# statistical-feedback — mechanism spec Six voices and a silence share one pulse. Nothing here rolls dice for each of them in turn: every option keeps a ledger — turns it has taken, less the turns its weight entitled it to — and the next turn goes to whoever the books say is furthest behind. The ledger is on the screen, the books always sum to zero, and one switch decides whether selection reads them at all. This document is the toy's other medium. It carries the mechanism, its parameters, its audio graph and its timing at the level a reader needs to rebuild the instrument somewhere else — another language, another audio framework, hardware. Where a choice is arbitrary this file says so, and where something was measured it gives the number. **Licence: CC0.** Copy it, port it, sell it, no attribution required. Attribution is nonetheless welcome: this page lives in the Playable Papers repository — `REPOSITORY URL TO BE SET BEFORE PUBLISHING` — and its author at [noisewrangler.art](https://noisewrangler.art). --- ## 1. The claim **Dice have no memory, and the ear hears the difference.** A weighted random stream gives you clumps and droughts; a fair rotation gives you a machine. Statistical feedback is the third thing: choice with a ledger. Charles Ames' fair-share principle — "the resources falling most behind their fair share of usage receive first consideration going forward"[^ames-tutorial] — is one line of arithmetic, and out of it come four things this page is built to make audible rather than argue. Ames' own published account of the method is "Statistics and Compositional Balance".[^ames-pnm] - **Variety with balance, and the balance is exact.** Over any long stretch each option's share matches its weight to a fraction of a percent (§8.1), and it does so while the order stays unpredictable. Neither dice nor a rotation can do both. James Tenney's dissonant counterpoint is the same bargain struck by hand, and reading it as statistical feedback is Polansky, Barnett and Winter's.[^tenney] - **Metre is something the ledger produces.** Stress scales what a turn *costs*, never its odds — so a weight buys a share of the stress spent, not of the turns taken, and heavy voices end up on the strong beats while light voices pay in many weak-beat turns (§2.5). Measured, `crash` at weight 0.5 reached the downbeat once in 2 500 of them against `hat`'s 856. Nothing is sequenced and no step is reserved for anyone. - **Silence is a citizen, not a gap.** It has a weight, it takes turns, it keeps a ledger. Its weight is renormalized live so the *probability* of silence stays where you put it however many voices are in the pool (§2.3) — so removing a voice changes who plays and not how much space there is. - **Removal is a weight of zero, and the ledger remembers.** An option at weight 0 is out of both loops, so its statistic **freezes**. Bring the weight back and it re-enters carrying the debt it went away with. There is no removal rule, no redistribution and no forgetting anywhere in this design — which is a claim you can check, because the bar stops moving. ## 2. The mechanism Options are indexed `m`: the six voices, then **silence**. Each carries a weight `W[m] ≥ 0` (a live control) and a statistic `S[m]` (its debt of attention). At tick `n`, with stress `A = A[n mod 8]` and heterogeneity `H`: ``` SELECT value(m) = S[m] + (A + H·U[m]) / W[m] U[m] ~ Uniform(-1,1), fresh per option, per tick m* = the option with the LOWEST value options with W = 0 are out CHARGE # every tick, whatever selected S[m*] += A S[k] -= A · W[k] / ΣW for every k with W[k] > 0 ``` `ΣW` is taken over options with `W > 0`. `S` starts at 0 for everyone. That is the whole law. ### 2.1 Why the statistic is a debt in turns, and the trap in scaling it `S[m]` is what Ames' fair-share principle needs it to be: **turns taken, less turns owed**. One turn is taken (`+A`), and the cost of that turn is shared out among the options in proportion to their weights (`−A·W[k]/ΣW`), so an option that has been chosen exactly as often as its weight says sits at 0. Two properties follow, and both are load-bearing. **Zero-sum.** The decrements sum to `A·ΣW/ΣW = A`, cancelling the increment exactly, so `ΣS = 0` forever. Measured over 10 000 ticks with weights moving under the law and the pool emptying and refilling mid-run, `max |ΣS| = 2.2 × 10⁻¹³` — float noise, and it does not accumulate. **The update alone pins the long-run shares.** At equilibrium every `S[m]` has zero drift, so with `f[m]` the selection frequency: ``` f[m]·A = A · W[m] / ΣW → f[m] = W[m] / ΣW ``` There is no selection rule in that derivation. **Any** rule that keeps `S` bounded lands on the same frequencies; what the selection rule chooses is the *order*, not the shares. This is the single most useful fact for a porter, and §2.6 and §8.5 are its consequences. **The trap.