A rhythm fast enough to be a chord

A drummer playing three against two and a violinist playing a perfect fifth are, on the page and in the ear, doing entirely different things. One is a rhythmic device with a feel and a groove; the other is a harmonic interval with a consonance and a function. They belong to different chapters of every book on the subject.

They are the same object at different speeds.

The lower rates are a rhythm at a walking tempo. The upper ones are 300 Hz against 450 Hz, which is D above middle C against the A above it. Nothing between them is a change of kind.

The boundary, and how sharp it is #

The transition is not a point, and pretending otherwise would be the easiest error to make here. Three things happen at different rates, in this order.

Below about 8 to 10 events a second the individual events are separately countable. A listener can follow which stream is which and can tap along with either.

Between roughly 10 and 20 the counting fails but the events are still individually audible: the stream becomes a flutter or a buzz rather than a sequence, and the two layers begin to fuse into one texture.

Above about 20 to 40 a pitch emerges. The lower bound of pitch perception is usually quoted at 20 Hz, and that number is honest about the phenomenon rather than precise about it: a 20 Hz tone is audible but its pitch is vague, and definite pitch takes hold nearer 40 or 50 Hz.

The interesting consequence of the middle region is that it is genuinely neither. A pattern at 15 events a second is not a rhythm anybody can count and not a pitch anybody can name, and it is exactly the region a drum roll, a flutter-tongued flute and a rolled R live in.

The arithmetic that does not change #

What survives the crossing is the ratio, and this is where the two subjects turn out to share a mechanism rather than merely an analogy.

Two streams of events at rates in the ratio p:q have a common period: the pattern repeats every q events of the fast stream and p of the slow. That is the definition of a polyrhythm and it is what makes three-against-two a twelve-step cycle with events on specific steps.

Three against two drawn as two cycles has a common period at the least common multiple, coincidences that fall where they fall, and a structure determined entirely by the two integers. That is the rhythmic end of the axis, and nothing about the drawing changes as the rate rises — which is the whole reason the transition this essay is about is invisible in the pattern.

For two tones the same arithmetic is the classical account of consonance: a simple ratio produces a combined waveform that repeats quickly, and 3:2 repeats after two cycles of the lower tone. That the two accounts are the same computation is not a coincidence at all — periodicity is periodicity, and the ear’s treatment of it is what changes.

Which means the shared arithmetic explains rather less than it appears to. Both domains are governed by the ratio, and what the ratio buys is different in each: in rhythm it decides how long the cycle is and where the coincidences fall; in pitch it decides whether the partials of the two tones land on each other.

How much less can be measured, and the measurement is one correlation. Rank every coprime ratio inside an octave by its cycle length, which is what it costs as a rhythm, and by its roughness, which is what it costs as an interval. Over twenty-three ratios the two rankings correlate at 0.629.

That is a substantial agreement and a long way from an identity, and the disagreements have a shape:

The two simplest ratios agree exactly and the agreement decays from there, and it decays in a direction with a name. The ratios that are rhythmically simple and acoustically rough are the narrow intervals — the major third, the minor third, the whole tone; the ratios that are rhythmically complex and acoustically smooth are the wide ones, 11:7 and 12:7 and 10:7.

The reason is that consonance has a term the rhythm has no analogue for. Roughness depends on how far apart two partials land in hertz, so a wide interval is smooth almost regardless of its ratio and a narrow one is rough almost regardless. A cycle length knows nothing about width — three against two is the same cycle whether the two streams are a fifth apart or a fifth plus three octaves.

So the identity this essay is built on is exact at the top of the list and approximate below it, and what breaks it is the one thing that is genuinely different about the two ends of the axis: a rhythm has a ratio and no register, and an interval has both. The shared arithmetic is real and it is 0.629 of the story.

The third thing that lives on the same axis #

Between a rhythm and a pitch there is a middle phenomenon this site has already met from the other direction, and putting it on the same axis completes the picture.

Two tones a few hertz apart beat at their difference, and the beating is a slow pulsation — an event rate of a few per second, which is squarely in the rhythmic band. So a pair of pitches can produce a rhythm, by a mechanism that has nothing to do with speeding anything up.

