer arranges the spectra to match across it.

A different instrument’s register break, which shares only the word #

The word “register” appears twice in this collection with two entirely unrelated referents, and the coincidence has misled enough people that it is worth putting the two side by side.

A sung note starts rather than fades in. There is a threshold pressure below which the mechanism does not run, which is why a singer crossing the seam has to arrive at the new mechanism already above its own threshold — the seam is not merely a change of register, it is a place where one oscillator has to be stopped and another started.

On a clarinet the register break is a change of which mode of the resonator is being sounded. The tube has a ladder of resonances and the player, with a small hole opened near the top, persuades the standing wave to move up one rung of it. The valve is unaffected. The pitch jumps by a fixed interval — a twelfth for a stopped cylinder, an octave for a cone or an open pipe — and the interval is arithmetic about where the ends are.

In the voice, the resonator does not change mode at all. The tract’s resonances sit at five hundred and fifteen hundred hertz whatever is happening at the folds; the note is set by the source, and it is the source that changes regime. The pitch does not jump by a fixed interval — it need not jump at all — and what changes is the spectrum.

Two mechanisms, two words, one term. A clarinet’s break is a boundary condition and a voice’s is a bifurcation, and the only thing they have in common is that a player has to get across them.

What the tract does about it, which is not nothing #

The source changes regime and the filter does not, but the filter is what a listener hears, so a great deal of what is called managing the break is done with the mouth.

The technical term for this in voice pedagogy is vowel modification, and the reason it works is the source–filter independence the previous rung established: the two systems can be moved separately. A change in mechanism moves energy out of the upper partials; a change in tract shape can move a resonance down onto a partial that is still strong. The listener hears a continuous timbre across a discontinuous source.

This is worth being careful about, because it is where the pedagogy and the physics are usually run together. “Mix” is not a third mechanism. The electroglottographic measurements find two, and what is called a mixed or blended production is one of the two with the tract and the adduction arranged so that the seam is inaudible. That is an entirely respectable thing for training to achieve; it is just not a third stable regime, and no measurement has found one.

The measurement the reader can make #

The site’s habit is that every claim is given a test it could fail, and this one has a test that needs no equipment.

Sing a slow slide upward through the middle of the range until the voice changes, and note where. Then slide slowly downward from above and note where it changes back. If the break is a threshold, the two notes are the same. If it is a bifurcation, the upward one is higher, and by a musically obvious amount — a whole tone in the measurements drawn above, and between one and three semitones across the published sets.

The failure condition is clean. A voice for which the two crossings coincide, repeatedly and at several tempos, is a voice for which this account is wrong. Nobody has reported one, and the effect is large enough that it would have been noticed.

Two secondary predictions come with it and both hold. The gap should shrink as the slide gets slower, because a system near a bifurcation is more likely to be pushed out of a marginally stable state by any perturbation, and a slow slide gives more time for one. And the same hysteresis should appear in an excised larynx blown by a pump, with no singer in the experiment at all — which is where Švec and his colleagues found it, and is the observation that rules out any explanation in terms of what the singer was intending.

Whose music uses it on purpose #

A break in the middle of a held note is a defect only under a particular aesthetic, and a great deal of music treats it as a resource. Naming those repertoires is not decoration; it is the check on the claim that this is a property of larynxes rather than of a technique.

Alpine yodelling is a rapid deliberate alternation between the two mechanisms, usually on wide leaps, so that the change of mechanism coincides with the change of note and the seam is placed rather than concealed. Central African yodelling — among the Aka and Baka — does the same thing inside dense polyphony, and does it with the mechanism change as a rhythmic event. Both traditions are old, both are geographically nowhere near each other, and both exploit the same bifurcation.

Closer to the European art tradition, the Tyrolean idiom entered nineteenth-century popular song and from there the American recording industry, which is why a country singer’s deliberate break has a name — the hillbilly or cry break — and a documented lineage. Blue yodelling is that lineage.

And the eighteenth-century Italian tradition the concealment aesthetic comes from was not unanimous about it either. The falsettisti who preceded the castrati sang the top of the range in M2 by design, and the seam was a normal part of a technique rather than a failure of one.

The generalisation worth drawing is the one this collection keeps arriving at from different directions. A physical discontinuity is not a musical fact until a practice decides what to do with it, and the same discontinuity has been treated as a fault, as an ornament and as a metrical device by different traditions using the same larynx.

Whose voices, and where the numbers came from #

The frequencies drawn here are for an adult male voice, and the whole picture shifts upward for higher voice types — the break in a soprano sits around the same scale degree of her range rather than at the same frequency, which is one of the reasons the two literatures on this took so long to agree.

The mechanism scheme is Roubeau, Henrich and Castellengo’s, built on electroglottography, which measures the electrical impedance across the neck and so reports contact between the folds directly rather than inferring it from the sound. That is what makes the open quotient a measurement rather than an estimate. The hysteresis figures are from that work and from Švec, Schutte and Miller’s videokymographic and excised-larynx studies.

The historical vocabulary is much older and much worse. Falsetto means false, which is a value judgement embedded in a technical term; chest and head name places where singers feel vibration rather than places where anything happens; and the registers of the eighteenth-century Italian tradition were counted variously as two, three or four depending on the teacher. The measurements find two mechanisms and a great deal of variation in how each is used, which is consistent with all of those schemes and identical with none.

What the picture cannot show #

The frequency ranges are one voice’s, and they vary enormously. Where a mechanism’s range ends is a property of a particular larynx and of training. What does not vary much is the existence of an overlap and the sign of the hysteresis, and those are what the argument uses.

Nothing here explains why the mechanisms differ in mass. The body–cover model of the fold — a stiff muscular body under a loose mucosal layer — is the standard account, and the mucosal wave travelling over the body is what supplies the phase difference that sustains the oscillation at all. Which layers are entrained is a consequence of the muscular setting, and this essay takes the two settings as given rather than deriving them.

The pulse shapes are model shapes. They are the two-parameter family the previous rung introduced, at parameter values typical of each mechanism, not inverse-filtered recordings of a particular singer. The spectral consequences drawn from them are robust — the fundamental’s dominance in a nearly-sinusoidal flow is not a delicate result — but the exact decibel figures are the model’s.

And the hysteresis is not measured here. It is quoted from published work, drawn to scale against the ranges, and named as a measurement in the caption. This site has no larynx and no electroglottograph.

The ladder from here #

Both mechanisms put almost all of their energy below a kilohertz, which is exactly where an orchestra puts its own. The next rung asks how a single voice is heard over ninety players, and finds that the answer is not level at all.