A cycle that says where it is

What the clave buys with its unevenness asked what a timeline is for, given that the Euclidean patterns it produces are not always the ones the world plays. Its answer was that a timeline tells a listener where in the cycle they are, and that the cost of doing so is evenness — the son clave sits on the frontier between the two.

The mechanism there is rotational asymmetry. A pattern locates a listener because its rotations are distinguishable, and the rotation the necklace cannot see is the essay about what happens when they are not: a pattern with a rotational symmetry cannot locate anybody, ever, however long they listen.

There is another way to do it, and it has nothing asymmetric in it at all.

Putting the code in the instrumentation #

A colotomic structure is the organising principle of Javanese gamelan and of a good deal of other cyclic music, and it is worth describing before it is measured: a cycle of a fixed number of beats, marked by punctuating instruments at nested periods. The gong ageng sounds once a cycle; the kempul divides it; the kenong divides that; and so on down. The cycle length is a property of the piece, the periods are fixed by convention, and the instruments are named for the position they mark rather than for what they sound like.

Nothing about those layers is a pattern in the sense the Euclidean ladder means. Each is a plain pulse. What carries the position is which of them coincide, and coincidence at nested periods is a binary counter: the step where all three sound is step one, the step where two sound is the halfway point, and so on.

That is a completely different design from a clave. A clave is one part whose shape is the information; a colotomy is several parts whose combination is, and each part is as dull as it is possible for a part to be.

What it buys and what it costs #

The two designs can be measured the same way, which is what makes this a rung rather than an observation. Take the same number of strokes — seven, in a cycle of sixteen — and ask two questions of each.

How much does one step tell a listener? The layered code leaves 2.81 bits unknown after a single step; the best single line of seven onsets, chosen from all 715 rotation classes, leaves 3.01. The layered code says more immediately.

How long until they are certain? The layered code needs eight consecutive steps; the single line needs five. The single line resolves sooner.

So it is a genuine trade rather than a win, and the shape of it is about where the information sits. The layered code puts it in a single instant — hearing one stroke of the gong is unambiguous, and needs no memory at all. The single line puts it in a sequence, so a listener has to hold several steps at once before any of it pays — and holding several steps at once is a memory this collection has a limit for.

Why the layered code cannot finish #

The reason the layered code is slower to certainty is worth stating, because it is a structural limitation rather than a matter of degree.

Nested layers at 16, 8 and 4 sound only on steps divisible by four. Every other step — one, two, three, five, six, seven — has the same empty signature, and no amount of listening at those steps distinguishes them. The code locates the listener among the structural positions and says nothing whatever about the ones between.

That is not a defect. It is precisely what a colotomy is for: a listener needs to know where the cycle’s articulation points are — which is the beat’s own hierarchy arriving as a set of instruments rather than as an inference, and a player of the elaborating instruments needs to know which stroke is coming, and neither needs a unique label for step six. That is the same division a cycle cannot cadence draws between what a cyclic form marks and what it leaves unmarked. The design answers the question it is asked and declines the rest.

A single-line timeline cannot make that distinction. Its information is spread over its onsets and its silences equally, so it locates every step or none.

The nesting is not what makes it work #

Nested periods are one choice among many, and sweeping the alternatives says the nesting is not the reason the code performs as it does. Holding the cycle at sixteen and varying which periods the layers run at:

Coprime periods beat nested ones on both measures at once. Sixteen against eleven against seven uses one fewer stroke than the traditional set, tells a listener exactly as much from a single step, and reaches certainty in six rather than eight. Sixteen, seven and five reaches certainty in five — matching the best single line, which is the comparison this rung opened with — and is more informative after one step as well.

So a colotomy at 16, 8 and 4 is not the arrangement of three layers that locates a listener best. It is a good deal worse than several arrangements that were available, and the ones that beat it are exactly the ones whose periods do not divide one another.