Push the difference upward and the beating follows the same path everything else on this axis does. At 3 Hz it is a countable pulse. At 15 Hz it is roughness — the sensation of grating that the whole consonance model is built on. Above about 20 Hz the two tones separate and are heard as two pitches, and the beating disappears because the ear has resolved them.

So roughness is beating too fast to count. That is the same statement as this essay’s, arrived at from the other end, and it means the boundary appears twice in the theory of consonance: once as the lower edge of pitch, and once as the point at which a difference frequency stops being a rhythm and becomes a texture.

Where the analogy breaks #

It is tempting to go further and say that consonant intervals are consonant because they are simple polyrhythms, and that a 3:2 rhythm should feel as settled as a 3:2 interval. Several writers have said something like this. It does not hold up, and the reasons are worth setting out because they are the reasons pitch is not just fast rhythm.

The ear performs a frequency analysis and the mechanism is different above the boundary. Above about 20 Hz the cochlea resolves a complex tone into partials, and what reaches the brain is a list of frequencies rather than a waveform shape. Consonance is then decided by how those partials interact — a spectral matter with no rhythmic counterpart at all. Below the boundary there is no such analysis: the events arrive as events.

Simplicity ranks differently. In pitch, 3:2 is more consonant than 5:4, which is more consonant than 7:5. In rhythm, three-against-two is easy, four-against-three is harder, five-against-four is harder still, and seven-against-five is a specialist skill — the ordering happens to be similar because both track ratio complexity, but the reason is completely different. Rhythmic difficulty is about subdivision and motor control; harmonic roughness is about partials in a critical band.

The tolerance is different by two orders of magnitude. A fifth mistuned by twenty cents is noticeably out. Twenty cents is about 1%, so at 300 Hz it is 3 Hz. The corresponding error in a rhythm — a beat placed 1% late — is completely inaudible, and the deviations that make a groove are twenty to fifty milliseconds, which at these rates is an enormous fraction of a beat.

Two tones two hertz apart are audibly out of tune as a pitch pair and beat twice a second; as a rhythm — two streams whose rates differ by 0.7 per cent — the same relationship would take a minute to drift through a cycle. One relationship, two descriptions, and the units are what decide which one a listener is given.

Lower down the same axis there is a second transition, and it is one a listener makes rather than one the rate makes for them.

So the axis this essay runs along has more than one boundary on it. Long before a rhythm becomes a chord, it stops being one rhythm and becomes a choice between two readings — and that crossing moves with tempo while the pitch crossing does not, because one of them is set by a preference and the other by the ear’s own limit.

The one place the continuum is used #

Rhythm and pitch being one axis is a fact with almost no compositional application, which is worth saying plainly because it is a fact that invites overreach.

The exception is electronic music, where the parameter is directly available. A sequencer’s rate control taken continuously upward turns a pattern into a tone, and the effect has been used deliberately since Stockhausen’s Kontakte (1958–60), which contains the transformation as an explicit structural event: a pulse accelerates, loses its identity as rhythm, and arrives as pitch. Stockhausen’s theoretical writing of the period — the essay “…how time passes…” — proposes building an entire compositional system on the unified axis.

The system did not take, for the reasons in the previous section: the two domains share an arithmetic and not a perception, so a structure derived on one side does not transfer to the other. What survived is the effect, which is now a standard piece of studio vocabulary.

The natural version is the drum roll, and it is worth noticing that a snare roll is deliberately kept in the ambiguous band. Slow enough to be a texture rather than a pitch, fast enough not to be counted. Players find that band by ear and it has no name.

What the boundary is made of #

The rate at which counting fails is not arbitrary and it is not a single mechanism, which is why it is a band rather than a line.

The lower edge — around eight to ten events a second — is a limit on attention and motor tracking rather than on hearing. The events are perfectly resolved by the ear well above that rate; what fails is the ability to assign each one a position in a sequence. It is the same limit that caps how fast a person can tap, and it is why a drummer playing sixteenth notes at 180 beats per minute is executing a learned pattern rather than placing each stroke.

The upper edge — pitch emerging around 20 to 40 Hz — is a limit of a different kind. Below it, the auditory nerve’s response follows the individual events and no periodicity mechanism has enough cycles in its window to extract a rate. Above it, the periodicity is fast enough to be encoded as such, and what emerges is a pitch whose definiteness improves as the rate rises.

