A newcomer learns a cycle's landmarks in four hearings

A slow gong cycle is a long time to count. In a sixteen-step cycle of 160 seconds, the strokes that mark its structure come every ten seconds or more, and a listener who loses their place has only their sense of elapsed time to find it again — a sense that grows vaguer the longer they count. A cycle that says where it is measured how much of the cycle’s position such a listener knows, in bits, and later essays added recognisable points: an elaborating part’s distinctive arrival, a vocal entry, a drum call, anything a listener can recognise and know the position of.

Two landmarks in a cycle help only if they can be told apart placed two such points and found that what matters is whether the listener can tell them apart. Two points told apart — at steps 6 and 13 at best — cut the bits of position unknown from 0.646 to 0.094. Two points that sound alike, so that hearing one says only that one of the two has just passed, cut it only to 0.219, and placed half a cycle apart, alike points did worse than a single point.

Every listener in that essay either could or could not tell the points apart from the moment they arrived. A newcomer is neither. Two marks that differ only by where they fall in the cycle sound alike the first time; only after a few cycles does a listener notice that one always comes a little after the halfway stroke and the other well before the gong. The question the earlier essay left was how many cycles that takes, and which points serve a newcomer best in the meantime.

One hearing’s evidence #

What a newcomer can use is the one thing they always have: the gong. Every cycle begins with it, and from it the listener can time each mark. The timing is the same timing every clock in these essays has used, with a Weber fraction that is small for a few seconds and grows to about three-eighths for intervals of tens of seconds, so a step near the gong is placed precisely and a late step loosely. At 160 seconds a cycle, step 1 is placed within a sixth of a step, step 6 within two and a quarter steps, step 13 within nearly five.

One hearing of a pair therefore gives evidence that they are two different events in proportion to how far apart they are, divided by how loosely each is placed — a d′, in the language of signal detection. Steps 6 and 13 are seven steps apart and placed within 2.25 and 4.88, which gives a d′ of 1.30 a hearing. Steps 1 and 2 are a single step apart but both placed so precisely that a d′ of 2.98 comes from one hearing. Steps 4 and 12, eight steps apart but both well into the cycle, give 1.69.

Hearing the same pair again in the next cycle gives independent evidence, and independent evidence adds in d′ squared. A listener who decides the two marks are different events once their evidence passes a criterion, with criteria varying across listeners, comes to name the pair at a rate set by that d′. The model takes the share of listeners naming the pair after c cycles as one minus the exponential of half of c times d′ squared — so a pair with a d′ of 1.30 is named by nine listeners in ten after 3.7 cycles, and one with 2.98 after 1.5.

The clock, cycle by cycle #

In the first cycle every pair is heard alike, and the clock is the alike clock. From there each pair improves towards its learned value at the rate its d′ allows. Steps 6 and 13 start at 0.219, reach 0.117 in the third cycle and 0.098 in the fifth, within a few thousandths of their learned 0.094. Steps 1 and 15, the pair whose first point sits right after the gong, start at 0.301 and are learned almost at once — 0.255 in the second cycle — but their learned value is 0.253. Steps 4 and 12 start worst of all, at 0.445: two alike marks half a cycle apart, which the earlier essay found worse than one. Their d′ of 1.69 teaches them quickly, and they reach their learned 0.191 by the fifth cycle.

So the pair that is learned fastest is not the pair that makes the best clock, and the pair that makes the best clock is not the best for the first cycle. At 160 seconds the newcomer’s first cycle is best served by steps 7 and 13, whose alike clock is 0.180; but 7 and 13 are learned more slowly, d′ 1.08, and settle at 0.102, a little above 6 and 13.

The quickest to learn are the worst clocks #

The 105 pairs make a clear picture. The pairs with the largest d′ — 21 of them above 2, a dozen with a point on the first step after the gong — are the ones a newcomer learns within two cycles, and none of them is a good clock: the fastest, steps 1 and 2, leaves 0.360 bits even when named. A point close to the gong is placed precisely because the gong has just been heard, and for the same reason it adds little: it tells the listener where they are at the moment they least need telling.

