How long a note has to be

The smallest audible difference in pitch is about five cents in the middle of the range, and that number does a great deal of work in this collection. It is what decides which commas are music and which are arithmetic. It is what every temperament argument is measured against. It is the reason the schisma is a curiosity and the syntonic comma is a problem.

It is also a measurement made on tones that last as long as the listener wants. Wier, Jesteadt and Green’s figures are for steady tones presented in a quiet room to trained listeners with no hurry. A note in a piece of music is a quaver.

The bound that has nothing to do with hearing #

Before anything about the ear, there is a bound from the note itself. A tone that lasts T seconds is not a frequency; it is a band, and the band is about 1/2T wide. Nothing a listener does can make it narrower, because the information is not there.

In cents, at 440 hertz, that is 19.6 for a tenth of a second, 38.9 for fifty milliseconds and 95.7 for twenty. Those are not small numbers — 95.7 cents is very nearly a semitone.

Four hundred and eighty-six milliseconds is a long note. A quaver at 120 beats a minute is 250; at 200 it is 150. Almost every note in almost every piece is on the wrong side of the crossing.

It is worth being careful about what kind of claim that is. The bound is not a fact about ears; it is a fact about signals, and it would apply to a machine with a perfect spectrum analyser inside it. A fifty-millisecond tone does not have a frequency accurate to a cent that the ear fails to recover — it does not have one at all. Everything the ear could conceivably do is bounded by it from above.

Which makes it a different sort of limit from the difference limen, and the difference matters for how the two combine. The limen is a performance figure that improves with training, attention and practice. The Fourier bound improves with nothing. So the crossing point is the note length below which no amount of skill helps, and it is the more robust of the two numbers by a long way.

Where the crossing is, by register #

The crossing moves with pitch, and it moves the wrong way for music.

A bass note needs nearly a second to be worth arguing about and a high one needs a seventh of that. The reason is the same one in both halves: the Fourier bound goes as 1/(2Tf), so a low note has fewer cycles in the same time, and the steady limen is also worse down there.

What this does to the commas #

The commas this site argues about are between 2 and 24 cents, and putting them against the duration bound sorts them by tempo rather than by size.

At 440 hertz: a 250-millisecond note gives an effective limen of 7.9 cents. The Pythagorean comma at 23.5 and the syntonic at 21.5 both clear it comfortably; the schisma at 2.0 does not, and did not clear the steady limen either. At 100 milliseconds the limen is 19.6 cents and the syntonic comma barely clears it at all — on this reckoning a passing note in a fast passage can hardly be heard to be mistuned by a comma.

Those two figures are for a pure tone, which is the only thing a bare Fourier bound describes. A section near the end of this essay gives the note its partials and recomputes them at 8.1 cents and 4.9, which is the correction that keeps this paragraph’s conclusion from being much stronger than the evidence supports.

This does not overturn the tuning ladder. It bounds it, in a direction the ladder itself has been feeling for. A comma under the threshold established that a temperament’s errors have to clear the limen to be a musical fact at all; what this rung adds is that the same error clears it at one tempo and not at another, so a temperament’s audibility is a property of the music played on it.

It also explains something otherwise puzzling about practice. Historical temperaments are argued about in terms of held chords and sustained harmony, and their differences are described as “key colour” — a quality of a passage rather than of a note. On this account that is not vagueness. The interval that carries a temperament’s error has to be held for something like half a second before the error is available to be heard at all.

There is a musical consequence of that ordering which is easy to state and hard to notice. Fast passages in the bass are, as far as pitch is concerned, less determinate than the same passages in the treble — not because a bass line is harder to follow but because there is less pitch information in it. A walking bass at 120 beats a minute is quarter-second notes around 100 hertz, where the bound is about 12 cents; the same rhythm played two octaves up carries a bound of 3.

Which is one reason a bass line can be doubled at the octave without sounding out of tune, and a soprano line cannot.

The other end: how few cycles is a pitch #

The Fourier bound is about resolution. There is a second question, which is when a tone has a pitch in the first place, and its answer is in cycles rather than in seconds.

A tone of a few cycles is a click with a colour. The pitch becomes definite over something like four to eight cycles, and because that is a count rather than a duration it converts into wildly different times across the compass.

At C2 — 65 hertz — a fifty-millisecond note is three cycles. At C6 the same fifty milliseconds is fifty-two cycles. So a fast passage in the bass is doing something categorically different from the same passage two octaves up, and every model in this collection that takes a pitch as given treats them alike.

The one place the effect is used deliberately #

If a short note’s pitch is imprecise, then a deliberately short note is a way of putting a pitch into a texture without committing to it — and there is a whole class of ornament that does exactly that.

An acciaccatura is written as a note with a line through it and played as fast as possible: at fifty milliseconds its pitch is specified to about 25 cents at the top of the treble staff, 39 at A440, and a great deal worse below. So the dissonance it makes with the chord it decorates is not a pitch relation a listener can price. It is an event with a colour, and the notation’s instruction — as short as possible — is an instruction to stay under the bound.

The same reading applies to a fast trill, whose alternation rate is such that neither of its two notes reaches the crossover. A trill is one object with a band rather than two pitches alternating, which is why its notated interval matters far less than its speed, and why a trill on a major second and one on a minor second are so much more alike than the two intervals are.

Which computation produced the numbers #

The Fourier bound is 1/2T converted to cents at the stated frequency: 1200·log₂(1 + 1/2Tf). The factor of two is the ordinary convention for the half-width of a rectangular window’s main lobe and it is a convention — a different windowing gives a factor between about 0.5 and 1.5, which moves the crossover by that factor and moves nothing else.

