A band that follows a curving rise

A counter that cannot be slid found that the index-free chain counter — the one that follows each family of spirals into chains without knowing the order the organs arrived in — reads a growing stem only inside a band of widths. Too narrow, and it drops the finer family and returns the rung below; too wide, more than about a third of a rung of rise, and it holds a little of two lattices and returns a pair on no rung. A band that follows the rise then built a counter that chooses its own width: it fits the decay of the rise through the logarithms of its own band’s whorl spacings, takes the band spanning a third of a rung, and refuses any reading too narrow to have shown the rung above. On seven stems it chose 49, 97 and 193 organs at rates of 150, 300 and 600 without being told, and never returned the rung below.

Every one of those stems had a rise falling by a constant factor per organ. The decay was one number for the whole stem, the logarithms of the spacings a straight line, and a straight-line fit exactly the right instrument. A real shoot’s rise falls because its apex changes size, and nothing says that change is geometric: a cone has a rise that falls found a cone’s rise falling as one over the distance from its tip, which is a power law, not an exponential. The question the counter’s essay left was whether a straight-line fit through the log spacings still gives a usable local decay on a curving profile, and whether a band sized from it still spans a third of a rung, or systematically more at one end of the stem.

Two stems that do not fall geometrically #

The stems are grown by the same free placement rule as every jugate stem here, from a rise of 0.05 down to 0.0005 — a hundredfold, as before — at one and two organs a whorl. Only the profile changes. The cone stem’s rise falls as a power of the organ index, $h = h_0 (1 + i/i_0)^{-p}$ with $p = 2.57$ and $i_0 = 385.6$, so its rate of fall, $p/(i + i_0)$, slows from one part in 150 per organ at the base to one in 900 at the tip, over 1,929 organs. The ogive’s rate does the opposite, speeding up linearly from one in 900 to one in 150 over 1,185 organs.

Both stay within the rates the counter has already been tested at — the fastest geometric stem fell at one in 150, the slowest at one in 600 — so anything the counter does differently here is the curve’s doing and not a new rate’s. And both change their rate sixfold along a single stem, which no fixed width can match.

The band, read along one cone #

Read band by band up a cone with one organ a whorl, the self-sized counter is right on 94 per cent of its 201 bands and refuses the rest; it returns neither the rung below nor a pair on no rung anywhere. The refusals are spread along the stem, in each third six per cent, all of the same kind: the search for a width that spans a third of a rung did not settle in the six revisions it is allowed.

A fixed width of 49 organs — the width the counter chose for itself on the fastest geometric stem — reads the first two thirds of the cone perfectly and then fails on 96 per cent of the last third, every failure the rung below. A fixed width of 193, the slowest stem’s choice, fails on 93 per cent of the first third with a pair on no rung and reads the last third perfectly. On one stem, the widths that suited the fastest and the slowest geometric stems are each right at one end and wrong at the other, and the end each is wrong at is the one where the rate has drifted furthest from its own.

Two organs a whorl, and a width that fails sooner #

The cone grown two organs a whorl tells the same story with the failure arriving earlier. A fixed width of 49 organs is right on every band of its first third, on 45 per cent of its middle third — the other 55 the rung below — and on none of its last. A fixed width of 193 is a pair on no rung on every band of the first third, right on 85 per cent of the middle and on all of the last. The self-sized counter is right on 94, 94 and 88 per cent of the three thirds, and of the twelve per cent it misses at the tip, nine are refusals and three are pairs on no rung.

So the band that suits a stem changes not only along it but with how the stem builds its whorls, and a person choosing a width would have to know both the rate and the jugacy at every height to choose it right. The self-sized counter needs neither: it reads its width off the positions.

A straight line through a curve #

The first question has a clean answer. On all four stems, at every band, the decay the straight line gives is within half a per cent of the true decay at the band’s middle organ: within 0.23 per cent on the cone with one organ a whorl, 0.37 with two, 0.20 and 0.40 on the ogives. The cone’s fit runs very slightly high and the ogive’s very slightly low, and both drift towards one as the stems climb, but the bias never reaches half a per cent.

That is because a band is short against the curve. A band spans a third of a rung of rise, a fall of about 38 per cent, and over that the cone’s rate of fall changes by a few per cent at most. A straight line through the log spacings averages the rate over the band, and the average and the middle value differ by the curvature over the band, which is small. The method assumed a geometric rise and does not need one; it needs only that the rate change little across one third of a rung.

A third of a rung, at both ends #

The second question has the same answer. The share of a rung of rise each chosen band actually spans — measured from the stem’s true rise at the band’s two ends, not from the counter’s estimate — runs from 0.324 to 0.333 across every band of every stem. There is no end at which the bands systematically span more: the counter’s width tracks the rate wherever the rate goes.

So the width moves along each stem. Up a cone it widens from 84 organs at the first band to 272 at the last, more than threefold; up an ogive it narrows from 139 to 58. A stem with two organs a whorl chooses the same width in organs as one with one, as the geometric stems did — 86 to 270 on the cone, 138 to 58 on the ogive. At every band the width is, to the organ, a third of a rung of rise divided by the rate there.

Better than a geometric stem at the same rates #

The cone stems are read better than the geometric stems whose rates they pass through. A geometric stem falling at one in 300 is read right on 82 per cent of its bands with one organ a whorl and 95 with two; one falling at one in 600, on 93 and 92. The cone’s read section runs from one in 261 to one in 847, and it is read right on 94 and 92 per cent. Nothing about a curving rise makes the counter worse; the part of a cone that can be read at all is the slow part, and the counter reads slow stems well.