The best clocks sit in the middle of the scatter, with d′ around 1 to 1.5: points far enough into the cycle to be useful, and far enough apart to be distinguishable eventually, but placed loosely enough that one hearing is not enough. The clock’s value and the speed of learning pull in opposite directions for one reason. A point is worth having where the listener’s own count has become vague, and a point is quick to learn where it has not.

That answers the earlier essay’s question in a form it did not anticipate. It asked whether the points that serve a newcomer best are the ones that are distinct at once or the ones that sit furthest apart and reveal their difference soonest. The points that reveal their difference soonest are neither far apart nor well placed; they are near the gong. And the points that are far apart reveal their difference slowly, because distance from each other comes with distance from the gong.

What a newcomer should be given #

Averaged over the first few cycles a newcomer hears, the best pair is 7 and 13 for any span up to four cycles and 6 and 13 from six cycles on. The difference is never large — 0.180 against 0.219 over the first cycle, 0.142 against 0.147 over four — because the two pairs are close neighbours, one step apart in their first point. Moving the first point from 6 to 7 makes the alike reading better, because the two marks are then closer together and a listener who confuses them is confused about a smaller stretch of the cycle, and makes the named reading slightly worse.

So the earlier essay’s best placement is also, to within a few hundredths of a bit, the best placement for a newcomer.

The reason the two placements nearly coincide is in the alike reading. A pair of alike marks is a liability when it splits the listener’s belief between two distant parts of the cycle, as 4 and 12 do, and a mild asset when its two members are close enough that confusing them costs little. The best named pair, 6 and 13, sits seven steps apart and its alike reading is already better than either of its points alone — 0.219 against 0.256 for the better of the two — so a newcomer who cannot yet tell its marks apart is still better off than one with a single landmark. An elaboration is a clock hand where the skeleton has none found the elaboration’s own regular strokes doing the timing where the skeleton is silent; the marks here are the moments when that hand points somewhere recognisable, and they help a newcomer from the first cycle on, if less than they help a listener who knows them. That was not obvious and is worth stating: a pair of landmarks placed for the experienced listener does not leave a first-time listener stranded. It leaves them, for the first cycle, at the alike clock’s 0.219 bits — already a third of what the skeleton alone leaves — and within four cycles at the experienced listener’s level.

Four hearings, at any slow tempo #

The number of cycles depends on the timing’s Weber fraction, and above about eighty seconds a cycle the fraction no longer changes: both points are timed over intervals long enough that their spreads are a fixed fraction of the time counted. So the d′ for 6 and 13 is 1.30 at 80, 160 and 320 seconds alike, and the pair takes 3.7 cycles to learn at all three. In minutes that is 4.9, 9.9 and 19.8.

At quicker cycles the earlier steps are timed over shorter intervals where the Weber fraction is smaller, the d′ rises, and the pair is learned in fewer cycles — 3.3 at 40 and 60 seconds, 1.4 at 20 and 30. But at quick cycles the pair was hardly needed. The cycle that outruns the memory set a listener’s memory span in seconds against a cycle’s length, and at quick cycles the skeleton’s own strokes come often enough that little is left unknown: naming the pair is worth 0.018 bits at 40 seconds against 0.125 at 160.

So the time a newcomer needs to learn a cycle’s landmarks grows in proportion to the cycle’s length, at exactly the tempi where the landmarks matter. A ten-minute stretch of a slow gendhing — some slow pieces run far longer, and some are over in less — passes before most newcomers can use two alike marks as two. An expectation cannot rescue a cycle too slow to time carried a listener’s learning of the cycle’s structure forward from one hearing to the next; the learning here is of a different kind, and runs on the same clock of hearings.

