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Design notes — Part VII — Phase 8: Suzu, the synth (step 55b, the field as displacement, then the synth inside Voxo — steps 56–59, 59b–59c; formerly `DECISIONS_7.md`)

Ambiguities resolved during Phase 8 (steps 55b, 56–59 and 59b–59c of ROADMAP_5.md, now ROADMAP.md Part 7): the field stored as a displacement first, then Suzu — the symplectic synth inside Voxo, the orbit trace, the lab on the web. The author re-cut the arc before step 56 (Suzu became Phase 8, the instruments Phase 9), which is why this part’s title once read “Instruments”. References written as DECISIONS_7 #n mean this part. The specs are PROJECT_SPEC.md (SPEC §n; its §10–§12 drafted in specs/TO_PROJECT_SPEC.md), specs/SYNTH_SPEC.md (SYNTH §n — in git history since the day after the close, git show 5111748:specs/SYNTH_SPEC.md; its content as shipped is specs/TO_PROJECT_SPEC.md §13, #35), specs/INSTRUMENT_SPEC.md (INSTRUMENT §n) and specs/QUALITY_OF_LIFE_SPEC.md (QOL §n); where an entry here and a spec conflict, the entry is the record of what shipped — and the conflict is flagged to the author, who owns the specs. Step 55b’s seven entries were merged here at its fold (2026-09-29, the author’s call — the pre-release tag v2.0.0-alpha.2, cut at the fold, covers Phase 7 and this step), earlier than the phase’s end; the rest of the phase accumulated in DECISIONS_7.md from #8 and merged here at the close (2026-10-04, #35). The roadmap’s Phase-8 header pointed the name’s confirmation at “DECISIONS_7 #1”; step 55b came first, so it is the first entry of step 56 (#8); the fingering confirmation is Phase 9’s DECISIONS_8 #1. The evidence left the tree at the folds: docs/evidence/step55b/ is in git history at 6d1e034, the boxes’ gate-55b/ at fd7a196 (Linux) and 8f0cdb0 (Windows); steps 56–59b at df110c9 (the profiles and the charts, the rotor’s video step59/rotor_anod.mp4, the lab’s screenshots), step 59c’s at 5111748; CHANGELOG.md’s step entries are their condensation, and tools/strike_compare.py, tools/sound_profile.py, tools/chaos_chart.py, tools/mode_splitting_chart.py and the two Suzu gates the scripts that outlived them.

Step 55b — The field as displacement (the Mac)

Section titled “Step 55b — The field as displacement (the Mac)”

  1. The field stores each texel’s DISPLACEMENT, not its pre-image; the fixture moved once, by construction; the Sumi print did not move at all. The payload of tex_A/tex_B (SPEC §4.1–§4.2) is now (u − x, v − y, ink, aux): the pre-image coordinate minus the texel’s own, then the phase and the selector as before. Every pass computes its inverse lookup P_src from its own texel exactly as it did, reads the field there and re-bases what it read — d′(P) = d(P_src) + (P_src − P), which is u(P_src) − P, the same pre-image the absolute payload carried — through ONE helper (sumi_fetch in deform.glsl, a shared @block); the identity, the ingress rule and a drop’s interior write zero. Bilinear filtering commutes with the change (the weights interpolate the texel centres exactly), so the map is the same map to rounding — and the rounding is the point: the half-float quantum of a stored value follows the displacement (2^-24 canvas heights at rest, 2^-15 at a sixteenth of the canvas, 2^-11 only for content that has travelled half the sheet) where the coordinates carried 2^-11 over the outer half of the sheet whatever the motion. THE ONE SEMANTIC TRAP, found by the fixture: the drop, the tine, the vortex and the swirl have no ingress branch and lean on the sampler’s clamp-to-edge for a source past the edge, which under the absolute payload meant “the edge texel’s pre-image”; re-basing with the raw P_src would instead extrapolate the edge’s displacement off the canvas (a 0.09-canvas difference in the corners of the canonical script, a different map at the rim). So the helper re-bases with the coordinate the sampler actually read — P_src clamped to the edge texel centres, [½/W, 1 − ½/W] per axis, from textureSize — and the value is the 1.1.0 value exactly. THE INDEPENDENCE that makes the fold cheap: no pass reads the coordinates to decide where to sample, so the ink and aux channels never depended on how the coordinates were stored — the re-captured field_512_metal.bin is bitwise the old fixture in ink and aux (0 of 262,144 texels differ in either), its pre-images agree with the old ones to 1.6e-3 at worst (seven passes of the OLD rounding, 2^-12–2^-11 each) and to 1e-4 over the bulk, and the composite fixture composite_512_metal.rgba — a Sumi print, which reads only ink and aux — is bitwise untouched, as the roadmap promised. What reads the coordinates: the Anod composite (the strain estimator now rebuilds J = I + ∇d from the displacement, the fresh class is “no phase and under half a texel” with the ULP term gone, the rounding bias is taken at the displacement’s magnitude, the water’s grid reads the payload directly), the harness (its reader rebuilds u = dx + (x + ½)/W once in float32, so every check keeps reasoning in pre-images; the raw reader and the dumps keep the stored payload), the two gate tools (channel names dx/dy, the tolerances carry: a difference in dx is the same difference in u) and the print ledgers, which keep a field only within a session (no shell persists one) — so nothing converts and sumi_read_field / sumi_export_begin simply hand the new payload over; libsumi is 1.2.0 for that change of bytes (no signature moved). MEASURED on the Mac: the field gate green on Metal against the new fixture with its negative control, a second dump bitwise; the composite gate bitwise; every harness self-test green (the 17 suites, 86 checks) with no threshold touched; the Anod four of the palette test moved (they read the strain off a field 60× finer) and Metal’s column is recaptured with this entry’s evidence — the GL and D3D11 columns and tiers are the boxes’ to re-measure (the handoff); THE WEB TIER against the new fixture: max 7.6e-5 / 1.2e-4 in dx / dy, mean 3.9e-9 — under the old payload the same Chrome sat ~6e-4 from the Apple fixture “through half-float rounding alone” (DECISIONS_4 #20); the rounding that separated the backends was the coordinates’. SPEC §4.1–§4.2’s “(u, v, ink, aux)” and “initial state: identity coordinates” are the author’s to transcribe (specs/TO_PROJECT_SPEC.md, the pointers); §4.6’s regression test is unchanged in kind. The phase invariant restarts from this fixture (DECISIONS_5 #12, #87). Evidence: docs/evidence/step55b/.