The ogive is the opposite case. Its last third runs faster than any stem but the fastest geometric one, and it is read about as well as that one was: the self-sized counter refused 64 and 89 per cent of bands on the geometric stems falling at one in 150, with one and two organs a whorl, and refuses 92 and 67 per cent of the ogive’s last third. A curving stem is read, band by band, the way a geometric stem at the local rate is read.

Every reading, by stem #

Set side by side, the four stems separate into two pairs. On the cones the self-sized counter is right on 94 and 92 per cent of bands and refuses 6 and 7; on the ogives right on 68 and 78 per cent and refuses 31 and 22. It returns the rung below on none of the four, and a pair on no rung on at most three per cent of one. The fixed widths are right on 48 to 99 per cent of a cone’s bands and 0 to 93 per cent of an ogive’s, and when they are wrong they return the rung below or a pair on no rung, never a refusal.

One fixed width does better than the counter on the cones: 97 organs is right on 99 per cent of both. That is not a virtue of 97 in general. The part of the cone that is read — the first 320 organs are below the first band, where the rate is fastest — falls at rates from one in 261 to one in 847, and 97 organs happens to be a width whose two edges, the coverage floor and a third of a rung, both hold across that whole span. On the ogives the same width is right on 62 per cent.

Two kinds of refusal #

The counter refuses for two different reasons, and the two profiles divide them cleanly. Every refusal on the cones is of one kind: the search for a width spanning a third of a rung did not settle within the six revisions it is allowed, twelve bands on the stem with one organ a whorl and seven on the stem with two. A revision re-measures the decay on the newly sized band, and on a curving stem a wider band sees a slightly different decay, so the search can step between two neighbouring widths. Every refusal on the ogives is the other kind, the counter’s own check: the band the rise allows is narrower than the coverage floor of the rung above, so the pair it counted cannot be told from the pair above it. At the ogive’s tip the band is 58 to 61 organs, at a rise near 0.0028, and the floor the pair above needs is wider than that.

The first kind is a cost of the search and could be bought back with more revisions; the second is the instrument working as designed.

Where it fails, it refuses #

The ogives’ refusals are all at one end. In the first two thirds of each ogive the self-sized counter is right on 94 to 100 per cent of bands. In the last third, where the rate of fall reaches one part in 193 to one in 153, it refuses 92 per cent of bands on the stem with one organ a whorl and 67 on the stem with two. That is the geometric stems’ limit, reached by a different road: on the fastest geometric stems, at one in 150, it refused 64 and 89 per cent. Too fast a fall leaves a third of a rung narrower than the coverage floor the counted pair needs, the two bounds cross, and the counter has nothing it can return.

The fixed width of 49 organs reads that fast tip better — 78 and 44 per cent right — and returns the rung below on the rest, 22 and 56 per cent. A count that is the rung below is a perfectly good pair with nothing on it to say it is wrong, which is the silent failure counting without an index built the index-free counter to expose. The self-sized counter gives up those correct readings to avoid every one of the wrong ones.

What the curve does not change #

The counter’s three numbers — a third of a rung, the coverage divisor of 0.45, and the check that refuses a band too narrow to have shown the rung above — were all measured on geometric stems. None of them needs revising for a curving rise. The straight-line fit inside a band was the one piece of the method that assumed a geometric profile, and it turns out to need only that the rate change little over a third of a rung, which on these stems it does by a factor of about forty. The window nobody varied and how many organs a pair needs set the two edges of a good band; on a curving stem they move along the stem, and a counter that moves with them is the instrument. A plateau the instrument should have had found the same two bounds on the settling table; here they are bounds that travel.

What a fixed band costs on a real shoot #

A person reading a real shoot with a fixed band — a window of so many organs, chosen once — is reading a stem whose rate almost certainly changes along it, because its apex is growing or shrinking. On these stems the narrow and wide fixed widths each failed at one end of the cone, and the middle width, which read the cones almost perfectly, failed on more than a third of each ogive — by returning the rung below or a pair on no rung. The failure is not visible in the readings: a rung below looks like a reading, and a pair on no rung looks like a strange plant. The only reading on these stems that never misled was the one that sized itself and refused when it could not. A counter that sees no positions found the same trade in a counter handed only angles: it refuses more often, and where it refuses it would have been wrong.

What these stems assume #

Two profiles, both falling a hundredfold, both inside the rates the counter was built at; a cone rise with a stated power and an ogive with a stated linear acceleration; one and two organs a whorl; the free placement rule. The first 320 organs of every stem are not read, which removes the cone’s fastest section — the counter’s widest allowed band needs that many organs below it. Nothing here measures a real shoot’s profile.

What would withdraw it #

A band whose fitted decay is off the local decay by a per cent. A band spanning more than 0.34 of a rung of rise at either end of a stem. The self-sized counter returning the rung below anywhere on a curving stem. Each is checked at every band above.

Still open: a rise that turns round #

Both profiles here fall all the way, faster or slower. A shoot that is starting to flower can do something neither does: its apex swells, the rise falls, and then as the apex is used up the rise climbs again — a profile with a minimum. Near that minimum the rate of fall passes through nought, a third of a rung becomes enormous in organs, and the counter’s widest allowed band is the limit rather than the rise. The measurement is a stem whose rise falls and then recovers, read by the same counter, asking whether it reads the pair correctly through the turn, where the stem climbs back down the ladder it came up, and whether its widest band, rather than the rise, sets what it can see there.

Measured, a counter built on a falling rise: through a turn the band of widest width costs nothing, and the rise’s minimum is read correctly. What fails is the climb: a negative decay was treated as no decay, the counter stays at its widest band, and it is right on one band in eight until the band is sized from the decay’s magnitude, when it is right on 94 per cent. The stem itself goes back down the transitions it came up, at the same rises.