What being new costs, summed #

The curves can be added up. Over the cycles it takes to learn a pair, the newcomer’s clock sits above the learned clock by a shrinking amount each cycle, and the sum of those differences is the whole cost of being new to that pair, in bits of position times cycles. For steps 6 and 13 it is 0.125 in the first cycle, 0.053 in the second, 0.023 in the third, and a little after: about a fifth of a bit-cycle in all. For steps 4 and 12, whose alike reading is so poor, it is about a third. For steps 1 and 15, learned at once, it is a twentieth — but that pair is worse than 6 and 13 even in the first cycle, 0.301 against 0.219, and costs 0.16 more in every cycle after, so its quick learning buys nothing at all.

That is a general shape and it is worth putting plainly. A landmark that is quick to learn is cheap to be new to and dear to know; one that is slow to learn is dear to be new to and cheap to know. Over a long listening the second wins, because the cost of being new is paid once and the cost of a poor clock is paid every cycle. Repetition buys least where it is needed most found the same asymmetry from the other side: in the slowest cycles, where a settled listener most needs the repetitions, repetition helps least. Here the slowest cycles are where learning the landmarks takes longest in minutes, and where the landmarks matter most.

The pattern also bears on what kind of cycle a tradition builds. The best gong cycle halves at every level found that the colotomic structure which places a slow listener best is the one in which each layer of strokes is twice as fast as the one above it — a structure whose strokes a newcomer can tell apart at once, by their instruments. A cycle whose landmarks differ only in position would ask the newcomer for four hearings before it pays; a cycle whose landmarks are distinct instruments asks for none.

What the learning model assumes #

The marks differ only in where they fall. The model gives the newcomer no other cue: the two points are acoustically alike, and the only way to learn they are two is to time them. Real landmarks usually differ in other ways too — the earlier essay’s example was a descending run before one arrival and a rising one before the other — and every such cue would add evidence and shorten the learning.

Timing is from the gong, and the gong is never missed. The strokes between the gongs are a clock found that an experienced listener does not time a cycle as one interval but resets at the last gong, kempul and kenong. A newcomer who already did that would place late points more precisely, and the d′ for pairs late in the cycle would rise. The gong alone is the conservative reference, and the right one for a listener who has not yet learned which stroke is which either.

Evidence adds across cycles without forgetting. A cycle of 160 seconds is a long time to hold one hearing’s impression of where a mark fell; a listener who forgot part of each hearing would need more cycles. The figures are the fastest learning the timing allows.

The population’s criteria are spread exponentially. The form one minus the exponential of half of c·d′² is a convenience: it gives the share of listeners naming the pair as evidence accumulates, with the steepness set by d′. A different spread of criteria would move the cycle counts, but not their ordering across pairs.

What the clock cannot say #

Whether listeners learn this way. The model is a statement about the evidence available, and what an ideal learner could do with it. Whether a real newcomer notices two marks’ positions, or simply never distinguishes them, or distinguishes them at once by cues the model omits, is a question for listening experiments.

How a repertory places its landmarks. The clock prices every placement. A tradition’s own choices — where elaborating parts converge, where vocal entries fall, where the drum calls — would say which placements newcomers actually meet, and that needs the notations of real pieces rather than an enumeration.

Whether the slow learning is a cost. A listener who cannot yet tell two marks apart is still listening. The bits of position unknown measure one thing a listener might want; they say nothing about the pleasure of not knowing exactly where in a long cycle one is.

Still open: a newcomer who learns from the marks’ sounds as well #

The marks here differ only by position, and the arithmetic says that is slow evidence: four cycles at best, and the pairs it serves best are learned slowest. A real elaboration rarely arrives at two points in the same way, and a listener can learn the difference in sound as well as in place. The computation is this learner with a second source of evidence each cycle — a d′ of sound alongside the d′ of position, the two adding — and it would say how different two marks have to sound before a newcomer names them in one cycle instead of four, and whether at that point the best pair for a newcomer becomes the best clock outright, with the slow geometry of the cycle no longer mattering.