  2. The emission floors, re-derived from the displacement’s quantum: a quarter of the old step, kept as a pass economy — and the #61 experiment repeated says the rounding-back they were built against is gone. Three floors were the coordinate quantum’s: the burst’s and the spark’s 5e-4 canvas heights (one 2^-11), the stir’s 1e-3 (two). The displacement’s spacing follows the displacement — 2^-24 at rest, 2^-15 at a sixteenth of the canvas — so ONE quantum is defined for the floors: SUMI_FIELD_QUANTUM = 2^-13 = 1.22e-4, the spacing at a QUARTER canvas of displacement, which a stirred playable sheet does not exceed; a pass whose peak reaches it lands whole anywhere. The burst’s and the spark’s floors are one quantum, the stir’s two (voice_mapper.h), each a quarter of what it was: a strike’s tail deals four times as many, four times finer stages, the stir’s clock on the small-disc layouts (the Jankó, the fifths — every 50–60 ms before) is now the frame. MEASURED, #61’s test on the chromatic grid (150 frames of 0.0125 rad at 512², the harness): under 1.1.0’s coordinates with no floor the ring turned 0.07 rad (nothing accumulated — #61); under the displacement with NO floor it turns 1.36 rad and the ring’s radial scatter |log r′/r| reads 0.05; with the old floor (a pass every third frame on this grid) 1.45 rad and 0.03; with the new floor (a pass every frame here, as with none) 1.36 and 0.05. So the floors no longer protect the motion — they trade resampling scatter for temporal resolution: three times the passes cost six percent of the rotation and twice the ring’s radial blur on this grid, the bilinear medium’s erosion per pass summed. FOR THE AUTHOR: the roadmap asked for the re-derivation and “may go”; they could go to zero (the same numbers as the new floors on the grid, more passes on the tails) or back up to the old values purely as an erosion economy — the choice is now about the look and the GPU budget, not about correctness, and this entry holds the three measurements to make it with. Every episode-driven self-test (the burst, the spark, the torsion sweep, the gestures, the Anod prints) passes with no threshold touched; the canonical script has no episodes, so the fixtures and the palette hashes do not see the floors. Evidence: docs/evidence/step55b/floors/.

