Because Anneal's sound is not a function of the current knob position. It is a function of a state variable, the lattice's frozen configurational deviation, that only updates when the material is hot enough to move and stops updating the instant it is not. Set TEMP to 0.30 by quenching from molten in 30 milliseconds and the lattice freezes with 17.28 cents of scattered detune still in it. Set TEMP to the same 0.30 by cooling over 8 seconds and the lattice has time to relax nearly all the way to order, freezing at 0.95 cents. Same final knob position, roughly eighteen times more disorder in one path than the other, because the sound is a record of the journey, not the destination.
The distinction
A memoryless map versus a system that carries state.
Most synthesizer controls are memoryless. A filter cutoff knob at 800 Hz sounds like 800 Hz whether you turned it up from 200 or down from 4,000; the transfer function only cares where the knob is now. This is such an ingrained assumption in synthesis that it is rarely stated, because it is almost always true: oscillators, filters, and most modulation sources are stateless with respect to history, even when they have internal state (a filter's reactive elements), because that internal state settles to the same place given enough time at a fixed setting.
A glass-forming system breaks that assumption on purpose. Its defining feature is a state variable, fictive temperature \(T_f\) (introduced in note 09), that does not track the current control setting. It tracks how much time the system spent able to relax at each temperature it passed through on the way here. Below the glass transition, \(T_f\) stops updating entirely: it is locked to whatever it was when the material crossed \(T_g\), regardless of anything that happens to the knob afterward, short of reheating past \(T_m\). This is what "path dependence" means operationally: the present state is a functional of the entire trajectory, not a function of the present input.
For an instrument, this means the same TEMP value can correspond to a whole family of possible sounds, distinguished only by which path got you there. That is a real cost in predictability, the kind of nonlinearity most instrument designers avoid. Anneal keeps it because the cost is also the entire expressive point: it is a knob whose sound depends on how you moved it, which is a genuinely different kind of control surface from anything memoryless.
The measurement
Two paths, one destination, two sounds.
The test is designed to eliminate every other explanation. Both runs start molten and end at the identical final setting, TEMP 0.30. Both are loudness-matched afterward so a listener cannot distinguish them by level. The only variable that differs is how fast the cool happened.
- Anneal (8 s cool): frozen spread 0.95 cents, \(T_f\) 0.526, crystallinity 0.996, median beat 0.86 Hz.
- Quench (30 ms cool): frozen spread 17.28 cents, \(T_f\) 0.874, crystallinity 0.288, median beat 8.48 Hz.
- Log-spectral distance between the two end states: 17.59 dB, against a 3.09 dB re-phase floor measuring ordinary take-to-take noise, roughly a 5.7× gap beyond that noise floor.
- End-state loudness matched to within +0.49 dB, so the spectral gap is not a level artifact.
Crystallinity here is a normalized measure of how close the lattice sits to its ordered centerline: 0.996 is nearly fully ordered, 0.288 is mostly disordered. Median beat rate, the rate at which nearby detuned partials interfere audibly, moves from under a hertz in the annealed case to nearly 8.5 Hz in the quenched case, the difference between a stable pad tone and a fast, chorus-like flutter. Every one of these numbers is measured at the identical TEMP knob position.
The control
A macro built specifically to fail this test.
The obvious objection to any claim of path dependence in software is that it might just be a static lookup dressed up with a physics story: perhaps the "memory" is actually just a slow parameter smoother, and a sufficiently clever memoryless macro could fake the same effect. Anneal's design process included building exactly that adversarial control on purpose: a memoryless detune macro that reads the current TEMP, looks up the equilibrium disorder map for that TEMP, and applies a fixed detune pattern scaled by it. No integration over time, no relaxation clock, no fictive temperature. Just a static function of the current knob position, engineered to be the best-case fake.
It scored 0.0000 on every path-dependence metric tested; run it with a fast cool and a slow cool to the same final TEMP and the two outputs are numerically identical, because a static lookup cannot distinguish paths by construction. Its hysteresis loop area, discussed next, is also exactly zero, for the same reason. This is not a weakness of the adversarial control; it is the control working as intended. It demonstrates that whatever margin Anneal's real path-dependence numbers show above zero cannot be explained away as "a chorus effect with a physics-sounding name," because the actual best attempt at that alternative explanation produces a flat, measurable zero on the same tests where the real lattice produces 0.95 versus 17.28 cents.
Interactive lab
Sweep the temperature. Watch the loop trace itself.
The axes are TEMP on the horizontal and frozen spread in cents on the vertical. As the lab sweeps TEMP through a triangle wave, up then down, a dot traces the lattice's actual spread at each instant. With the real state-carrying model, the down-leg (cooling) and up-leg (heating) do not retrace the same curve. The down-leg lags behind, because relaxation cannot keep up with the falling temperature, so it freezes in more spread than the up-leg accumulates on the way back through the same temperatures; the result is a visible loop rather than a single line, matching the documented 1.13 to 1.33 cent-times-temperature loop area and the 1.75 to 2.75× down-leg-over-up-leg asymmetry at TEMP 0.60. Check Memoryless control and the same sweep collapses the loop to a single-valued curve with zero enclosed area: the visual form of the 0.0000 score from the previous section.