That is the sharpest form of the section above. The nesting is not paying for locating power, because coprime periods buy more of it for the same or fewer strokes. What the nesting buys is the thing coprime periods destroy: every layer’s strokes fall on a subset of the layer below’s, so the instruments form a hierarchy in which the gong’s stroke is also a kempul stroke and also a kenong stroke, and a player of an elaborating part knows that a kenong stroke is a subdivision of something rather than an independent event.

A coprime colotomy would locate its listener faster and would present three unrelated pulses crossing one another, with no articulation points and nothing for the elaborating instruments to subdivide. The traditions chose the worse locator and the better skeleton, which is a design decision this rung can now price rather than describe: two steps of certainty and one stroke, spent on making the layers nest.

The two designs are answers to different questions #

Put that way the two traditions stop looking like alternative solutions to one problem.

A clave-based music has one part carrying the cycle and everybody else playing against it. That part has to be a shape, because a shape is the only thing one part can be, and the shape has to be asymmetric or it carries nothing. The frontier the son clave sits on is what a single part costs.

A colotomic music has several parts carrying the cycle between them and the elaborating instruments playing against all of them. No single part has to be a shape, so none of them is, and the asymmetry lives entirely in how periods that are each perfectly regular line up with one another.

The second design needs more players and buys a code available at any instant. The first needs one player and buys a code that resolves completely. Which is available is a fact about the ensemble before it is a fact about the music, and that is the sort of constraint this collection keeps finding underneath a stylistic difference.

A deeper stack, and where it stops paying #

Adding a layer is cheap in strokes and dear in players, so the obvious question is what the fourth one buys.

Layers at 16, 8, 4 and 2 give five distinct signatures rather than four and cut the bits after one step from 2.81 to 2.13, at a cost of eight more strokes a cycle and one more player. Adding a layer at every step buys nothing at all: it sounds everywhere, so it is constant, and a constant carries no information.

That is the general shape of it. Each layer halves the ambiguity among the positions it can distinguish and does nothing for the rest, so the returns fall off exactly as the periods approach one — and the last useful layer is the one at period two, after which the next would sound on every step.

A cycle of sixteen therefore supports four useful punctuating layers, which is very close to the number the traditions use. That is not evidence of anything, and it is a pleasant arithmetic coincidence to have found: the design has a natural stopping point and the traditions stop at it.

Run the same comparison at four layers and something happens to the other design rather than to this one.

The trade the previous section drew has therefore inverted, and not because the layered code improved. Fifteen strokes in sixteen steps leaves one rotation class: a line that sounds almost everywhere is almost constant, and a constant says nothing about where in the cycle it is. The single-line design has a density at which it stops working, and the layered design does not — which is the sharper form of the same point, that one design puts its information in the instrumentation and the other in the sequence, and only the second can be saturated.

The cycle as a circle, which is where both designs live #

Both designs are about a cycle rather than a line, and the reason the question is interesting at all is that a cycle has no beginning. Rhythm is a circle is this ladder’s first rung and it is about exactly that: a pattern drawn on a line has a first event and a pattern drawn on a circle does not, so where am I is a question a listener genuinely has to answer.

Drawing the layered code on the circle makes its structure visible in a way the strip does not. The layers are concentric rings, each with its own number of equally spaced points, and the position code is which rings have a point at the same angle — which is a picture of a set of nested regular polygons sharing a vertex.

Which computation produced the numbers #

The layered code is a signature per step: which of the layers sound there, as a string of ones and zeros. Two quantities are read off it.

Bits after one step is the same measure positionCode reports for a single line: given that a listener has heard one step, the average log of how many entry points remain consistent with it. A signature that occurs once leaves zero bits; one that occurs four times leaves two.

The locating length is the smallest number of consecutive steps whose sequence of signatures is unique for every entry point. For a single line that is positionCode’s own locate; for the layered code it is the same definition on the signature sequence.

The comparison is at equal onset count, and the single line is the best of its class rather than a typical one — positionCensus enumerates every rotation class of seven onsets in sixteen steps, all 715 of them, and ranks them by the same measure. Giving the single line its best case is what makes the comparison a comparison.