  3. The soak’s pre-image tier was the coordinate payload’s freeze; it is re-derived to 32 texels, and the (b) gate now measures what it says. The first soak under the new payload went red on (b) for every exact operator — tine 11.9 texels of pre-image deviation after 500 (+k, −k) pairs against the 8.0 tier (was 1.6), pinch-saddle 13.2 (2.3), swirl 12.1 (1.5), the vortices 14.2 / 16.1 (1.8 / 2.7), the torsion 16.4 (5.5), the Chladni stir 22.1 (5.0), the spark shear 20.3 (3.6), the ripple bake 9.1 (2.6) — while the ink-mass half of (b) and all of (c) and (d) were BITWISE the Phase-6 numbers (the ink never read the coordinates). A growth-law hook (--pair-drift <op>: the deviation after 1, 2, 5 … 500 pairs and its 8×8 map) run on both payloads — the old from a worktree at 7ceb111 — says why: the single-pair residual is the same on both (the tine 0.194 vs 0.184 texel; the stir 2.18 vs 2.17) — the resampler’s smoothing of the pre-image map, deterministic, the same sign every pair; under the coordinates it FROZE after ~50 pairs (0.54 at 10, 1.10 at 50, 1.37 at 100, 1.60 at 500; the map: 0.01 texel in the water, 5–9 in the ink) because a smoothing step of ~0.02 texel is under half a 2^-11 quantum and rounds back — DECISIONS_3 #37’s “protective property”, named then as such; under the displacement it accumulates as ~N^0.7 everywhere (0.77 at 10, 2.35 at 50, 3.84 at 100, 6.28 at 200, 11.87 at 500; the map 6–23 texels, water and ink alike). The non-inverse control, the crossed pinch, grows linearly from its first pair (1.48, 8.2 at 10, 36.0 at 100, 72.9 at 500; the mass +25 %) on either payload. So the 8-texel tier — “eight keeps the oscillatory family green with a 25× margin” — was calibrated on frozen numbers and never measured the resampler; the roadmap’s “the tiers re-measured” is read to include it. SOAK_DEV_EXACT = 32: 1.45× the worst exact operator, 0.44× the failure it must catch (--soak-negative’s negative-inversion control stays red at 73). FLAGGED for the author: specs/TO_PROJECT_SPEC.md §10.2 quotes the tier as “the pre-image within 8 texels over 500 strong pairs” — 32 from here, with this entry as the reason. And a consequence to carry: the pre-image map now erodes as honestly as the ink does under long stirs — an Anod charge’s strain reading will fade with hundreds of resampling passes where the frozen coordinates held it — the same physics the ink always had, no longer masked. Evidence: docs/evidence/step55b/drift/, the two soak reports.

  4. The Tab: the Adreno’s loss was the coordinates’, and the six strikes now print alike on the Tab and the Mac. The GLES field tier against the new Metal fixture (the fieldDump intent, 512²): dx max 1.1e-3, dy 1.2e-3, ink 1.5e-2, aux 4.9e-4, mean 7.9e-5 — under #88’s coordinates the same Tab sat at max 1.51e-2 / mean 6.08e-4, so the coordinate channels are 14× closer to Metal and the mean 7.7× (the ink channel’s 1.5e-2 at a rim texel is the Adreno’s bilinear on the phase, as before; PASS at the mobile tier 2.5e-2 / 1e-3). THE JUDGEMENT the roadmap named: #88’s frame-locked script (six velocity-100 MPE strikes 18 frames apart at 1/120 s, 150 frames, the dip) at 1024×1024 through a temporary --es strikes 1 [--ef sparkShear f] intent on the Tab and the bench’s --anod-strike-render on the Mac, both under the Tab’s own session (--preset, new on the bench: the author’s preset with anod_drop 0.33, no grid, the phosphor palette), the prints compared by lit blobs (tab/strike_compare.py, 60 levels over the glass, ≥ 200 px): under the session’s shear 0.6 the Mac keeps 6 charges of 6 423 … 5 014 px (3.25 % lit) and the Tab 6 of 6 263 … 4 964 (3.18 %); under the harsh shear 2.0 of #88 — where the Tab had kept six with two thinned to 1 253 and 763 px against the desktop’s 10–16 k — the Mac prints 6 of 8 986 … 5 301 (4.08 %) and the Tab 6 of 8 684 … 4 802 (3.80 %): every charge within 10 %, none thinned. The strike’s charge back at the thin proportions (#88’s plan B undone) is the author’s call to make with these prints (mac/, tab/); nothing in the core waits on it. THE THIN STRIKE ITSELF, run as the author asked (“change the tablet to the spark version to see if it really helps”): a lab switch (sumi_debug_set_anod_strike_thin, a mapper flag off by default, dev only — the bench’s --strike-thin, the hook’s --ei strikeThin 1) restores #71’s composition — the small charge alone, the shear’s band and kick on the Sumi radius, no burst — and the same script and session ran again on both machines. At the session’s shear 0.6 the Mac prints 6 charges of 9 600 … 4 256 px (4.24 % lit) and the Tab 6 of 7 651 … 2 680 plus a 209-px splinter (3.31 %): the threads survive, the two thinnest at half the Mac’s. At #88’s harsh shear 2.0 — where the coordinates left the Tab TWO fragments of 151 and 127 px, 0.02 % lit, against the desktop’s six at 1.93 % — the Mac prints 6 of 5 642 … 409 px (1.50 %) and the Tab 6 of 2 201 … 302 (0.68 %): six where there were two, 34× the lit share, at 45 % of the Mac’s — the displacement field carries the thin strike’s coordinates through the Adreno; what still thins it is the phase channel’s own resampling on that GPU (the ink’s, which no payload touches: the field tier’s ink maximum, 1.5e-2, is the same as ever). So it helps, decisively at the author’s shear and by a large factor at the harsh one, without making the Tab the Mac. The classic spark of #88 stays the default; the switch stays in the core, documented, for the author to judge the thin strike by eye and restore it if they want (a one-line change in the mapper: the thin branch as the only branch). The hook is removed and the Tab carries the hook-free build, as after #88. Evidence: docs/evidence/step55b/tab/ (the *_thin prints), mac/session_thin*/.