** The obvious-looking variant — scaling the whole update by the selected option's weight, `S[m*] += A/W[m*]` with `S[k] -= (A/ΣW)·(W[k]/W[m*])` — is *also* exactly zero-sum, and it is wrong. Redo the drift calculation and it gives `f[m] ∝ W[m]²`. Measured on the shipped weights it is not subtle: `hat` (W 3.0) takes 35.4% where its weight says 25.9%, `crash` (W 0.5) takes 0.99% where its weight says 4.31%, and silence set to 0.25 arrives at 33.1%. Zero-sum is necessary and nowhere near sufficient; a port should check §8.1 rather than trust the invariant. ### 2.2 What the two extra terms in `value` actually do Neither term can move the shares (§2.1). Both shape the order, and they do different jobs. **`A/W[m]` is a per-option offset that breathes with the stress.** At a constant `A` it does nothing at all — a fixed offset per option just shifts where that option's statistic sits and leaves the dynamics alone. It earns its place the moment `A` varies: on a hard beat a light option is pushed far up and a heavy one barely at all, so **strong beats fill with heavy voices and weak beats level the field** (§2.5). Nothing is sequenced. **`H·U[m]/W[m]` is a regularity control, and nothing else.** It scatters the comparison, and scatters a light option further than a heavy one because its `1/W` is larger. What that buys is measured rather than assumed (§8.5), and it is not what the shape of the term suggests: across `H = 0 … 2` every option's share holds to within **0.01 percentage points** — the update pins them and no selection rule can move them (§2.1). What moves is how predictably each option's turns arrive, and it moves unevenly. From `H = 0` to `H = 2` the coefficient of variation of the gaps rises **×2.5** for the heaviest voice and **×21.6** for the lightest, from a near-metronomic 0.03 to 0.70 — against about 0.9 for pure dice. So `H` sweeps the *texture* from clockwork to nearly-dice with the proportions held exactly throughout. It does not make rare voices more frequent; it makes them unpredictable. **Ties are broken by index, within `1e-9`.** This is not a rounding nicety. At `H = 0` with equal weights every option is exactly level at the end of each round, and a bare `<` breaks that tie on floating-point residue, which differs between engines — the clockwork would be a different rotation on a different browser. With any `H` at all the epsilon never fires. ### 2.3 Silence is normalized live ``` W_silence = p / (1 - p) · ΣW_voices p = the silence control, 0 .. 0.95 ``` so that `W_silence / (W_silence + ΣW_voices) = p` for any number of voices at any weights. **Silence is a probability that holds**, and that is the point: take three voices out of the pool, or double one weight, and the amount of space in the music does not move. Measured across three very different pools and three settings of `p`, every combination lands within 0.0006 of `p` (§8.2). At `p = 0` silence has no weight and is out of the game exactly as a voice at weight 0 is — including the freeze. The control is clamped just under 1 (**0.95**) because `p/(1-p)` diverges; the last twentieth of the control's travel is therefore dead, which is the honest cost of this formulation. A port that dislikes it can reach the identical probabilities with `W_silence = p·ΣW` and voice weights scaled by `(1-p)`, which is finite at `p = 1` — at the price of the ledger's units moving with `p`. ### 2.4 Stress scales the charge, not the odds `A` multiplies what a turn **costs**, and it appears nowhere in a probability. A voice taken on a heavy step pays more and stays away longer; an offbeat turn is cheap and can recur straight away. Nothing is scheduled and no step is reserved for anyone: `A` enters the law twice, once as the charge and once through §2.2's offset, and that is its entire involvement. ### 2.5 The weight is a share of charge, not a share of turns This is the least obvious thing about the instrument and the most important, because it is what makes the metrical claim exact rather than atmospheric. The equilibrium condition of §2.1, taken over a whole stress cycle rather than a single tick, is ``` Σ_n f[m](n) · A[n] = (W[m] / ΣW) · Σ_n A[n] ``` — the weight fixes each option's share of the **stress spent**, not of the turns taken. So when the stress varies, the two part company. Light voices, pushed onto weak steps by §2.2's offset, need *more* turns to pay the same charge, and heavy voices need fewer. Measured (§8.3), `crash` at W = 0.5 takes **6.64% of the turns and 4.31% of the charge** — and 4.31% is its weight, to 0.10%. What that produces is a metrical hierarchy nobody wrote down. Who takes which step, as a share of that step's turns, over 20 000 ticks at `H = 1`, `p = 0.25`, shipped stress, seed 7: | A | hat | clave | tom | pluck | acid | crash | silence | |---|---|---|---|---|---|---|---| | **1.00** (step 0) | 34.2% | 16.4% | 11.3% | 4.2% | 1.0% | **0.0%** | 32.8% | | 0.70 (step 4) | 27.8% | 18.7% | 12.8% | 7.8% | 3.5% | 1.1% | 28.3% | | 0.50 (steps 2, 6) | 24.7% | 18.7% | 13.2% | 9.3% | 7.0% | 4.3% | 23.0% | | 0.25 (the offbeats) | 17.3% | 15.7% | 14.4% | 12.9% | 11.9% | 10.9% | 16.9% | Read the first row against the last: on the downbeat the field is steeply ordered by weight, and on the offbeats it is nearly level. In raw counts, **`crash` reached the A = 1 downbeat once in 2 500 of them, against `hat`'s 856.