What a listener would have to be doing #

The measure used here — bits still unknown after one step — assumes a listener who knows the design: how many layers there are, what their periods are, and that they are nested. A listener who does not know that is not decoding anything.

That is a strong assumption and it is the same one the Euclidean ladder makes about a timeline. A listener who has never heard a son clave cannot be located by one either; what locates them is a pattern they recognise, and recognition is learning rather than decoding.

The difference between the two designs under that assumption is worth stating, because it is the practical form of the trade. Learning a timeline means learning one arbitrary pattern of a stated length. Learning a colotomy means learning a rule — the periods, and that they nest — which generalises to every cycle length the tradition uses and to instruments the listener has never heard.

So the layered design is not only faster to a first reading; it is cheaper to acquire, because it is a system rather than an item. That is a claim about learning and this collection cannot test it, but it is the direction the arithmetic points and it fits what the two traditions actually teach: a clave is taught by rote and a colotomic structure is taught as a set of rules with names.

Where the model stops #

Every stroke is the same stroke. Real colotomic instruments differ enormously in timbre, register and decay — a gong ageng and a kenong are not remotely alike — and a listener distinguishing them is doing something much easier than distinguishing a coincidence of anonymous clicks. The model treats the layers as a set of labelled channels, which is generous to the layered code in one way and stingy in another: it gives the listener perfect discrimination and gives them no help from anything else.

The layers are exactly nested. Real colotomic structures are not always powers of two, and several traditions use periods that are not divisors of one another at all. A non-nesting set would produce more distinct signatures and a different trade.

Nothing here is about tempo. A cycle of sixteen at a slow gamelan tempo lasts the better part of a minute, which is far longer than the perceptual present, so a listener holding eight consecutive steps is holding something a good deal longer than they can hold — and the single-line code’s five steps may be past that boundary too.

And a listener does not enter at random. Both measures average over every entry point, which is the right thing for a listener arriving at an unknown moment and the wrong thing for one who has been present since the beginning — and most listeners have.

What the picture cannot show #

It cannot show the elaboration. A gamelan’s punctuating layers are a skeleton and the music is what the elaborating instruments do over it, which is where nearly all the notes are. The code is what the skeleton carries and it is not what the piece is.

Nor can it show entrainment. A listener who has heard three cycles is not decoding at all; they have a metre, and the beat is inferred is the ladder about how. Everything measured here is about a cold start, and a cold start is a small fraction of any listening.

And it cannot say the arithmetic is why. Nested punctuation is a design with a great many things going for it — it distributes the labour, it gives an ensemble a hierarchy, it makes a cycle audible as a shape — and its coding properties are one of them and are not evidence that anybody chose it for them.

Whose music, and when #

The nested-period structure is Javanese and Balinese first of all, where it is explicit, named and taught as such, and the cycle lengths are powers of two by convention. It appears with different names and different periods across a wide band of South-East and East Asian practice, and something structurally similar happens in West African bell-and-support ensembles, where the timeline is a single asymmetric line and the supporting parts are isochronous at nested rates.

That last case is the interesting one for this rung, because it uses both designs at once: an asymmetric timeline for the fast, complete code and nested regular parts for the slow, instantly available one. If the two designs really do trade in the way the figures say, an ensemble using both is buying immediacy and certainty separately, with different instruments — which is a prediction about what the parts are for rather than a description of them.

Where this ladder goes next #

Three rungs. Rhythm is a circle; a cycle cannot cadence; and now a cycle can say where it is without any of its parts being able to.

What the ladder owes after this is the elaboration. Everything above is about the punctuating skeleton, and the parts that play over it are the music — they are dense, they are derived from a fixed melodic outline by rules that differ by instrument, and the rules are stated in the tradition’s own terms as functions of the cycle position. That makes them a second position code, computed from the first, and asking how much of a listener’s location comes from the skeleton and how much from the elaboration is a question this collection has the machinery for and no corpus to run it on.