  5. The thin strike is the strike again — the author’s call on #4’s prints — and the lab switch goes. “The thin strike as default.” So the Anod strike is #71’s composition once more: the charge at anod_drop × the Sumi radius (the default back to #71’s 0.33 from #88’s 0.57) and the spark shear episode with its band and kick on the SUMI radius, no burst — in the mapper’s strike and in sumi_gesture_tap alike (#75’s tap follows the strike); ANOD_STRIKE_BURST_D leaves the mapper a second time; burst_order_by_class stays in the ABI, unused, as #71 left it. #88’s plan B — the classic spark on a bigger charge, so that a lossy renderer had thick streamers to keep — answered a loss the displacement payload has since removed (#4: six charges on the Tab under both shears where the coordinates left two specks). The switch that ran #4’s experiment (sumi_debug_set_anod_strike_thin, the bench’s --strike-thin) is removed: the tree carries one strike, the entry carries the measurements. Tests: test_medium_binding_tables expects no burst piece and the shear’s band at twice the SUMI radius (a new check), the charge at 0.33; the bench’s tap check reads “a third of the Sumi drop”. The user’s sessions keep whatever anod_drop they hold — the author’s Tab already held 0.33. A LOOK, MEASURED SO THE PAYLOAD IS NOT BLAMED FOR IT: under the author’s iPad session (shear 2.0, τ 0.5 — the harsh preset of #88, with the thin strike) the six strikes leave ONE bright charge on the Mac; the pre-#88 bench at 76e2f5e, the same thin strike on the COORDINATE payload, built in a worktree and given the same session, leaves the same one (6 239 px against 5 829); at τ 0.25 the old payload keeps three and the new four; at shear 0.6 both keep six. The tearing at that shear and time constant is the composition’s own, on either payload — a setting to choose by eye (shear, τ), not a regression; the Tab’s session (shear 0.6) prints all six. The author, with the build on the iPad: “they look as expected, this is working quite nicely.” Evidence: mac/ipad_session/. FLAGGED for the spec: specs/TO_PROJECT_SPEC.md §10.5’s strike row and §10.6’s tap are rewritten to this; MEDIUM §4’s strike row as #71 had it (“a small charge + the spark shear”) is true again.

  6. The GL tier under the displacement payload, measured on the Linux box; the GL column’s Anod four recaptured. Step 55b’s handoff, run at 6d1e034 (libsumi 1.2.0; ctest 9/9). The §4.6 field gate on GL (NVIDIA RTX 5090, driver 610.43, GNOME on Wayland) against the re-captured Metal fixture: dx 2.44e-4, dy 3.66e-4, ink 3.91e-3, aux 0, mean 2.26e-6 — green at the unchanged 1e-2 / 1e-4 tier, where the coordinate payload sat at max 1.5e-2 / mean 6e-4 on the same box (DECISIONS_5 #44, #83): the mean fell some 270×, the max 4×, the same picture as the web tier’s fall on the Mac (#1). The composite gate holds its tier (max diff 1, the 8-bit step of #78/#83). The palette test’s Sumi four matched without an edit; the Anod four moved as predicted and their hashes are the gl column now (gate-55b/palette_hashes_gl.txt: 7bc658dd87238d03, b3121824077f8525, bc0fc6df5b03f797, eb3e16793e8608db; 5/5 after). The seventeen self-tests pass. The soak at the 32-texel tier: 47 of 48, the crossed pinch’s pair the known red (73.10 texels), the exact operators’ pre-image deviation 8.11–21.98 texels, each a few tenths under Metal’s; the ink-mass pairs bitwise this box’s Phase-6 ones for burst, chladni and chladni-field, the spark’s moved (−8.66 → −8.12 %) with its emission floor (#2). Two things learned about running the soak on this box: it steps frames through the bench window’s swaps, so it runs at the display’s pace (~7 min per operator under the step-55 pacer) and a bench window hidden behind another gets no Wayland frame callbacks and stalls — one window at a time, uncovered; and stdout block-buffered into a file lets stderr’s lines splice a SUMMARY line (stdbuf -oL avoids it). The D3D11 column waits for the Windows box. Evidence: gate-55b/ at the repo root, as the handoff asked.

