A low note cannot start on time

Every attack time in this ladder is a single number per instrument. A bowed violin is ninety milliseconds, a piano is eight, a marimba is three; the first rung tabulated them, the second computed what a mixed ensemble’s spread of them costs, and the third found that the dynamic moves each one through a range of three or four to one.

All three treat the number as a property of the instrument. It is not, quite. There is a second thing setting it, and the third rung’s last paragraph named it:

What the ladder still owes is the thing all three rungs have held at one value: the pitch. Every envelope here rises as fast as its instrument allows, and a low note cannot rise faster than its own period.

The floor, which is arithmetic and not a measurement #

An amplitude is a property of a waveform over some stretch of time, and a stretch of time shorter than one cycle does not contain a waveform. So there is a hard lower bound on how fast a note can be said to have an amplitude, and it is the period.

The bound that matters is a few periods rather than one, for the same reason the duration limen needs a few: one cycle is enough to have a peak and not enough to establish that the peak is repeating. This site’s own arithmetic for how long a note has to be before it has a pitch at all puts the requirement at several periods, and four is the middle of that.

Four periods at 27.5 hertz is 146 milliseconds. At 41.2 hertz — the bottom of a double bass — it is 97. At 82.4, the bottom of a cello, it is 49. At 440 it is 9, and at 1,319 it is 3.

That is the whole model, and its entire content is that a period is a period. What makes it interesting is what it collides with.

The instruments where it bites, which are the fast ones #

The obvious guess is that this matters for slow instruments in the bass, and the obvious guess is exactly backwards.

A bowed double bass has an attack of about ninety milliseconds. Its bottom E has a floor of 97. The floor is barely above the instrument’s own attack, so the correction is a few milliseconds and nothing much has happened.

A piano’s bottom A has a floor of 146 milliseconds and a hammer contact of about eight. The floor is eighteen times the instrument’s own attack. Everything about the heard moment of that note is set by its period and nothing by the action.

So the effect is largest exactly where nobody would look for it: on the fast, percussive, precisely-triggered instruments, in their bass registers. A pianist playing a two-handed chord across the instrument cannot make it sound together, and the reason is not the action, the touch, the regulation or the player. It is that the bottom note takes 146 milliseconds to become a note and the top one takes three.

What forty-three milliseconds is, in this collection’s units #

The spread across a piano is worth putting beside the numbers this site has established for what it is competing with.

It is twice the twenty milliseconds at which two events stop being one. It is nearly twice the twenty-four milliseconds an accent moves a bowed note by. It is about half the ninety-millisecond spread a mixed ensemble carries between its attack families, and it is inside one instrument. And it is comfortably larger than the eight to twenty milliseconds of motor noise a skilled player produces, which is the band the microtiming ladder works inside.

So it is not a subtlety. It is one of the largest timing quantities on this site and it is entirely mechanical.

It also has a fixed sign, which is what makes it different from noise. Every low note is late and every high note is early, always, on every instrument, in every performance. An averaging procedure does not remove a bias.

What a player does about it, and what the notation does not say #

There is a well-known thing pianists do that this model gives an account of.

Rolling a chord upward. A wide left-hand chord played strictly together sounds bottom-heavy and late; played with the bass fractionally early it sounds together. That is taught as an expressive device and as a period practice, and there is a large literature on the historical rolling of chords that treats it entirely as a matter of style.

The arithmetic here says that a bass note played 40 milliseconds early is heard with a treble note played on the beat, on a modern piano, at any tempo. Which does not make the stylistic account wrong — players roll chords for many reasons, and the historical practice extends to places this effect does not reach — but it does mean the device has a mechanical floor underneath it, and that a pianist who plays a wide chord exactly together is producing a sound that arrives unevenly.

Notation has no way to say any of this. The stave is not a ruler vertically and it is not one horizontally either; two noteheads on a stem are simultaneous by definition, and the definition is about the instruction rather than about the result.