** No rule anywhere forbids a light voice a strong beat; it simply cannot afford one often. Under a flat stress pattern all of this collapses — turns and charge are the same quantity and §8.1's table is the whole story. The metre lives entirely in the shape of `A`. ### 2.6 Memory ON / OFF — the experiment ``` ON select by the law above OFF select with probability W[m] / ΣW # plain weighted dice ``` **The books are kept identically either way.** The toggle decides only whether selection *reads* them, and that asymmetry is the experiment. While memory is off, dice already realize the equilibrium frequencies, so the ledger loses its restoring force entirely: `S` becomes an unbiased random walk and the debts grow as `√n`. Measured, worst debt after an off-stretch: 3.42 at 100 ticks, 8.72 at 800 — a factor of 2.5 over a factor of 8 in length, which is `√8 = 2.83` inside the noise of single runs. That is why the bars visibly walk apart while nothing sounds wrong, and why turning memory back on pays them off in a burst of exactly the starved voices (§8.4). A port that "optimises" by skipping the charge while memory is off has removed the experiment, and the page has nothing left to argue. ## 3. Every gesture goes through the same books | gesture | what it is | |---|---| | **tap a voice's name** | it sounds now, off the grid, **and is charged exactly as a turn**: `CHARGE(m, A)` at the most recent tick's stress. At weight 0 it sounds and the books are untouched. From a stopped instrument it starts the pulse first and then sounds, because a hand asking for a voice is a hand asking for sound. | | **turn a weight to 0** | the option leaves both loops. It cannot be selected, it receives no share of anyone's payment, and **its statistic freezes** at the value it held. | | **turn the weight back up** | it re-enters carrying that same statistic — so a voice muted while deep in debt comes back and is paid first. | | **memory off, then on** | §2.6 | **The tap is the one that answers "what happens when a pooled item is invoked again".** A gesture from outside enters the same ledger the machine keeps for itself, so the machine absorbs it rather than ignoring it. Measured on §8's pool: `crash`'s next turn arrives 11.7 ticks after an arbitrary moment; after one tap, 17.4; after three, 38.8; after ten, 121.5 — 18 seconds of deliberate absence at a 150 ms turn, bought by hammering a key for a second and a half. The same charge, run ten times, does exactly what running it once does ten times. The freeze deserves one more sentence because it is easy to get wrong by being helpful. Nothing redistributes a departing option's debt, and nothing decays it. `ΣS` therefore stops being 0 over the *live* options while one is parked — the invariant is over all options including the frozen ones — and that is correct: the parked option's debt is still owed. ## 4. Parameters | control | range | what it does | |---|---|---| | **weight** `W[m]` | 0 – 4 per voice, defaults 3.0 / 2.0 / 1.5 / 1.0 / 1.6 / 0.5 (hat, clave, tom, pluck, acid, crash) | that voice's share of the stress the pattern spends. 0 removes it and freezes its ledger. | | **silence** `p` | 0 – 0.95, default 0.34 | the probability of a silent turn, held constant against the pool (§2.3) | | **tempo** | 40 – 300 BPM, linear, default 120 | one turn is an **eighth note**, `30 / bpm` seconds — 250 ms at the default — so one cycle of the eight-step stress loop is one bar of 4/4. The law counts turns, not seconds, so this changes no proportion at all. | | **heterogeneity** `H` | 0 – 2, default 0 | how loosely the ledger is read: 0 as regular as the pool allows, 2 near-dice — with the shares unmoved (§8.5) | | **stress** `A[0..7]` | 0.05 – 1 each, default `[1, .25, .5, .25, .7, .25, .5, .25]` | the cost of a turn at each step of an 8-step loop | | **memory** | on / off | whether selection reads the books. They are kept either way. | | **level** | 0 – 1.5 per voice, default 1 | the voice's output level. A mix control, outside the mechanism: it is how loud, where the weight is how often. | | **mute** | per voice | that voice's output to zero. It is **still in the pool** — it takes its turns and its ledger goes on moving; what stops is the sound. | Every one of these is read **live**, at the moment it is used, and none is cached into the ledger. Moving a weight mid-run is therefore a continuous act and not a reset — the statistics that exist go on meaning what they meant. Level and mute are the two that touch no part of the law. They are on the voice's row because they belong to the voice, and each says which question it answers when it is touched before the pulse has started. A port may leave them out entirely; the mechanism does not know they exist. There is no MIDI, no reseed and no sample loading in this round. The eight-step stress length is fixed. The PRNG seed is fixed and unexposed: nothing on the page reveals it, so a random one would buy nobody