  7. The D3D11 tier under the displacement payload, measured on the Windows box; the D3D11 column’s Anod four recaptured — and the bench’s paper dip found to take a variable number of frames on D3D11, which leaves the Anod prints frame-dependent (a bench finding for the author, not fixed here). Step 55b’s handoff, run at fd7a196 (libsumi 1.2.0; ctest 10/10, abi_c_compile_static the tenth). The §4.6 field gate on D3D11 (NVIDIA RTX 5090, driver 616.64, MSVC 14.44 Release) against the re-captured Metal fixture: dx 2.44e-4, dy 3.66e-4, ink 3.91e-3, aux 0, mean 2.26e-6 — green at the unchanged 1e-2 / 1e-4 tier. On this box the coordinate payload sat at max 3.9e-3 (the ink) / mean 6.85e-6 (DECISIONS_5 #83 — the D3D11 field was bitwise Step 11’s, not the 1.5e-2 / 6e-4 of the GL box): the ink’s maximum is the same 3.906e-3 (ink is bitwise the old fixture, its D3D11 deviation unchanged), the mean fell 3× with the coordinates. The four channel maxima land on the SAME texels as the Linux box’s GL run (#6: (359,228), (335,152)) and the means agree to four digits (2.2579e-6 here, 2.2582e-6 there) — the same silicon’s arithmetic through two APIs. The composite gate holds its tier (max diff 1, 25 617 samples of 1 048 576 differ; GL: 25 672). The palette test’s Sumi four matched without an edit; the Anod four moved as predicted and the first run’s hashes are the d3d11 column now (gate-55b/windows/palette_hashes_d3d11.txt: 9a3d35df09e0c37f, 1b72baa0345056a9, 1024f97d03e48f64, 493b66a2e532a203). But the rerun after the rebuild said 4/5 with four NEW hashes, and 25 runs in all show each Anod case at its column value about four times in five and at one of two other values otherwise (diag/: p0 21 : 4, p1 20 : 5, p2 20 : 5, p1-morph 22 : 2 : 1), the cases deviating independently of one another; the Sumi four never moved. Traced (diag/trace_palette_*.txt, a temporary env-gated print since removed): the bench’s t19_dip_print steps frames until sumi_read_print reports the dip’s print, and on D3D11 that takes 5 frames usually, 4 sometimes, 3 for a fresh instance — the readback is a staging CopyResource polled with Map(DO_NOT_WAIT) (§5.3), so the frame it lands on is the GPU’s timing. A Sumi scene does not care how many idle frames precede it; an Anod scene does (its state runs on the clock: the strain-glow’s phase, the episodes), so a 4-frame dip before an Anod case prints a different picture. The recorded column is the 5-frame outcome, the majority; a run that draws a 4 in an Anod case says != recorded for that case. GL and Metal poll the same way (a fence at timeout 0, a completion handler) and happened to be constant on their boxes. The same variance reaches the gesture test: its Sumi check compares a scene after the FIRST dip (4 frames, the fresh instance’s) with the same scene after a later one (5), and the two fields differ at 1–18 samples by a payload ULP (FAIL … (1 samples differ); diag/trace_gesture_*.txt, five runs); its Anod-twist check differs at a constant 78 samples (row 0 and three texels near (254,33), up to 3e-3), the cause not pinned in the timebox — both D3D11-only, both 0 on the Linux box. Proposed, not done: the bench’s dip helper should cost a fixed number of frames on every backend (step until ready, then pad to a constant), which makes the Anod prints and the gesture pairs deterministic wherever the readback is asynchronous — it would move the Metal and GL Anod columns too if their boxes’ constant differs from the pad, so it is the author’s edit with a Metal recapture, not this box’s. The other fifteen self-tests pass with 0 ^FAIL. The soak at the 32-texel tier: 47 of 48, the crossed pinch’s pair the known red (73.10 texels), the exact operators’ pre-image deviation 8.11–21.98 texels — the table is the Linux box’s to the second decimal in every (b) and (c) cell (tine 11.84, pinch-saddle 13.16, torsion 16.36, chladni 21.98, spark-shear 20.21; the erosion rates differ in the last digit for four operators), the ink-mass pairs bitwise the Phase-6 ones for burst, chladni and chladni-field, the spark’s −8.12 % with its floor (#2); the three negative controls red as required. The soak ran 22 minutes here (stdout to its own file, stderr to another — no spliced lines). Evidence: gate-55b/windows/ (the Linux box’s files sit at gate-55b/; a folder per box keeps both).

