How small a difference is audible

The tuning ladder on this site is nine essays long. Every one of them argues about a discrepancy of a few cents to a few tens of cents, and every one of them has silently assumed that those discrepancies are audible. That assumption is a claim about a listener, it has never been tested here, and it turns out to be true of most of the ladder and false of one rung.

The curve is a fit published by Wier, Jesteadt and Green in 1977, from a large set of two-alternative discrimination trials. Expressed in hertz it rises steadily with frequency; expressed in cents it has a shallow minimum in the middle of the singing range, at about three and a half cents at 1 kHz, rising to about five at 250 Hz and five and a half at 4 kHz.

What the number sorts #

Applying that curve to the site’s own quantities produces a list, and the list is the essay’s argument.

The Pythagorean comma, 23.46 cents. Six or seven times the limen. The gap between twelve fifths and seven octaves is not a technicality; it is a large, obvious, unmissable interval, roughly a quarter of a semitone, and every listener in every tuning argument for two and a half thousand years has been arguing about something plainly audible.

The syntonic comma, 21.51 cents. The same. The second comma, arriving by thirds rather than by fifths, is equally audible and equally consequential.

The schisma, 1.95 cents. Below the limen at every frequency on the curve. The difference between the two commas is not audible as a pitch difference by anybody, and this is why a whole family of temperaments could treat them as interchangeable without anybody complaining.

The spread of thirds in Werckmeister III, 17.6 cents. Well above. Key character in an irregular temperament is not a poetic claim about the affect of a key; it is a difference in interval size five times larger than the discrimination limit.

Equal temperament’s major third, 13.7 cents sharp of just. Above, by a factor of four. The compromise every keyboard makes is audible, which is the entire reason the argument about it has lasted four centuries.

The difference between 53-tone equal temperament and just intonation, under 2 cents on most intervals. Below. Fifty-three divisions of the octave closes the syntonic comma to within a fraction of the limen, and the claim that it is “effectively just” is not rhetoric but a statement about a listener.

That list is the single most useful thing this essay produces. It draws a line through the site’s own subject matter, and everything above the line is an argument about something a listener can be asked about.

Two different questions with the same units #

The number above is the discrimination limit and it answers a laboratory question: two tones, one after the other, differing only in frequency, which was higher? That is a much easier task than any musical one, and the essay would be dishonest if it left the impression that five cents is the resolution at which music operates.

The musical quantity is different in at least four ways, and each one makes it larger.

Musical intervals, not tones. Asked whether an interval is in tune, listeners are far less precise than they are at comparing two tones. Published thresholds for mistuning of a melodic interval are in the range of ten to thirty cents, depending on the interval and the training of the listener.

In a context, not in isolation. A note in a phrase is judged against a key, a harmony and a memory, and all three are far coarser instruments than a direct comparison.

Complex tones, not pure ones. With harmonic tones, some judgements get better — two simultaneous complex tones beat, and beating is detectable far below the limen for pitch — and some get worse.

And the categories interfere. Interval size is heard in boxes, and a difference that stays inside a category is systematically underestimated.

The last two pull in opposite directions and it is worth being precise about how. For simultaneous tones the operative limit is not this curve at all: two harmonic tones a fifth apart, mistuned by two cents, beat at about half a hertz, and half a hertz of beating in a sustained chord is obvious to anyone who has tuned an instrument. That is why a piano tuner works to a fraction of a cent while a singer does not — the tuner is counting beats rather than comparing pitches, and beat counting is a completely different measurement with a completely different limit.

So the honest summary has three numbers, not one. Two cents is audible in a sustained chord, through beating. Five cents is audible between two successive isolated tones. Twenty cents or more is what it takes for a melodic interval in music to be reported as out of tune. Every claim about tuning belongs to one of the three regimes, and confusing them is the standard error in the field.