The steady limen is dlfHz, the same fit every acuity figure here uses, and the crossover is where 1/2T equals it: T = 1/(2·Δf). At 440 Hz the limen is 1.03 hertz, so the crossover is 486 milliseconds.

The effective limen is the larger of the two, which is the simplest combination and almost certainly optimistic — two independent limits normally combine to something worse than either.

The cycle counts are arithmetic and the four-to-eight figure is quoted. How many cycles a tone needs before it has a definite pitch is a measurement from the literature, it varies with what the listener is asked to do, and nothing here computes it.

The two corrections this essay names and does not make #

Two of the caveats below are quantitative and both can be run. They point in opposite directions and they are nearly the same size, which is the reason to do both or neither.

Combining the bounds properly makes everything worse. Taking the maximum of the two limits is the optimistic combination; adding them in quadrature is the ordinary one. Under quadrature the effective limen is above the steady value at every duration, so “the crossover” has to be defined as where the degradation reaches some fraction — at ten per cent, it moves from 486 milliseconds to 1,060 at 440 hertz, and by the same factor of 2.2 at every register. On that reading essentially no note in any music is long enough.

And giving the note partials makes everything better. A harmonic complex is not one measurement of a frequency but several of the same fundamental. The Fourier width is an absolute number of hertz and it is the same for every partial, so partial n, sitting at n·f₀, fixes the fundamental to that width divided by n — the high partials are the precise ones. Weighting each by its amplitude and combining, the pure-tone bound improves by the square root of the sum of (n·aₙ)², which for the string spectrum this site uses is a factor of 2.81. It is 3.02 for a clarinet and 3.91 for a bell; a spectrum falling as 1/n makes every partial contribute equally, so the factor is very nearly the square root of the number of audible partials.

They nearly cancel #

The combined column is 0.78 times the published one at every register — the two corrections are each about a factor of two and they very nearly cancel, so the headline survives to within a quarter and does so for entirely different reasons than the ones given.

What does not survive is the verdict on the middle of the three groups below. The effective limen for a string tone at 440 hertz, corrected:

The correction is largest exactly where the essay’s conclusions are drawn. At 150 milliseconds a syntonic comma clears the bound by a factor of 3.5 rather than 1.6, and at 100 milliseconds — where the essay concludes the comma “no longer clears it” — it clears by 2.6.

So the melodic-intonation arguments do not weaken after all, and the reason is the one thing a single-tone Fourier bound cannot see: a real note is a stack of partials and the high ones are where the pitch information is. The bound is right about a gated sinusoid and music does not contain any. The three-way split below stands, but its middle row moves from “weakens” to “survives with a factor of three rather than four to spare”.

Whose listeners, and whose music #

The steady limen is trained listeners in a laboratory, and this rung inherits that. What it changes is the stimulus rather than the listener: the same trained ear, given a shorter tone.

The tempo marks are ordinary Western ones, and the argument they support is general. Any tradition whose intonation distinctions are fine — the comma-sized differences between two theories of one maqam, the fine inflections of a raga — is making a claim that requires held notes, and the traditions that make those claims are the ones whose practice is built on long notes over a drone. That is a suspiciously good fit and it is offered as an observation rather than as a finding, because nothing here measured a raga.

What the picture cannot show #

Two bounds are combined by taking the maximum, which is a placeholder. The real combination is some quadrature or worse, so the effective limen near the crossover is understated and the crossover itself is later than drawn — the section above prices it at a factor of 2.2, and prices the partials correction that offsets it at 2.81.

Nothing here is measured on short tones. The literature on the frequency limen at short durations exists and reports a rise, and this essay computes a bound rather than reproducing that data. The claim is that the bound holds, not that it is tight.

A musical note is not a gated sinusoid. It has an attack, a decay and partials, and the partials give a listener several independent estimates of the same fundamental. That factor is computed above and is 2.8 for a string spectrum, in the direction of making short notes better than the bare bound says. What is still not modelled is the attack and the decay, both of which shorten the interval over which the note is actually steady and therefore work the other way.

And a note in a phrase is not heard alone. A listener hearing a scale has the surrounding notes as context, and pitch judgements in context are far better than in isolation. Everything above is the isolated case, which is the same limitation the category-counting rung ran into from the other side.

What it means for the rest of the collection #

Three groups of essays here rest on the five-cent figure, and the correction lands on them differently.

The temperament arguments survive. They are about held chords: a wolf fifth sustained, a third in a cadence, a chord left ringing while a key is established. Those are long notes by construction and the steady limen is the right one.

The melodic-intonation arguments weaken less than the bare bound suggests. The leading note pulled in two directions by a comma is an argument about a note that is usually short, and on the single-tone bound a 21.5-cent difference on a 150-millisecond note is only 1.6 times clear of it. Counting the note’s partials, as the section above does, puts it 3.5 times clear — weakened from the four-fold margin a long note gives, and not at the edge.

And the drift arguments are unaffected in a way worth stating. A comma of drift per circuit accumulates across a passage; it is not a property of any one note, so no note’s duration bounds it. The pitch that has drifted is heard against the pitch at the start, and both of those are long-run averages rather than measurements of a quaver.

That three-way split is the useful output of the rung. A limen is not one number applied to a collection of arguments — it is a number per argument, set by how long the note carrying the difference lasts, and the collection has been using the most favourable value throughout.

Where this ladder goes next #

Two rungs of this ladder have asked how finely a pitch can be heard and found the answer is three different numbers depending on the question and that the octave is not where it should be. This one adds a fourth axis and finds the whole family is a function of note length, with a crossover that most music sits below.

The rung after it is the one this makes obvious: the same duration bound applied to interval rather than to pitch. Two notes in succession are two short tones, and the error in their difference is the sum of two errors — so a melodic interval in a fast passage has a limen worse than either note’s, and that is a computable number this site has never produced.