anything and cost the ability to reproduce a report of what the instrument did. **Saved state is the settings, not the books** — the six weights, the stress shape, tempo, heterogeneity, silence, the memory switch, every voice's level, mute and pan, and the master chain. The ledger is what *this run* has done, and restoring a debt against a pool that has changed underneath it would be a fiction. A saved session carrying a `period` instead of a tempo restores no tempo and opens at 120 BPM. **Reset** is what that persistence obliges, and it does two things: it puts every setting in the list above back where it opened — weights, stress shape, tempo, heterogeneity, silence, the memory switch, and every level, mute and pan — and it **clears the books**, so no option is owed anything. It keeps the master chain, which is the session's setting rather than the instrument's state, and it does not touch the transport: reset while the pulse is running and the pulse goes on, from a ledger of zeros. It asks twice, and any write inside the instrument between the two presses cancels the arming. **The arrival state.** The page opens **stopped and silent**, with every setting where the last visit left it and every ledger at zero: settings persist, the books do not, and nothing that makes sound reopens making it. There is exactly one discovery between arrival and the first sound, and the page names it in a line above the instrument: press Play. A first visit opens on **`default.json`** beside the page (JP's state of 2026-10-09), which is also the factory: six voices spread six to one, silence at 0.34, 120 BPM, heterogeneity 0, the shipped stress shape — so the proportion law is audible on the first press without anything being set up, and reset lands exactly where the page opened. A returning visitor's own settings win over the file; a page opened off a disk cannot fetch it, opens on the identical factory values, and says once that it is opening bare. The file's silence is a thumb position (0.3410…) and is rounded to the control's step, 0.01, on the way in. ## 5. Sound Six voices from the collection's shared palette, chosen to be tellable apart in a fast stream rather than to make a kit — six families rather than six drums, because at a fast turn a listener gets one event and then the next, and two members of one family arriving in that window read as a single voice with a wobble: | voice | weight | event gain | what it is | |---|---|---|---| | `hat` | 3.0 | 1.00 | the metallic bank behind a highpass, ~50 ms | | `clave` | 2.0 | 1.00 | an impulse into a bandpass at 2.5 kHz, Q 42, 100 ms | | `tom` | 1.5 | 0.31 | struck sine, 340 → 90 Hz in 110 ms | | `pluck` | 1.0 | 1.00 | a Karplus–Strong string, 500 ms | | `acid` | 1.6 | 0.65 | a saw through a resonant lowpass swept by its own envelope | | `crash` | 0.5 | 0.85 | the metallic bank spread wide over noise, 900 ms | ### 5.1 Choosing the six, and the bar it had to clear Which six is a measurement rather than a taste, because the toy's whole claim rests on a listener hearing *which* voice took a turn. Every synth voice in the palette is rendered under this toy's envelope, **cut at a 150 ms turn**, and compared as a log-mel spectrogram in three time slices — the attack, the body, and what is still sounding when the next event lands. The distance between two voices is the mean absolute difference in dB across that feature, and a set is scored by its CLOSEST pair, since a pool is only as separable as its worst confusion. Of the 1716 ways to choose six of the thirteen, this set is one of the two furthest apart. **The threshold is not invented.** It is the pool that shipped before this one, measured the same way: 10.1 dB between its closest pair, against 14.0 dB here. **The comparison is run twice, with level and without it.** A scalar gain shifts every band of the feature equally, so subtracting a voice's own mean removes level and leaves shape — and a set that separates only because one voice is louder leans on a distinction any mix can erase. Both readings are reported; the level-free one is the one that counts. Tuning the gains to maximise the score is possible and is deliberately not done: a search over the musical range reaches 18.2 dB as heard but *falls* to 13.4 dB on the reading that matters, which is the signature of fitting the scorer rather than the ear. The shipped trims are not that fit, and they clear the bar by about 4 dB on both readings. **What the trims do not do is equalise per-hit energy.