Step 56 — Suzu cells (the Mac; Voxo only)

Section titled “Step 56 — Suzu cells (the Mac; Voxo only)”

  1. Suzu is the synth’s name, and the cell is the leapfrog with its orbit re-based on retune, its filter the Chamberlin core tuned for the damped ring, both in a 2× section; every element declares its class. The author’s working name from SYNTH_SPEC — 鈴, the small bell — stands unless they override it here (the roadmap points this confirmation at “DECISIONS_7 #1”, which 55b’s entries had taken; it is this entry). Suzu is a SOURCE beside the sampler (voxo_set_source, VOXO_SOURCE_*; Voxo 0.8.0, additive): the same callback, the same normalizer-fed voice model (a SuzuVoice inside each Voice), the same bus and master gain; switching ends every voice as an instrument swap does; the desktop’s Sound section grew the source row (INI sound_source), the tablets consume it in their shell steps. THE CELL (voxo/src/suzu.h, the class table at its top): x −= ε·y; y += ε·x with the UPDATED x — the magic circle, det = 1 — at ε = 2 sin(π f / fs) so the pitch is exact (the oscillator measured within 0.000 cent across MIDI 21–108 at 44.1 k and 48 k, the struck note through the ABI within 0.001); the strike a DRIVE that sets the orbit (x = A, y = εA/2, the point where x peaks on the invariant); the release a DECLARED conformal contraction per sub-step (a T60 in the patch — contraction_for); the output the cell’s x through an attack ramp (a mixer stage, not a map on the cell) and the pressure’s level. THE MPE CORRECTION, measured (SYNTH §2.1’s [ITERATE]): the invariant of the update is x² + y² − ε·x·y, on which x peaks at A = sqrt(E/(1 − ε²/4)); a retune with the state kept changes E by −Δε·x·y, so the SHIPPED form re-bases the state onto the orbit of the same amplitude under the new ε (Cell::retune, one rsqrt per retuned sample, none while ε holds) — what the spec called the Gordon–Smith form is this recurrence (Gordon & Smith’s own is the staggered update itself); the plain form is the lab’s retune_mode 1, and 2 steps the pitch once per block without the ramp, SYNTH §2.1’s feared case. THE NUMBERS: a scripted −48 → +48-semitone glide over 2 s ripples 0.097 dB re-based (the gate 0.5) — and 0.057 plain under Voxo’s per-sample pitch ramp, 0.062 plain and stepped: at a sweep’s rate Δε per step is tiny and x·y averages out, so the feared modulation does not appear; it appears at a JUMP — −48 → +48 in one block, eight phases of the orbit: 0.082 dB re-based (the decimator’s roll-off alone) against 0.630 dB plain, the invariant’s −Δε·x·y at whatever phase the jump lands. The re-base ships because it is exact at any rate of change; the ripple that remains is the 2× section’s decimator, a 2-tap average whose gain is cos(θ/2) — −0.08 dB at C8 — the [ITERATE: budget] of §2.4 carried (a half-band FIR when the budget says). THE DRIFT TEST: the bare cell at A = 0.5 for ten minutes at 48 k — amplitude +0.0001 dB, frequency +0.000 cent; the negative control, the “simultaneous” update x′ = x − εy, y′ = y + εx (det 1 + ε²), spirals past the 0.1 dB bound within the first second and reaches +690 dB by the tenth minute — proven red. THE FILTER (§2.2’s [ITERATE], decided by the tuning test): the Chamberlin SVF at the oversampled rate, its (low, band) core the magic circle, q = 2 − 1.9·resonance the declared dissipation, CC 74 scaling the cutoff by 2^((t − 0.5)·6) as the sampler’s default timbre map; the exact tuning f = 2 sin(π fc / fs) holds for the UNDAMPED core only — with damping the update is [[1, f], [−f, 1 − fq − f²]] and its ring sits sharp of fc by cos θ = (2 − fq − f²)/(2√(1 − fq)): measured 2.3–2.5 cents at C8 for Q = 50, over the 2-cent gate. So the coefficient is solved for the damped ringing frequency per block (svf_f_for: the same equation in its cancellation-free form through 4 sin²(θ/2) and fq/(1 + √(1 − fq)), in double, three passes of a quadratic — the first form drowned at 35 Hz in float): the ring lands within 0.008 cent across MIDI 21–108 at both rates; the thematic form kept, the trapezoidal escape hatch unused; the classic form’s fs/6 ceiling sits at fs/3 of the output rate, the cutoff clamped under it. THE SHEARS (§2.4): x += g(y), cubic (g·y³) or a triangle fold, in the 2× section, the gain clamped to 1 — the fold’s ×4 waits on the budget. FTZ/DAZ (§2.4, §4): set on the rendering thread at its first block (suzu.cpp: MXCSR on x86, FPCR.FZ on arm64 — verified by a read-back, reported in voxo_stats.ftz), the 64-voice decay-tail stress flat to the end (48 ms of render per second of tail, first and worst alike); the control with FTZ off shows no cliff on this Apple CPU — the cliff is the x86 boxes’ to show, the gate is the flat line. HEADROOM: sixteen Suzu voices with the SVF and a shear, 128 frames at 48 k — 0.06 ms mean, 0.09 worst, 3 % of the callback. The suite is tests/voxo_suzu_tests.cpp (ctest, 10 suites now); the storm on the real output (--voxo-source suzu --voxo-storm) is the evidence’s other half. Carried: the quadrature y as a stereo width, the drift’s 10-minute figure re-run per box, the spec’s companions (SOUND_SPEC.md, MEDIUM_SPEC.md) which are in git history and specs/TO_PROJECT_SPEC.md §10/§12 now (the spec itself moved to specs/SYNTH_SPEC.md, where the roadmap points). Evidence: docs/evidence/step56/.