And all three carry a fourth number that the units hide, which is how long the note lasts. The five cents is a figure for a tone long enough that the ear rather than the signal is the limit, and the section on Weber’s law below finds where that stops — about half a second at A440, below which the limen rises as the reciprocal of the duration and passes every quantity in the list above within a few hundred milliseconds. So the three regimes are three regimes for sustained notes, and there is a fourth régime below them in which none of the distinctions can be made at all.

The point that ruler makes most sharply is that equal temperament is not uniformly a compromise. Its fifths are essentially perfect as far as any listener is concerned. Its thirds and sixths are audibly out. A description of it as “everything slightly wrong” is a description of the arithmetic; a description in terms of what can be heard is “the fifths right and the thirds wrong”, and that is a different system to argue about.

Weber’s fraction, and why the curve has the shape it does #

The underlying regularity is that the smallest detectable change in a quantity is roughly proportional to the quantity — Weber’s law, which holds approximately for many senses and approximately here.

For frequency the fraction is around 0.2 to 0.5 per cent over the middle of the range. A constant fraction is a constant number of cents, because cents are logarithmic, and that is why the curve above is nearly flat when drawn in cents and steeply rising when drawn in hertz.

That the ear’s resolution is roughly constant in cents rather than in hertz is not a small fact. It is the reason a musical interval can be the same interval anywhere on the keyboard — the reason transposition is possible at all — and it means the whole logarithmic apparatus of cents, semitones and octaves is not a mathematical convenience imposed on the subject but a description of the measuring instrument.

The departures from Weber’s law are where the interest is. The fraction is worst at the very bottom of the range and at the very top, and it is best between about 500 Hz and 2 kHz, which is where speech and most singing sit. It is also considerably better for tones lasting more than about 200 ms than for short ones, and that limit is worth computing rather than mentioning, because it puts a tempo on every entry in the list this essay opens with.

A tone of duration T cannot have its frequency specified more finely than about 1/2T, whatever the listener is like, so below about half a second at A440 the limen is set by the note’s length rather than by the ear. Solving for the length at which each quantity drops under it:

The tempered major third’s error stops being audible at a semiquaver at a hundred and four beats a minute, which is not a fast tempo and not an unusual note value. So the argument this whole ladder is about is an argument about sustained notes: at three notes a second nobody can hear which temperament a keyboard is in, and at seven nobody can hear a comma.

That is a much sharper version of the qualification the section above makes. It is not merely that a musical judgement is coarser than a laboratory one — it is that the coarsening has a rate attached, and the rate is inside the repertoire rather than beyond it. A slow chorale and a fast toccata on the same instrument in the same temperament are two different tuning arguments, and only one of them is an argument.

It also explains a practice that is otherwise a matter of taste. Tuners, theorists and the listeners who claim to identify a temperament all work on held chords, and every demonstration recording of a historical temperament sustains its chords for seconds. That is not a rhetorical choice about how to show something off; below about a sixth of a second there is nothing to show. And the same arithmetic says which repertoire a temperament can be argued about from: music whose harmonic rhythm is slow enough that each chord is held, which is most of the repertoire the temperaments were designed for and rather little of what came after.

There is one more comparison worth making explicitly, because it is the case where the limen decides an argument that has been had many times. The claim that a particular historical temperament is identifiable by ear is a strong claim, and it is true for the thirds and false for the fifths of every system in that figure. Anyone claiming to identify a temperament is identifying it from its thirds, and a test that presented only fifths would be a test nobody could pass.

Which computation produced the numbers #

The limen curve is a published fit and the rest is arithmetic done here.

Wier, Jesteadt and Green’s expression gives the detectable frequency change in hertz as a function of frequency; converting it to cents is a logarithm. The comma values are computed from their definitions rather than quoted — the Pythagorean comma is twelve fifths against seven octaves, the syntonic is four fifths against a major third and two octaves, and the schisma is the difference between them — so the three horizontal lines in the figure are exact and the curve they are compared against is not.