** Equal energy per hit was the principle they were reasoned from, and it is not what they achieve — the intent must not be read as the result. Measured as ∫x² over one 150 ms turn, each voice rendered under this toy's envelope at its own event gain, the six span **11.2 dB**: | tom | acid | clave | pluck | crash | hat | |---|---|---|---|---|---| | −30.5 dB | −31.7 dB | −34.9 dB | −36.1 dB | −40.2 dB | −41.7 dB | `crash` sits **7.1 dB below the median** of the other five and `tom` 9.7 dB above it. Two things that spread is *not*. It is not the cymbal's length: the shared envelope falls 79 dB over `crash`'s 0.9 s decay, so the whole hit's energy exceeds the first turn's by 0.1 dB, and nothing is being counted outside the window. And it is not this toy's 0.85 — `crash` carries a −12 dB trim inside the library's own voice, underneath that gain, and without it the voice would sit 4.9 dB *above* the median rather than 7.1 below. Whether the spread is right is an ear's ruling and it has not been made; every gain in the table above is a guess until it is. This paragraph is here rather than in §8 because §5 is where a port reads the gains, and a port that took the principle for the measurement would equalise nothing and believe it had. A port that changes the palette re-runs this, and the bar for it is whatever its own previous pool measured. The weights are deliberately unequal, spread six to one, so the proportion law is audible on the first press. Which voice gets which weight is not arbitrary: **`crash` is the rarest and also the longest and most distinctive**, so its drought is the one an ear notices, and it is what makes the memory switch a demonstration rather than a statistic. **Event gains are fixed and are not the level control.** The table above is the gain each voice is *played* at — chosen as §5.1 describes, and measured there — and nothing on the page moves it. The level on a voice's row is the output bus underneath that — an ordinary mix fader, outside the mechanism (§4). The two knobs on one row are a real risk of confusion and the page answers it rather than avoiding it: press either from cold and it says which question it is, *how loud* against *how often*. **One selection per tick, and tails may overlap.** There is no busy state and no voice stealing. At the default 250 ms turn a `crash` rings over the next three turns and that is correct — a voice that refused a turn would be a second scheduling rule quietly fighting the first, and the shares would stop matching the weights. ``` per hit: rendered buffer (or synth) -> shared envelope -> panner -> the voice id's bus -> master shared: master gain -> soft-clip drive -> glue compressor -> [dry + small room] -> limiter (-3 dB, 20:1) -> out ``` The shared master chain is not part of the mechanism and a port may replace it with anything, including nothing. ## 6. Timing The law runs inside the collection's lookahead scheduler: a coarse 25 ms timer asks for every tick up to `now + 150 ms` (widened to about 1.6 s while the page is hidden, because background timers are throttled to around a second), and each selection is scheduled at its own absolute audio-clock timestamp. Only the *scheduling* rides the timer. The tick period, `30 / bpm`, is read **live, once per tick**, so a move of the tempo alters only ticks not yet scheduled — no phase jump, and no accumulated cursor that could disagree with the grid. **The ledger is released against the audio clock, not drawn when it is computed.** Every tick queues a copy of the books stamped with its own timestamp, and the animation frame releases them when `currentTime` reaches it. Anything else puts the see-saw up to 150 ms ahead of the sound — more than half a turn at 120 BPM, a whole one from 200 BPM up — and the mechanism-on-screen would be showing the future. The running-step marker on the stress row is released the same way, for the same reason. **A stop is a master fade**, so ticks already inside the lookahead window are silenced — but they have been charged, and the books are the books. The display is flushed on stop so that what is drawn equals what is recorded, rather than lagging the truth by the width of the window. **A tap lands at `currentTime`, off the grid deliberately.** The record quantizes it to the nearest cell and draws a deviation triangle, so a visitor can see that their gesture was not on the pulse and was counted anyway. ## 7. The surface The visual design is incidental. These four properties are not. **The ledger is the mechanism, drawn — not a decoration.** One centre-zero bar per option, right of the line for credit already spent and left of it for a debt owed, updated every tick. The see-saw has to read: one bar jumps as its option pays, and every other bar sinks a little in the same instant, because zero-sum means one can only rise by the others falling. A brief flash marks the row that was chosen. **The scale adapts, slowly and asymmetrically.** It opens at about 0.55 s so a debt built during a memory-off stretch stays inside the cell, and closes at about 3.2 s so the bars do not twitch between ticks, with a floor of **1.5** — the band the law's own steady state occupies (§8.6), so an idle ledger does not magnify float dust into a full bar. **Hue means "this option" and nothing else.** The swatch in the rack, the ledger bar beside it and the lane in the record wear one colour because they are three views of one option. Every control stays greyscale. Silence is the one grey row, since it is the citizen with no sound. **Silence draws nothing, and the empty column is the mark.