The marimba, which is the clean test #

The prediction has a shape — the effect scales with the period and inversely with the instrument’s own attack — so the cleanest way to test it is an instrument that is fast everywhere and covers several octaves.

A marimba’s attack is about three milliseconds and does not vary much across the instrument. Its crossover, by the arithmetic above, is at 1,333 hertz. So every note of a four-octave instrument is pitch-limited, and the heard moment of each one is set by nothing but its own frequency: the bottom C at 65 hertz has a floor of 61 milliseconds and the top C four octaves up has one of 3.8.

The five-octave instruments reach C7 at 2,093 hertz, which is above the crossover, so their top half-octave is instrument-limited like a piano’s treble and the curve flattens there. That is a boundary inside one instrument’s range rather than outside it, and it is drawn in the figure below — the last two notes sit on the flat part.

That is a fifty-seven millisecond spread in the rise, and eighteen in the heard moment, on an instrument whose mechanism is identical at both ends and is played by one person with two hands. If the model is right, a marimba roll across two octaves in rhythmic unison with anything else is systematically bottom-late by an amount that grows toward the bass, and a player who has learned to compensate has learned a curve rather than an offset.

The eighteen milliseconds is the number to carry, because it is the one a listener would be judging. A rise is a stretch of time and a heard moment is a point inside it, and the criterion this ladder uses puts the point about a third of the way up — so a fifty-seven millisecond difference in how long two notes take to arrive is an eighteen millisecond difference in when they are heard to. Eighteen is still comfortably above the twenty milliseconds at which two events stop being one when it is doubled by an ensemble, and it is above the eight to twenty of a skilled player’s motor noise, so the effect survives being converted into the units the argument is about. But it is not fifty-seven, and the two numbers describe different things: one is a property of the envelope and the other of the listener reading it.

It is also the reason the effect is separable from everything else. On a piano the bass notes have thicker hammers, longer strings, more inharmonicity and a different action regulation, so a measured lateness could be attributed to several things. On a marimba those confounds are absent and the period is nearly the only variable left.

Which computation produced the numbers #

The floor is four periods, and four is the only asserted number in the essay. Its origin is stated above and it is asserted rather than derived: this collection’s own duration limen says a pitch needs several periods, and four is the middle of the published range. Sweeping it is one loop and it separates what the number decides from what it does not.

The spread and the ordering behave as claimed and the crossover does not. Every number scales linearly with the cycle count, so three periods shrinks the spread by a quarter and six grows it by a half; and the lags remain monotone in pitch at every value, so no note ever overtakes another. But the crossover is f₀ = 1000 × cycles / rise, which is proportional to the cycle count — it moves two octaves, from C4 to C6, across the range of values nobody can exclude.

That matters because the crossover is the qualitative claim. At two periods a piano is instrument-limited over most of its compass and this essay is about its bottom two octaves; at eight it is pitch-limited nearly everywhere and the essay is about the whole instrument. The sign of the effect is a theorem, the size is proportional to a guess, and the reach is proportional to the same guess — and the third of those was previously stated as though it were the first.

The rise used for each note is the larger of the instrument’s own attack and that floor, and the lag is that rise put through this site’s own perceptual-centre criterion — the fraction of the way up an exponential rise at which the note is heard, which the first rung established and which is the same criterion every figure in this ladder uses.

The crossover is where the two are equal, at f₀ = 4,000/rise in milliseconds. For a piano’s eight milliseconds that is 500 hertz, which is B4 — a little above the middle of the keyboard, and visible in the hero figure as the point where the curve flattens.

One thing has to be checked before any of it stands, because every number above was produced under one choice of criterion and the literature offers three.

That is the reassuring answer and the worrying one at once. The criterion cannot change which note is later, because proportionality preserves order — so the register finding above is safe whichever criterion a reader prefers. What it changes is the size, by a factor of twelve, and every millisecond figure in this essay is therefore a number attached to one criterion rather than to the ear. The ordering is a result; the forty-three milliseconds is a result and a convention.

