A cycle cannot cadence

A rhythm drawn as a circle has no beginning, which is the site’s oldest claim about cyclic music and is where this ladder started. The obvious next question is whether it has an ending, and the answer turns out to be no in a stronger sense than expected.

Not “the ending is subtle”. Not “the ending is marked differently”. The measurements this site has built for closure return zero on cyclic music, at every setting, for as long as it runs.

Why the vector is empty #

Closure is five components: the root moves by a fifth, the goal is the tonic, the leading note resolves, the arrival lands on a strong position, and the goal is held longer than what came before.

Take a form built on a repeating cycle and go through them.

The first three need a chord change, and there is not one. A vamp on a single chord, a drone, a bell pattern with no harmony at all — in each case the harmonic content is constant, so a root cannot move by a fifth, a goal cannot be the tonic in any sense that distinguishes it from every other bar, and there is no leading note to resolve.

The fourth needs a position to be strong relative to others. There is one, at the cycle level — a listener locks to the cycle and hears its first step as a downbeat — but every cycle’s first step is equally strong, so the component cannot single one out.

The fifth needs a duration that stands out. Every cycle is the same length as every other by construction.

So the vector is all zeros, and it is all zeros not because the music is bad at ending but because the components are all comparisons with what came before, and in cyclic music what came before is the same thing.

The music is not shapeless #

That result is a fact about the measurements, and it would be a poor essay that stopped there. A drum ensemble, a gamelan piece, a techno track and a jazz vamp all have shape, and listeners agree about where it is.

What varies is how many things are playing. Layers enter, layers drop out, and the density of onsets rises and falls. That is a quantity, it is computable from the onset patterns and the entry and exit points, and it is what the hero figure draws.

Five layers over thirty-two cycles, each with its own pattern and its own span. Sum the onsets of the layers active at each point, divide by the cycle length, and the result is a curve — a rise, a plateau, a fall, and a ratio of about three between the thinnest and the thickest passage.

Nothing about that curve is harmonic and nothing about it is a cadence. It is a description of texture over time, and in this music it is the form.

Why density and not something else #

Onset density is not the only thing that could vary, and it is worth saying why it is the right quantity rather than an arbitrary one.

It is what a listener can count. Adding a layer is a discrete, unmissable event; removing one is the same. Unlike a change of harmony it needs no learned system to interpret, and unlike a change of timbre it is not a matter of degree.

It is also the one variable that a cyclic ensemble has in abundance. A West African bell-based ensemble is several players each with a fixed pattern; the composition is in who plays and when. A gamelan’s instruments elaborate a core melody at different densities, and the structure is which of them are active. A techno arrangement is built in a piece of software whose primary gesture is muting and unmuting a track.

So the constraint and the resource line up. Take away harmonic motion and what is left is who is playing, and the traditions that took away harmonic motion built their entire formal apparatus out of it.

The number that says how cyclic a passage is #

Before leaving the closure measurements behind entirely, there is one of the site’s quantities that does register on this music, and it registers in a useful direction.

Redundancy measured in new dictionary phrases per bar puts a repeating four-bar ostinato at 0.25 against 0.42 for a twelve-bar blues and 0.59 for a thirty-two-bar song form. That is a factor of two and a half, it is stable under every encoding tried, and it separates cyclic material from sectional material cleanly.

So the collection is not entirely blind to this music. What it can measure is how much repetition there is, which is a great deal, and what it cannot measure is where anything begins or ends, which is nothing. That division is exactly what would be expected from a set of tools built to find contrast.

A cycle does mark its own boundary #

There is a distinction to make here or the claim overreaches, because cyclic music is full of markers.

A gamelan’s cycle is punctuated: the largest gong sounds at the end of the gongan, smaller gongs at its subdivisions, and the resulting nested punctuation is one of the most explicit metrical structures in any music. A tala has a sam which is unmistakable. A bell pattern’s first stroke is heard as a first stroke once a listener has entrained to it.