The duration figures use the same steady-tone fit against the Fourier bound of one over twice the duration, taking whichever is larger — which is the convention this collection uses throughout and is a simplification, since two limits combined in quadrature would give a smoother crossing and slightly earlier thresholds.

The site’s gate asserts the comparison that matters: that the schisma falls below the curve everywhere on it and that both commas fall above it everywhere. If a future revision of the fit moved the curve enough to change either answer, the check would fail rather than the essay quietly becoming wrong.

The limit that is not a limit #

There is one more measurement worth putting beside the curve, because it is the case where a listener beats it by a factor of ten and the reason is instructive.

Two sustained complex tones tuned a fifth apart produce a coincidence between the lower tone’s third partial and the upper tone’s second. Mistune the fifth by one cent and those two partials sit about 0.23 Hz apart at a root of 262 Hz — a beat with a period of four and a half seconds, which is slow and completely unmistakable in a held chord.

The moral is general and it applies well beyond tuning: a perceptual limit is a limit on a task, not on a system. Reformulating a comparison as a detection of change, or of movement, or of a coincidence, routinely buys an order of magnitude. This is why every practical tuning procedure in history — laying a temperament by counting beats, tuning a unison until the beating stops, checking an octave against its own partials — is built around beats rather than around pitch comparison. The instrument makers found the loophole long before anybody measured the limen.

What the picture cannot show #

Individual variation, which is very large. The curve is an average over trained listeners. Untrained listeners are typically two to four times worse; the best trained listeners are better than the curve. A statement of the form “nobody can hear two cents” is false for some people and the figure has no way of saying so.

Training, which moves it. Discrimination improves substantially with practice on the task, over hours rather than years, and it keeps improving for a long time. The limen is not a fixed property of an ear, and any statement of the form “the ear can resolve five cents” is really a statement about a particular listener on a particular day at a particular stage of practice.

Duration and level, both of which matter. Short tones are harder; very quiet tones are much harder. The curve is for tones of moderate duration at a comfortable level, which is the best case.

And the whole question is about pure tones. The figure’s own subject is a pure tone, and there are almost none in music.

Whose music this bears on #

The claim that a listener can hear the difference between two temperaments is a claim about a specific practice, and the answer differs sharply between them.

Keyboard music of the seventeenth and eighteenth centuries. Here the essay’s numbers bear directly: the differences between meantone, Werckmeister and equal temperament are ten to twenty cents on the thirds, which is above every threshold discussed here. The historical arguments about temperament were arguments about audible things.

Music for instruments with flexible pitch. A string quartet or a choir adjusts continuously, and the question of which temperament they are in has no stable answer — they are not in one. The relevant limit for them is not this curve but the drift tolerance of a sustained chord, which is a beating measurement.

Singing, where the limit is not the ear’s. A trained singer’s pitch is stable to perhaps ten or fifteen cents on a sustained note, which is two or three times the discrimination limen. That gap is the interesting quantity: it means a listener can hear a singer’s own instability, and it means the question of what temperament a choir is in has no answer more precise than the choir’s own control. The whole apparatus of comma arithmetic is finer than the instrument it is being applied to, and it stops being a description of practice and becomes a description of an ideal.

And the contemporary microtonal repertoire. Music written in 19, 31 or 53 divisions is written with these thresholds explicitly in mind. A composer using 53-tone equal temperament is relying on its intervals being indistinguishable from just ones, and the schisma-sized errors involved are below the limen by design.

Where the ladder goes next #

This rung established that a listener’s pitch resolution is a few cents. The next asks a question that resolution alone cannot answer: what happens when the same listener is asked to set an interval rather than to detect a change in one. The answer is that they do not set it where the arithmetic says — an octave tuned by ear comes out wide, by ten to thirty cents, which is several times the limen and in a consistent direction.

That is a much stranger result than anything here, because it means the listener’s pitch scale is not the logarithm of frequency. This essay’s curve measures how finely the scale is divided. The next one measures whether the scale is straight.