** The record is one lane per voice on a grid of one turn — an eighth note — anchored on the ticker's most recent tick and carrying its tick *numbering*, so the heavy gridline falls where the stress loop starts. A silent turn leaves a gap in an otherwise regular column of events — which is what makes "silence is an option" visible rather than asserted. ### 7.1 Where things are The page is one rack and two time surfaces, in that order, because the weights are what the hands play and the rest is what the eyes read. - **The master strip** carries the transport, the master chain, the **tempo** in the collection's pulse slot, and the three controls that shape the whole pool rather than one option: **heterogeneity**, **silence** and the **memory** switch. None of them asks for sound, so none of them starts the instrument; each says so in its own words when it is touched before the pulse has begun, and names the transport. - **The rack** is one row per option, silence included as the grey row: **name · mute · level · weight · the ledger bar**. The name is a button — pressing it plays that voice and charges the books (§3) — and it is the one thing inside the instrument that *does* ask for sound, so from cold it starts the pulse first and then sounds. Silence has a name, a weight readout and a ledger, and no level, mute or tap: it is the citizen with no sound. - **The stage**, below the rack, is the two things a row cannot hold, both of them time: the **stress loop**, eight bars in a titled panel, and the **record**. - Below that, a session bar with **reset** and save/load, then the standing text. **The keyboard reaches all of it.** Tab walks the transport, the four master knobs, the tempo, the three pool controls, the hint toggle, then each rack row in column order — name, mute, level, weight — then the eight stress bars one at a time, then reset and the file buttons. Every one of those is operated by the arrows (Page, Home and End too); each stress bar is named for the step it is — *"step 3"* — inside a group named for the loop, and announces its value as *"stress 0.50"*, and the arrows move it by 0.05, Page by 0.25, Home and End to the ends of its range. The transport also answers the **space bar**, except while a button has focus, where space belongs to that button. **Targets.** The rack would scroll sideways rather than reflow, because column alignment is the component; measured, it never has to above the teaser width (670 px of rack at a 721 px viewport, with room to spare). The stress row is one continuous pointer surface: a press picks the bar under the finger and a drag paints the contour across all eight, so there is no dead space between targets to miss. The strip is 400 px wide at every desktop width, so each bar is 46.4 px wide and 48 px tall on a fine pointer. **Widths.** Above the teaser breakpoint the page declares no reflow of its own: one column at every width. Below it the page is the teaser. Measured, there is no horizontal overflow of the document or the body at 1440, 1320, 1200, 1000, 900, 800, 760, 721, 720, 700, 620, 560, 480, 390, 380, 360 or 320 px. **The teaser.** A phone shows **Play** and **heterogeneity**, alone, and the record; the rack, the tempo, the master chain, the silence and memory controls, the stress row and the session bar are the desktop's. The heterogeneity dial is a twin of the desktop one — two controls writing `H`, each showing the other's writes — and touched from cold it presses Play first and then turns. The phone opens it where the page opens, at 0: each voice comes round as regularly as its weight and the stress allow, which is weighted randomness read off the books. Turning it up loosens the grain toward pure dice while every share stays exactly where its weight puts it (§8.5). At 0 the stream is not clockwork: measured on the opening pool at `H = 0` with the shipped stress, 20 000 ticks, seed 7, the stream has no period up to 4 000 ticks (§8.6 gives the same finding for the measurement pool) — so the dial runs from *weighted* randomness to *pure* randomness, which is another way to hear what a motif is. It is the control, rather than the memory switch, because it is the one whose whole travel is audible at once on a phone, where the ledger that makes the memory switch legible is not shown. The stress row writes on a drag, so it is the desktop's; the record is kept because it is passive. At 390 px the dial's face is 56 px and Play and the share button are 44 px tall, with no horizontal overflow. ## 8. What the mechanism actually does Measurements, so a port can check itself. The core is a dependency-free block (`AMES BEGIN` … `AMES END` in `index.html`); everything below was produced by lifting it into a bare script with a seeded mulberry32 and no page around it. Unless stated, the pool is the weights 3.0 / 2.0 / 1.5 / 1.0 / 0.7 / 0.5 (hat, clave, tom, pluck, acid, crash) with `p = 0.25`, and seconds are at a 150 ms turn. That is a measurement pool, not the page's opening one (§4). ### 8.1 Proportionality 20 000 ticks, `H = 1`, flat stress `A = 