None of that is a cadence. It is metre — periodic marking of a repeating unit, exactly what a bar line does and exactly what the metric weight profile computes at whatever level it is applied. A gong at the end of every cycle marks every cycle equally, which is the definition of a metrical accent and the opposite of a cadence.

The test is whether the marker distinguishes one occurrence from another. A cadence does; a gong does not. And when such music does end, it ends by something outside the cycle altogether — a slowing of the tempo, an agreed signal, a drop to a single instrument. That is closure by a change in a global parameter, which is the only kind available when every local parameter is periodic.

Where the site’s own bell pattern comes into it #

The rhythm field’s Euclidean construction and this essay’s density curve meet at a specific place, and it is worth making the connection explicit because it changes what a layer is.

A Euclidean rhythm spreads a number of onsets as evenly as a number of steps allows, and the patterns it produces are the ones in use across four continents. Those patterns are what the layers of a cyclic ensemble are made of, and the reason an ensemble can add layers without the result turning to mud is that maximally even patterns interlock rather than collide.

The obvious next sentence is that two patterns each spread as evenly as possible over one cycle must therefore share as few onsets as their counts allow, so adding the second contributes nearly its full count to the density rather than doubling up on strokes already there.

That is false, and it is false in the opposite direction. Take every pair of Euclidean patterns on a twelve-step cycle with between two and seven onsets, lay each pair over the other starting at step zero, and count the shared onsets.

Eleven of the fifteen pairs overlap more than random subsets would, and the same holds on a sixteen-step cycle, where twenty-four of twenty-eight do. Maximal evenness is precisely the property that makes two patterns fall on the same steps: both are trying to be as close to a uniform spacing as the cycle permits, so both put strokes in the same places.

The extreme case makes it plain. E(5,12) is 0, 3, 5, 8, 10 and E(7,12) is 0, 2, 3, 5, 7, 8, 10 — the site’s own bell pattern. The five is a subset of the seven. Laid over it at step zero, a whole layer of five strokes contributes not one new onset to the density; the curve does not rise at all.

Shift that same five-stroke layer by one step and the overlap is zero — every one of its strokes lands in a gap. The full sweep by rotation runs 5, 0, 5, 2, 3, 4, 1, 5, 1, 4, 3, 2, so one pair of patterns spans complete redundancy and complete complementarity depending on a decision that costs a drummer nothing.

So the interlocking is real and it is not the construction’s doing. It is the rotation’s, which is the ensemble’s own variable — which rotation a part enters on is what distinguishes one drummer’s part from another’s, and this is what that decision is buying. The density curve rises smoothly as layers enter because the layers are entered at rotations chosen so that they do, and an ensemble that aligned every pattern at step zero would be playing a thinner texture with the same number of players.

One more consequence of that follows and it is a practical one. Because the rise depends on where each layer is entered rather than on how many are playing, an ensemble can get louder without getting denser and denser without getting louder — the two are separate controls, and a player joining at a rotation that duplicates what is already there adds level and no onsets at all. That is a real technique rather than an artefact of the model: doubling a part at the unison is how a West African ensemble marks a section without changing its texture, and shifting the double by one step is how it changes the texture without adding a player.

Entrainment is the other half of it #

The listener’s side of this deserves a paragraph, because it explains why the absence of an ending is comfortable rather than frustrating.

A repeating cycle is the ideal stimulus for entrainment: a listener locks to it, predicts it, and after a few repetitions is running an internal clock that continues whether or not the sound does. A beat is inferred and once inferred it is cheap to maintain, which is why cyclic music can afford to give a listener no new information for minutes at a time.

Two consequences follow. Attention is freed for the things that do vary — the density, the small timing deviations, the details of an individual player’s part — which is why listeners to this music attend so closely to things a sectional listener would call ornament. And departures become enormous: against a fully established cycle, one missing stroke is an event, where in music with a changing surface it would pass unnoticed.