Where the model stops #

A hard floor is a soft thing drawn hard. Nothing about a note becomes true at exactly four periods; the amplitude becomes progressively better defined as cycles accumulate. What that means for the heard moment is that the curve in the hero figure should be a smooth blend between the instrument’s attack and the period rather than a maximum of the two, and a blend would round the corner and change nothing else.

The criterion is still the one the third rung forked over. Playing louder is playing earlier found that a criterion set by the note’s own peak and one set by the surrounding music make different predictions once the level is a variable, and left the choice open. This rung uses the relative one throughout. Because every comparison here is between two notes at the same dynamic, the choice cancels — which is exactly the condition the third rung identified as the one where the two are indistinguishable.

The instrument’s own attack is treated as pitch-independent and is not. A piano’s hammers are heavier in the bass, its bass strings are more massive, and the mass ratio between hammer and string inverts across the compass — so the eight milliseconds used here for every note is itself a simplification, and one that runs in the same direction as the effect being computed. Separating the two would need the coupled contact model rather than a constant.

And the floor is about the note, not about the instrument’s onset. A piano’s hammer produces a wideband transient at the moment of contact, and that transient is audible and is not subject to any period. So a low piano note arguably has two onsets — the thump, which is on time, and the pitch, which is 146 milliseconds later — and which one a listener aligns to a beat is a question this model cannot pose. It would predict that a very percussive low note is heard earlier than this figure says, which is testable and is not tested here.

Whose music, and when #

The claim is about periods and applies wherever there are low notes. What is a claim about a repertoire is where low notes are asked to be rhythmically precise, and the answer has a date.

Before the nineteenth century the bass of most textures is a line — a continuo, a walking bass, a moving part — and a line’s notes are heard in sequence, where a systematic lateness affects every note equally and is not detectable. The thing this essay is about needs a bass note that has to coincide with something: a chord attacked together, a downbeat marked simultaneously across an ensemble, a low note in a rhythmic unison.

That texture is a nineteenth- and twentieth-century one, and it is at its most extreme in music with a deliberately low and deliberately percussive bass — which is where the effect is largest and where, if the model is right, the compensation is most necessary. It would predict that recorded bass parts in such repertoires sit systematically ahead of the grid, which is a measurement the microtiming ladder makes and which has never been made against pitch.

What the picture cannot show #

Whether a listener uses the pitch onset or the transient. The whole model assumes the heard moment is a criterion on an amplitude envelope, and a low piano note’s envelope has a click at the front of it. If a listener aligns to the click, none of this happens.

Nor whether the criterion itself has a frequency in it. Everything in this ladder takes the heard moment to be a fraction of the way up an amplitude envelope, with the fraction the same at every pitch. There is no particular reason to believe that: the ear’s temporal resolution is not uniform across frequency, and a criterion that involved a fixed number of cycles rather than a fixed fraction of a rise would produce a curve of a similar shape by a different mechanism and would be indistinguishable from this one on the evidence here.

And it cannot show the room. The bass of a hall is where the modes are sparse, so a low note’s envelope at a listener’s ear is not the envelope at the instrument — it is that envelope convolved with a few strongly-ringing modes, which lengthens the effective attack further and does so differently in every seat. The floor computed here is a lower bound on something the room only makes worse.

Where this ladder goes next #

Four rungs. A note is heard after it starts; an ensemble that mixes attack families carries a spread; the dynamic moves each attack through a range; and now the pitch puts a floor under all of it that no player and no instrument can get below.

The rung after it is the sum, and which notes have to be played early is where it goes. This ladder now has three separate contributions to one quantity — the instrument’s attack family, the dynamic, and the period — and every figure in it varies one and holds the others. A real passage varies all three at once: a sforzando low note on a piano against a quiet high one on a flute has the dynamic effect and the pitch effect pulling in opposite directions on one of the two instruments. Adding them is arithmetic this ladder has every term for, and what would come out is not another curve but a map: for a given scoring, which notes of a chord have to be played early and by how much, computed rather than rehearsed.