0.5`, seed 7. | option | W | W/ΣW | measured share | relative error | |---|---|---|---|---| | hat | 3.0 | 25.86% | 25.86% | 0.01% | | clave | 2.0 | 17.24% | 17.24% | 0.01% | | tom | 1.5 | 12.93% | 12.93% | 0.01% | | pluck | 1.0 | 8.62% | 8.62% | 0.01% | | acid | 0.7 | 6.03% | 6.04% | 0.01% | | crash | 0.5 | 4.31% | 4.32% | 0.11% | | **silence** | (p = 0.25) | 25.00% | **25.00%** | 0.0000 absolute | ### 8.2 Silence holds against the pool 20 000 ticks each, `H = 1`, flat stress, seed 4242. Worst deviation over the nine cells: **0.0006**. | pool | p = 0.10 | p = 0.25 | p = 0.60 | |---|---|---|---| | six voices, the measurement weights | 10.00% | 25.00% | 59.99% | | three voices `3.0, 1.0, 0.5` | 10.00% | 25.00% | 59.99% | | six wildly skewed `4, .05, .05, .05, 4, .05` | 9.95% | 24.95% | 59.94% | ### 8.3 Turns against charge 20 000 ticks, `H = 1`, **the shipped stress pattern**, seed 7 — the check behind §2.5. The weight is a share of *charge* to 0.10%; the share of *turns* tilts toward the light voices, which take more of them on weaker steps. §2.5 carries the per-step table this one is the summary of. | option | W/ΣW | share of turns | share of charge | |---|---|---|---| | hat | 25.86% | 22.56% | 25.86% | | clave | 17.24% | 16.89% | 17.24% | | tom | 12.93% | 13.53% | 12.93% | | pluck | 8.62% | 10.28% | 8.62% | | acid | 6.03% | 8.27% | 6.04% | | crash | 4.31% | 6.64% | 4.31% | | silence | 25.00% | 21.82% | 25.00% | ### 8.4 The drought, and the debt 20 000 ticks, `H = 1`, shipped stress, five seeds, memory ON against memory OFF on the same seeds. Longest gap between plays of `crash` (W = 0.5): | seed | 1 | 2 | 3 | 4 | 5 | worst | |---|---|---|---|---|---|---| | memory **ON** | 66 | 53 | 66 | 65 | 55 | **66** ticks — 9.9 s | | memory **OFF** | 188 | 165 | 159 | 183 | 147 | **188** ticks — 28.2 s | Every option, seed 1, with the regularity of its turns beside it (coefficient of variation of the gaps): | option | longest gap ON | OFF | gap CV ON | gap CV OFF | |---|---|---|---|---| | hat | 14 | 29 | 0.63 | 0.85 | | clave | 21 | 42 | 0.66 | 0.91 | | tom | 25 | 58 | 0.66 | 0.94 | | pluck | 37 | 88 | 0.67 | 0.94 | | acid | 44 | 138 | 0.68 | 1.04 | | crash | 66 | 188 | 0.72 | 0.97 | | silence | 14 | 31 | 0.64 | 0.88 | **The books under memory ON sit in a band**: over a 1 000-tick window at steady state the largest `|S|` averages 0.60 and peaks at 1.14 (§8.6 gives 1.26 over 20 000 ticks — a longer run reaches a little further, and that is all). An off-stretch walks out of the band, and turning memory back on walks it home: | memory off for | worst debt reached | ticks for ON to return it to the band | |---|---|---| | 100 ticks | 3.42 | 27 — 4.0 s | | 200 ticks | 2.69 | 78 — 11.7 s | | 400 ticks | 4.02 | 73 — 10.9 s | | 800 ticks | 8.72 | 95 — 14.3 s | Repayment is fast and roughly flat in the size of the debt, because the law does not repay gradually — it spends its next several turns on the starved options and is done. That burst is the toy's argument in one gesture. ### 8.5 Heterogeneity moves the grain, never the shares 20 000 ticks, flat stress, seed 31337. Shares first, and there is nothing to see, which is the finding: | option | target | H = 0 | H = 0.5 | H = 1 | H = 1.5 | H = 2 | |---|---|---|---|---|---|---| | hat | 25.86% | 25.86% | 25.86% | 25.86% | 25.86% | 25.86% | | clave | 17.24% | 17.24% | 17.24% | 17.24% | 17.23% | 17.24% | | tom | 12.93% | 12.93% | 12.93% | 12.93% | 12.93% | 12.93% | | pluck | 8.62% | 8.62% | 8.62% | 8.62% | 8.62% | 8.62% | | acid | 6.03% | 6.03% | 6.04% | 6.04% | 6.04% | 6.04% | | crash | 4.31% | 4.30% | 4.31% | 4.32% | 4.32% | 4.32% | | silence | 25.00% | 25.00% | 25.00% | 25.00% | 25.00% | 24.99% | The largest drift from `H = 1` to `H = 2` anywhere in that table is 0.01 percentage points. **`H` cannot shift a share, and neither can any other selection rule** (§2.1). What it changes is regularity — and it changes it unevenly, because the scatter enters as `H·U/W` and a light option's `1/W` is larger: | option | H = 0 | H = 0.5 | H = 1 | H = 1.5 | H = 2 | H2 / H0 | |---|---|---|---|---|---|---| | hat (W 3.0) | 0.229 | 0.346 | 0.458 | 0.533 | 0.571 | ×2.5 | | clave | 0.140 | 0.345 | 0.472 | 0.544 | 0.593 | ×4.2 | | tom | 0.088 | 0.369 | 0.479 | 0.558 | 0.634 | ×7.2 | | pluck | 0.201 | 0.372 | 0.503 | 0.569 | 0.635 | ×3.2 | | acid | 0.078 | 0.388 | 0.519 | 0.603 | 0.640 | ×8.2 | | crash (W 0.5) | 0.032 | 0.388 | 0.537 | 0.615 | 0.695 | ×21.6 | For scale, the same measurement with memory **off** — dice — gives 0.85 to 1.04. So `H` sweeps the texture from clockwork (`crash` at 0.03 is a metronome) to most of the way to dice, with the proportions nailed at every point. **This is the honest characterization, and it is not the intuitive one:** heterogeneity does not favour the rare with more turns, it makes the rare unpredictable. ### 8.6 Clockwork, and where periodicity actually lives `H = 0` with flat stress, `p = 0` and six equal weights gives a **strict rotation of period 6** — `0 1 2 3 4 5` repeating, and audibly a machine. That is the degenerate case the