Weber fraction — where it appears
Named by 19 essays across 5 fields — each of them below, with the objects they name alongside it.
How small a difference is audible
Every essay here about tuning has assumed a listener who can hear the difference between two systems. The assumption has a number: about five cents in the middle of the range. It clears the two commas fourfold and it does not clear the schisma at all, which sorts the whole subject of tuning into the part that is about music and the part that is about arithmetic.
The beat a tuner can actually use
There is a modulation index for every interval on every timbre, and the published threshold for detecting a fluctuation is a function of exactly that and its rate. Running one against the other turns a table of decibels into a window in cents — and reverses the ordering, because the interval with the shallowest dip has the widest window.
A proportion is only as fine as its two durations
Analyses of form measure proportions in bars and report them to three figures — a climax at 0.618, a section in the ratio 3 : 2. A listener has each part only as an estimate of how long it lasted, and a ratio of two estimates is blurred by both. Timed as well as anyone times a single second, eleven proportions fit between 1 : 1 and 3 : 1; timed from memory over minutes, two do. The golden section is told from 3 : 2 only below a Weber fraction of 5.4 per cent.
A count is not an estimate
Both established routes to a proportion are estimates — a duration timed, blurred by a Weber fraction, and a duration stored, biased by what was new. A listener who has induced a hypermetre has a third, and it is exact until it fails. It fails two ways that pull opposite: a slip miscounts one unit and its relative cost falls as the section lengthens, while a lapse loses the count entirely and its chance compounds. The mixture has a floor at about four units, where counting is 3.7 times finer than timing, and it is worth almost nothing past a hundred.
A form is sharp at the bottom and vague at the top
A movement is divided into sections, each into phrases, each into bars, and every level is a ratio of two estimates. Timed, the hierarchy is almost uniformly blunt — 2.1 distinguishable proportions at the top and 5.2 at the bottom, because a Weber fraction is a step function of duration and six of a piece's seven levels fall in one step of it. Counted, the same hierarchy runs from 2.2 to 12.2 and sharpens monotonically downward. At no level of either does timing separate 3 : 2 from the golden section.
A golden section is a coin toss with six coins
An analysis that reports a climax at 0.618 of a piece has not tested one prediction; it has looked at a piece with several defensible boundaries and reported whichever landed nearest. The rate at which that happens under no hypothesis is one line of arithmetic, and the tolerance it needs is not a number chosen on the page — it is the blur a listener's own timing puts on the judgement. Over a stretch of minutes that blur covers everything from 0.492 to 0.730 of the piece, which contains the halfway point, and six candidate boundaries produce a hit eighty per cent of the time.
Checkpoints sharpen the middle of a form, not its top
A listener who counts bars loses the count somewhere in a long section and is thrown back on timing the whole of it. A listener who also counts phrases can mend the lapse at the last phrase. If the two counts fail independently, the level a minute long goes from four distinguishable proportions to nineteen; the whole eight-minute piece goes only from two to four, because thirty phrases are long enough to lose a count as well. And if a fifth of lapses take both counts at once, three quarters of the gain is gone.
An expectation cannot rescue a cycle too slow to time
A listener who knows a piece arrives with an expectation of where in the cycle they are, and the size of that expectation was the number the last essay said nobody had. It can be given one: a listener who has been timing the cycle carries a spread of their Weber fraction times the cycle, which is the same number of steps at any tempo. Timed to ten per cent, a forty-second cycle would be placed better than a newcomer places a two-second one. But forty seconds is judged in the band where the Weber fraction is nearer forty per cent, and there the expectation is worth a tenth of a bit.
Only a lapse of attention keeps a form's top vague
A listener who keeps a count at every level of a form — bars, phrases, periods, sections — each mending the one below when it lapses, does not have a vague top at all. The whole eight-minute piece becomes the sharpest level, at seventy-six distinguishable proportions against two. Failures shared between neighbouring counts barely dent it, because a shared failure has to climb every level to reach the top. What puts the vagueness back is a lapse that takes every count at once, and one lapse in a hundred of that kind is enough.
The strokes between the gongs are a clock
An experienced listener arriving in a forty-second gong cycle was expected to be no better placed than a newcomer, because a whole cycle is timed in the band where a Weber fraction is near forty per cent. But nobody times a gong cycle as one interval. A listener who resets at the last gong, the last kempul and the last kenong times three short intervals instead of one long one, and arrives knowing their position to a tenth of a bit — better than a listener timing the whole cycle to ten per cent. A single timeline, whose strokes are all one instrument, offers nothing to reset on.