  2. Every synth ships with its sound profile; the shears act on the normalized orbit and calibrate their own detune; the bass is flat and the ear is not. The author, with step 56 on the desktop: “I like the sound; the bass notes have a strange amplitude — we will need a graph of the sound profile for every synth in the evidence and in the docs.” THE PROFILE: the bench’s --voxo-profile <dir> (with --voxo-source, --voxo-preset, --voxo-suzu-shear) strikes every MIDI note 21–108 at velocity 100 offline — held 0.5 s, released 0.3 s — and writes the run (profile.wav) with a row per note (profile.csv: the held window’s peak and RMS in dBFS, the first three harmonics by a Goertzel at f, 2f, 3f); tools/sound_profile.py draws the level across the keyboard, the harmonics per note and a spectrogram of the run. From here every source and instrument has its figure in the step’s evidence and, drafted for step 63, in the docs (site/drafts/suzu/). THE BASS, measured: Suzu’s level is flat to 0.09 dB from A0 to C8 (−24.27 → −24.36 dBFS peak), the sampler’s sine to 0.00 — the signal does not vary; the ear and the speaker do, and a pure sine at 30–60 Hz at −24 dBFS sits where the equal-loudness curve and a laptop’s cone give little back and unevenly. The remedy is timbre: the desktop’s Sound section gained the patch’s knobs (level, release, shear and its kind, cutoff, resonance — INI suzu_*) so the author can hear the cells with harmonics; the modal voice’s partials (57) are the instrument’s own answer. THE SHEARS, fixed twice on the way: as first written, x += g·y³ was a fixed kick per sub-step — at low pitch it dwarfed the tiny rotation per step and the orbit escaped (every note below C3 rendered NaN at g = 0.3, the profile’s first run); a shear is a FORCE, so it scales with ε (Cell::step_sheared: x −= ε·(y + s(y)), the hardening sense — confined: the worst peak across the keyboard at full gain and velocity 127 is 0.088 against the bare cell’s 0.25, every sample finite, both shapes). Scaled, it detunes: the cubic hardens the spring by sqrt(1 + ¾g) to first order and the note went 110 cents sharp at full gain under that compensation, the fold — not sine-like — 555; so the cell CALIBRATES ITSELF at create (shear_period_ratio: the sheared cell at unit amplitude, 80 cycles at a reference ε per gain of a 17-point table per shape, its period against the bare cell’s — cubic 1.081 / 1.144 / 1.241 at g = ¼ / ½ / 1, the fold 1.088 / 1.244 / 1.651) and the voice scales ε by the interpolated ratio: C2, C4 and C7 within 0.19 cent (cubic) and 0.02 (fold) at full gain — a gate now. And each shape acts on the orbit NORMALIZED by its amplitude (u = y/A, s = g·A·shape(u)): a fixed-scale shape either never reached its fold at A = 0.25 (the fold was a pure detune there) or swamped a soft orbit; normalized, a soft note is sheared as a loud one. What the shear buys, measured: a symplectic cubic is a gentle waveshaper — the third harmonic 33 dB under the fundamental at g = 0.3 and 23 dB under at g = 1 (−47 dBFS at C2), the fold 29 dB under at g = 1; the hard timbres are the kicked rotor’s and the Duffing cell’s (58). Odd shapes make odd harmonics (the suite’s witness is the third). Evidence: docs/evidence/step56/profile/, the suite’s log.