tie rule in §2.2 exists to make reproducible. Loosen any one of those and periodicity is gone. Still at `H = 0`, with the shipped stress and the measurement pool's unequal weights, **no period under 400 ticks exists**; add silence at `p = 0.25` and none under 800. The stream is fully deterministic and not remotely periodic — the eight-step stress loop and six incommensurable weights are enough. `H` is not what makes this instrument unpredictable; it only makes it *irregular*. Ledger spread, for anyone sizing a display: max `|S|` over 20 000 ticks is 0.73 at `H = 0`, 1.40 at `H = 1`, 2.56 at `H = 2`, and 1.26 under the shipped stress at `H = 1`. ### 8.7 Determinism Same seed, same stream: two runs of 5 000 selections are identical. Adjacent seeds diverge at the first selection. ## 9. What must survive a port Keep, or it is a different instrument: 1. **A statistic per option, updated on every turn, that is a debt in turns** — `+A` to the chosen, `−A·W[k]/ΣW` to everyone. Zero-sum is a *consequence* of that shape, not the specification. See the warning below before writing this line. 2. **Selection is "lowest value wins", not a sampled distribution.** The randomness enters as a perturbation of a comparison, never as odds. 3. **Silence is an option in the pool**, with a weight, a turn and a ledger of its own — and its weight is renormalized live so its probability is independent of the pool (§2.3). A toy that implements silence as "and sometimes nothing happens" has lost the claim. 4. **A weight of 0 removes an option from both loops and freezes its statistic.** No decay, no redistribution, no forgetting. 5. **Stress scales the charge and appears in no probability**, so the metre is something the ledger produces (§2.5) rather than something anyone sequenced. 6. **The memory toggle updates the books without reading them.** This is the experiment; skipping the charge while it is off deletes the toy. 7. **An external invocation is charged as a selection**, at the current stress, through the same function. 8. **The ledger is visible and released against the audio clock**, showing the see-saw at the instant it is heard. 9. **Every parameter is read live and none is cached into the statistics**, so a hand on a control is a continuous act rather than a reset. ### The W² trap — read this before writing the update There is one variant of the update that looks like the same law, passes the zero-sum check, and is wrong. It divides the whole update by the *selected* option's weight: ``` S[m*] += A / W[m*] # WRONG — do not ship this S[k] -= (A / ΣW) · (W[k] / W[m*]) ∀k ``` `ΣS` still stays at 0 to float precision, forever, through every tick. The selection frequencies come out **proportional to `W²`**: | option | W | intended share | what this variant gives | |---|---|---|---| | hat | 3.0 | 25.86% | **35.44%** | | crash | 0.5 | 4.31% | **0.99%** | | silence | (p = 0.25) | 25.00% | **33.11%** | A weight of 3 against a weight of 0.5 becomes 36 : 1 instead of 6 : 1, and the silence control stops meaning a probability. The weight is no longer a share of anything. **A porter will be tempted by it**, and the reason is structural rather than careless: `1/W[m]` genuinely does belong on the *selection* line, so putting it on the update line too looks like consistency. It is not. The division by weight normalizes a *comparison*; the update is an accounting entry and must stay in the units the accounting is done in. Two defences. Derive the equilibrium rather than trusting the invariant — set the drift of every `S[m]` to zero and read off `f[m]` (§2.1 does this in two lines) — and run §8.1's table, which catches it in the first thousand ticks. **Zero-sum is necessary and nowhere near sufficient.** Incidental — change freely: - The six sounds and their gains entirely, the weight range (0–4), the tempo range and the eighth-note reading of a turn, `H`'s 0–2 scale, the silence ceiling of 0.95 and the alternative formulation §2.3 offers for it, the fixed seed. - **The eight-step stress length**, and the default pattern. Any loop length works; 8 is a round-1 decision, and §8.6 suggests the *length* matters more to the stream's aperiodicity than the shape does. - The number of options, and whether the pool is voices at all — the law is indifferent to what an option does when chosen. - The tie epsilon's value, provided ties resolve deterministically rather than on floating-point residue. - The whole visual design: the bar geometry, the auto-scale constants, the flash, the record's window and colours, the rack layout. - Saving the ledger along with the settings, if a port wants a document rather than a performance. --- [^ames-tutorial]: Charles Ames, "Statistical Feedback", interactive tutorial, . [^ames-pnm]: Charles Ames, "Statistics and Compositional Balance", *Perspectives of New Music* **28**(1), Winter 1990. [^tenney]: Larry Polansky, Alex Barnett and Michael Winter, "A Few More Words About James Tenney: Dissonant Counterpoint and Statistical Feedback", *Journal of Mathematics and Music* **5**(3), 2011.