The best gong cycle halves at every level
A colotomic cycle is a clock whose layers are its hands, and a clock can be built many ways from the same number of strokes. Searched over every nesting of a sixteen-step cycle, the one that places a slow listener best is, at every budget of strokes, the one in which each layer is exactly twice as fast as the layer above it. A gong with one other layer does best at a ratio of four, and three halving layers with the same four strokes beat it by a factor of three and a half. A layer finer than the one whose interval is already timed to half a step adds almost nothing.
An elaboration is a clock hand where the skeleton has none
A colotomic cycle is a clock whose hands are its punctuating layers, and the finest hand a slow cycle can use runs out as the cycle slows. The elaborating parts that fill the cycle were predicted to supply that finer hand. They can, and a two-step figure recognised at every boundary is a kethuk exactly. But the figure that does most is not a fine one. It is a long one with a single recognisable point in the middle of the cycle, where no skeleton stroke falls: at eighty seconds a cycle it leaves a listener a fifth as uncertain as the skeleton alone, where a kethuk takes off a tenth — and it still does more than a kethuk when a listener catches that point one time in ten.
Two landmarks in a cycle help only if they can be told apart
One recognisable point inside a slow gong cycle, placed where the skeleton has no stroke, did more for a listener than a whole layer of kethuk strokes. A second point does as much again, a third nearly as much again — provided the listener can tell which point they have just heard. If the points sound alike, a second one is worth a fraction of that, seventy of the hundred and five ways of placing it leave the listener worse off than one point would, and two alike points half a cycle apart undo most of what the first one gave. Whether the listener keeps one timer or several turns out not to matter at all.
A gamelan's beating names its goal only if each goal beats at its own rate
When a gamelan's elaborating parts reach a goal tone early, a pair of instruments tuned slightly apart beats on that degree for the strokes before the goal — a swell of shimmer just before each structural stroke, which is exactly where a recognisable event does a slow cycle's clock most good. Whether it does that good depends on whether one goal's swell can be told from another's. Tune every pair to the same beat rate, the practice that keeps a gamelan's shimmer even, and every swell is the same event: it buys a third of what it could. Tune every pair the same number of cents apart and each degree beats at its own rate, fifteen per cent faster than the degree below, and the swells start naming which goal is coming.
A gamelan tuner can name the goals for nothing
A pair tuned so that each goal beats at its own rate lets the swells before a gamelan's structural strokes say which goal is coming, and the question left was what that costs in roughness. Held at the same average shimmer, nothing: a constant interval adds one per cent less roughness than an even beat rate, and a rate rising as the square of the pitch adds three per cent less while naming the goals for a far coarser ear. The even shimmer puts its widest pairs on the low degrees, where a cent of detuning costs most.
The octave that fails both checks least
A tuner who cannot pass both the fourth–fifth check and the third–sixth check settles on an octave that fails them equally, and a listener compares beat rates by ratio. Weighed that way, the compromise sits a ninth of the way from the 4:2 octave to the 6:3 on every note of every piano, because the fourth beats twelve times slower than the minor third and that ratio belongs to equal temperament, not to the strings. On a grand both checks then pass to a fifteen-per-cent ear up to middle C, and above it no octave passes both.
The progression of thirds votes for the bearing, not the octave
Added as a third vote beside the two contiguous checks, the progression of thirds was expected to push a stiff piano's octave wider. It pulls it narrower, because the octave it asks for is not set by the strings at all: it is whatever stretch the bearing already carries, plus an eighth of a cent. In an equal bearing that drags the compromise back onto the 4:2 and costs the top of the bearing its pass; give the bearing a fifth of the octave's own stretch and the third vote falls silent.
A gamelan's shimmer can slow to a fifth before its clock notices
Roughness falls steadily as every pair's beating is slowed by one factor, and the listener who times a goal's swell does not object until the shimmer is a fifth of the promised speed. The limit is not how many beats a swell holds — a ten-second step holds eight even there — but the once-a-second boundary below which a beat is waited out rather than counted, and it bites when the second-slowest goal degree crosses it. At the quick tempi where a swell could hold too few beats, the skeleton already carries the clock alone.
A newcomer learns a cycle's landmarks in four hearings
Two recognisable points in a slow gong cycle help a listener only if they can be told apart, and a newcomer at first cannot: two marks that differ only by where they fall in the cycle sound alike until the listener has heard enough cycles to notice. Timed from the gong, the best pair of points is learned by nine listeners in ten after about four cycles — the same four at any slow tempo, so ten minutes at 160 seconds a cycle. The pairs learned fastest sit right after the gong and are poor clocks once learned; for the first four cycles a newcomer is best served not by the best clock but by a pair whose confusion costs less while it lasts.
Named alongside it
The objects these essays reach for when they reach for this one.
DurationMemory decayCycleInformationBeatingMusical formProportionTimelineHypermetrePhraseSlendroTemperament