That is the same trade the tonal repertoire makes with harmony and in the opposite direction. Establish something thoroughly and small departures carry weight; keep changing and only large ones do.

The absent ending as a design #

It is worth resisting the reading in which cyclic music is tonal music with the endings removed.

A form that does not end is a form that can be stopped rather than concluded, and a great deal follows from that. It can be as long as the occasion requires, which is what music for dancing, for ceremony and for work all need. It can be entered and left by individual players without the whole being disturbed. And it puts the listener in a different relation to the music: not following an argument towards a resolution, but inside something that is going on.

The tonal repertoire’s obsession with endings is the unusual case, historically and geographically. It is a design in which a piece is a single object with a beginning, a middle and a conclusion, and it required an enormous apparatus — a hierarchy of cadences, a key plan, a whole theory of function — to hold together. The cyclic design needs none of it and buys something the other cannot have.

What can be borrowed back #

The density curve is not confined to cyclic music, and pointing that out is the useful direction rather than the other one.

Any music has an onset density and any music varies it. An orchestral crescendo with instruments entering is exactly the hero figure’s shape; so is the build in a dance track and the tutti after a solo passage in a concerto. What differs is whether that curve is the primary structure or a secondary one laid over a harmonic plan.

That makes it a variable worth measuring in both, and it is one of the few in this whole field that requires nothing but a list of onsets — no key, no chords, no metre, no notation. A recording gives it up directly, which is more than can be said for anything else in this phase.

What this cannot show #

The density curve counts onsets and weights them all equally, which is plainly wrong. A bass drum and a shaker do not contribute the same amount to a listener’s sense of how full a texture is, and any real measure would weight by loudness, by register or by both.

The overlap result above is a second and larger version of the same complaint. A density curve built by adding onset counts is an upper bound, and how far below it the real curve sits depends entirely on rotations the figure does not record: at the extreme found above, adding a five-stroke layer to a seven-stroke one raises the count by five and the sounding density by nothing. Every density curve in this essay assumes the rotations were chosen well, which is what an ensemble does and not what the arithmetic guarantees.

It also has nothing to say about which layers are playing, only how many onsets they produce between them. Two arrangements with identical densities and completely different instrumentation are the same curve, and the difference between them is most of what an arranger works on.

And the entry and exit points in these figures are stipulated rather than measured. They come from a plan written into the figure, not from an analysis of a recording, which means the curve demonstrates what the quantity does and does not measure any actual piece. Getting them from audio is a solved problem and not one this site does.

There is a subtler omission too. The curve treats a layer as present or absent, and a great deal of real cyclic music varies a layer’s intensity continuously — a drummer playing the same pattern harder, a shaker moving from the edge of the beat to the centre. Those are changes a listener registers as an increase in energy and the onset count does not move at all. What would capture them is a measure with loudness in it, and loudness is not amplitude, so that is a larger job than it looks.

A stopping is not an ending #

One more distinction is worth making because it is what a listener actually experiences at the end of such a piece.

A tonal piece finishes and a listener knows several bars in advance that it is about to. The cadence is prepared, the harmonic rhythm accelerates, the texture thins, and the arrival is the completion of a process that was visibly under way.

A cyclic piece stops. There may be a signal — a break, a call, a ritardando, a last stroke on the largest gong — but the signal is an instruction rather than a consequence, and it could have come one cycle earlier or ten cycles later without anything in the music being different.

That is a real difference in kind and not a matter of degree, and it puts a boundary on how far the closure vector can be generalised. It measures whether the music has arrived; it has nothing to say about whether the music has stopped, and for a great deal of the world’s music the second is the only question there is.

Where the ladder goes #

There is a kind of cyclic music in which the form is not even a matter of arrangement — where a single rule is applied to a single pattern until it exhausts itself, and the whole shape of the piece is the enumeration of the rule’s outcomes. The number of stages is then arithmetic, the composer chooses the rule rather than the shape, and the music at each stage is computed rather than composed.