The measurement
The loop is the signature of memory.
Hysteresis is what path dependence looks like when you plot it rather than just measure two endpoints. Sweep TEMP as a triangle wave, up from cold to molten and back down, and record frozen spread continuously. If the system were memoryless, the up-sweep and down-sweep traces would sit on top of each other; whatever spread corresponds to a given TEMP would be the same regardless of sweep direction. Anneal's lattice does not do this. The measured loop area is 1.13 to 1.33 cent-times-temperature units, and at TEMP 0.60 specifically, the down-leg (cooling through that point) freezes in 1.75 to 2.75 times more spread than the up-leg (heating through the same point) accumulated.
The mechanism is the same relaxation-time asymmetry from note 09: heating past \(T_m\) melts the lattice and its accumulated disorder is erased almost immediately, because relaxation is fast in the liquid regime. Cooling through the same temperature range is where the relaxation clock and the cooling clock compete, and where whatever disorder exists when \(\tau(T)\) exceeds the available time gets locked in. A memoryless macro, by construction, cannot produce this asymmetry: its output is a pure function of the instantaneous input, so its loop area is exactly zero, which is precisely what the adversarial control measures.
The measurement
A frozen glass keeps relaxing, slowly, if you let it.
Physical aging is the third proof of state-carrying behavior: even a glass that looks frozen continues to relax, just too slowly to notice on short timescales, and that slow relaxation is itself measurable and matches the same Arrhenius law governing everything else. Quench the lattice to a hard glass, then park TEMP at 0.45, comfortably below the melt point but with a relaxation time on the order of seconds rather than years, and the disorder heals over real time: frozen spread falls from 21.98 to 0.57 cents and median beat rate falls from 19.45 to 0.58 Hz over 25 seconds, with a fitted creep time constant of 6.5 seconds that matches the Arrhenius prediction for \(\tau(T)\) at that temperature exactly, not approximately.
Park the same quenched glass at TEMP 0.20 or below instead, and nothing happens: 0.0% relaxation over the same 25-second window, because \(\tau(T)\) at that temperature is many orders of magnitude longer than any musically relevant hold time. The ANNEAL creep band, roughly 30 to 42% on the TEMP knob, is the range where this self-healing happens on a timescale a performer can actually use: park there and a quenched, glassy pad audibly smooths itself out over a few seconds without anyone touching a control.
The measurement
Same statistics, different pattern, every time.
One more property follows from a stochastic state-carrying model rather than a deterministic lookup: run the identical settings twice, with two different random noise seeds driving the Ornstein–Uhlenbeck update, and the two runs share the same statistical description (the same expected spread, the same \(T_f\)) while landing on individually distinct patterns of which sites froze where. The measured correlation between two such runs is +0.149, close to what independent draws from the same distribution should produce, and nowhere near the +1.0 a deterministic function of TEMP would guarantee. Two quenches of the same patch are the same kind of glass, not the same glass.
Playing it
Path dependence as a performance surface.
Treat QUENCH as a performance gesture rather than a switch: hit it mid-phrase and the lattice locks in whatever disorder existed at that instant, a snap decision the lattice will hold until the next melt. DOWNBEAT ANNEAL, using SYNC to quantize a cooling ramp to the host grid over several bars, turns the crystallization itself into a compositional event: a pad that starts glassy and audibly settles into tune over eight bars as it reaches the downbeat. MODE ADDITIVE / MODAL interacts with the same state: in MODAL, an annealed crystal rings long and coherent, around 2.6 seconds, because ordered partials share damping behavior, while a quenched glass scatters into a faster decay near 1.7 seconds with higher modes damping first, so switching MODE turns structural order directly into audible ring time rather than a separate decay control.
Honesty
What this proof does and does not establish.
The thesis is structural isomorphism, not science as decoration, and the proof numbers above are exactly that: a state-carrying stochastic model producing measured, repeatable path dependence, hysteresis, and aging that a memoryless alternative, deliberately built to be the best fake available, cannot produce. It is honest to say the loop area, the aging time constant, and the quench-versus-anneal spectral gap are measured properties of the actual DSP, not narrative dressing on a static effect.
It is not honest to claim these numbers constitute a general-relativity-grade validation against real laboratory glass data; the constants (\(T_m = 0.80\), the Arrhenius prefactor, the STRESS-scaled disorder ceiling) are chosen for a musically useful range on a synthesizer, as described in note 09's boundaries section, not fit to a specific measured material. The adversarial control rules out one specific failure mode, "this is secretly memoryless," convincingly. It does not and cannot prove the model is the unique or best possible glass analog; it proves that the one built has memory where a static system would not, and that the memory produces numbers, not adjectives.
References
Source material.
- Scherer, G. W. (1986). Relaxation in Glass and Composites. John Wiley & Sons.
- Moynihan, C. T., Easteal, A. J., DeBolt, M. A., and Tucker, J. (1976). "Dependence of the fictive temperature of glass on cooling rate." Journal of the American Ceramic Society 59(1-2), 12-16.
- Tool, A. Q. (1946). "Relation between inelastic deformability and thermal expansion of glass in its annealing range." Journal of the American Ceramic Society 29(9), 240-253.
- See also note 09 for the governing fictive-temperature model and its own references.