The gap between the two edges is what produces the ambiguous band, and its width is the reason the transition is interesting rather than merely a fact. If counting failed at exactly the rate pitch began, a listener would pass directly from one description to the other and there would be nothing to hear in between.

Whose rhythm, and when #

The polyrhythms this essay treats as ordinary are not universal, and the ratios in use vary sharply by tradition. Three-against-two is common across west African drumming, in the Latin American traditions descended from it, and in European music from the sixteenth century onward as the hemiola. Five-against-four and seven-against-four are marked as difficult in European notation and are ordinary in some south Indian tala practice.

What is universal, as far as the record goes, is the boundary itself. The rate at which a sequence of events stops being countable is a property of the auditory system rather than of a tradition, and no musical practice anywhere counts at 30 events a second.

A second near-universal sits just below it, and it is the reason tempo is not free. The rate at which people spontaneously tap, walk and set a comfortable beat clusters around two per second — 100 to 120 beats a minute — with most music falling within a factor of two or three of that. The upper end of usable tempo is bounded by the counting limit above, the lower end by memory, since a pulse slower than about one event every two seconds stops being felt as a pulse and becomes a series of separate events.

So the rhythmic band is narrower than the pitch band by a wide margin. Pitch spans about ten octaves of usable frequency, roughly a thousand to one; tempo spans perhaps four to one before a listener stops hearing a beat at all. Both live on the same axis of rate, and the part of it music can use for rhythm is a very small piece of the part it can use for pitch.

That asymmetry is the honest summary. The arithmetic of ratios is shared between the two domains and is culturally variable in which ratios get used. The boundary between the domains is not shared with anything and is not variable at all.

The experiment a reader can run #

The claim is unusual in this collection in being directly demonstrable rather than merely computable, and the demonstration takes one control.

Take a repeating pattern — the buttons under the figures above will do — and listen to it at each rate in turn. Three things are worth attending to, in order.

First, at the slow rates, that the pattern is the thing being heard: which stream is which, where they coincide, how the coincidences fall. Second, in the middle, that the question “which stream is which” stops having an answer while the sound is still obviously two things happening. Third, at the top, that what is heard is an interval — a harmonic relationship with a quality — and that the coincidences that were the whole content of the pattern are no longer available to attention at all.

The last of those is the surprising one. The coincidences did not go away; they are still there in the signal, at the same relative positions, and they are now called phase relationships and are almost entirely inaudible. The ear discards phase, which is a fact usually introduced in connection with waveform shape, and this is the same fact seen from the rhythmic side: the information a listener was using at two events a second is thrown away at two hundred.

That is the deepest thing the continuum shows, and it is a loss rather than a transformation. Crossing the boundary upward does not convert rhythmic structure into harmonic structure. It destroys the rhythmic structure and supplies harmonic structure in its place, from the same signal, by a different mechanism.

Why nobody noticed for so long #

The continuum is obvious once stated and it took until the twentieth century to be stated, which is worth a sentence because the reason is instructive.

Nothing before electronic sound generation could traverse it. A drummer cannot accelerate to three hundred events a second, a violinist cannot slow a pitch to two events a second, and there was no instrument on which the rate was a single continuously-variable parameter. The two domains were separated by the physical apparatus that produced them long before they were separated by theory, and a distinction enforced by every instrument in existence does not look like a distinction anyone made.

What changed was the oscillator. Once a rate could be set by a knob and turned smoothly through the whole range, the continuity was unavoidable — and the first people to notice it were the ones building the equipment rather than the ones writing about music.

What the picture cannot show #

A logarithmic axis of rate compresses the thing that matters most, which is that a listener crossing the boundary experiences a change of category rather than a change of degree. The figure draws a smooth progression because the stimulus is a smooth progression, and the percept is not smooth at all.

It also cannot show what happens to the ratio in the middle band. At 15 events a second there is no pitch to be consonant and no count to be complex, and whether a 3:2 in that band sounds different from a 7:5 is a question with no published answer, largely because nobody has thought to ask it.

The ladder from here goes downward in rate rather than upward: to the question of how a listener decides where the bar line is, when the events are slow enough to count and the signal does not say.