Step 57 — The modal voice & the breath bow (the Mac; Voxo 0.9.0)

Section titled “Step 57 — The modal voice & the breath bow (the Mac; Voxo 0.9.0)”

  1. The voice is a chain of cells coupled by a shared-potential kick, its partials placed by preset and KEPT on their ratios by an exact compensation solved at patch load; the load gate has two conditions, and the compensation’s binds first. THE LATTICE (SYNTH §2.5): N magic-circle cells (voxo_suzu_params_t.modes, 1–16, default 8), each at f₀·r_k with its declared contraction, and per sub-step (1) the coupling kick y_k += (κ·ε₀²/ε_k)·(L·x)_k from the PRE-update positions (L the chain’s Laplacian: nearest neighbours — the [ITERATE] topology, resolved as the chain), (2) each cell’s own drift and kick, (3) the declared damping. κ is relative to the LOWEST mode’s stiffness (ε₀ its ε), not the nominal fundamental’s, so the bell’s hum (r = 0.5) keeps a spring of its own; the order is the standard drift–kick leapfrog with the full S = diag(ε_k²) + κε₀²L up to a half-step shift (two kicks in a row commute and merge), so the joint stability bound is exactly λ_max(S) < 4. A mode’s CEILING is the output Nyquist (MODE_CEILING = ¼ of the oversampled rate — above it a mode folds back through the decimator): muted at the strike and the moment a glide crosses it; the chain is the alive prefix; the alive modes’ weights normalize the mix, so the strike peaks at A on every note however many modes the ceiling left (the bar’s profile still spreads 4.8 dB across the keyboard — fewer partials, less held energy). The coupling’s stiffness κ·ε₀² is CAPPED at C8’s ε₀: a bend reaches MIDI 156 and the chain must not stiffen past what the gate saw. THE DETUNE, FOUND AND COMPENSATED: a spring between neighbours stiffens each mode it touches — the first lattice played C4’s fundamental 149 cents sharp through the ABI at the default κ with the swirl’s share (a harmonic chain’s fundamental goes sharp by κ/4 to first order) — so κ would have been a pitch control before a timbre control. Not in the spec; decided here: at patch load the normal modes of M(κ) = diag(ρ_k²) + κL (ρ_k = r_k/r₀) are solved (a cyclic Jacobi in double, n ≤ 16, with its vectors) and each mode’s own stiffness d_k is moved by Newton on the eigen-derived Jacobian (∂s_k/∂d_j = V_jk², a plain fixed point stalls near the edge) until the k-th normal mode lands on ρ_k²; the voice runs ε_k·c_k, c_k = sqrt(d_k)/ρ_k, from a 129-point table over κ ∈ [0, 4] (c² stored, the root after the lerp; a 33-point grid missed by 3.6 cents where the curve bends toward the edge, 129 leaves 0.2), interpolated per block as the swirl moves, and cached on the fields that define the ratios (preset, modes, stiffness: 2.4 ms to build at 8 modes, 13.4 at 16; a knob that keeps them, κ among them, 0.8–1.9 ms). The table is solved for the LINEAR ratios ρ_k = r_k/r₀, and a mode’s ε bends below linear toward the ceiling (ε = 2 sin(θ/2): C8’s third partial at 12.5 kHz sits 5 % under, and read 5 cents off with the swirl at full), so the table keeps J⁻¹ per κ point as well (264 KB per instance) and the voice corrects the solved stiffness to first order at its pitch, δd = J⁻¹·(ρ’² − ρ²) with ρ’_k = ε_k/ε₀ the true ratios — n² multiply-adds per voice per block, the residual second order in the bend. Muted modes are WALLS, not kicked and held at zero, so the top alive partial keeps both springs (the first cut ended the chain at the first muted mode and the top partial lost a neighbour’s stiffness). Measured: the lowest three partials of the harmonic, bell and plucked presets at C2, C4, C6 and C8 within 0.08 cent at κ = 0.05 and with the swirl at full; C8’s five alive partials within 0.00 cent but the fifth, at 20.9 kHz against the wall, +1.0; at the compensation’s far end (κ = 1.483) C4’s fundamental within 0.76 cent; the solve’s own residual 5e-8. Mode SPLITTING stays what §2.5 names: two cells at the same pitch split into f₊ − f₋ of the joint map — measured against the analytic split within 0.01 % across κ = 0.0125 … 0.5 (2.7 … 91 Hz at A3) and charted once (tools/mode_splitting_chart.py); with placed ratios the coupling’s exchange of energy between partials is the feature, their pitches are the preset’s. THE LOAD GATE, two conditions in voxo_set_suzu_params at κ_patch + the swirl’s 0.5: (i) the compensation is feasible — every d_k keeps at least 1 % of the mode’s own stiffness (else “a mode’s own stiffness would be gone”); (ii) λ_max(S) of the alive, compensated chain — as the voice runs it, corrected per pitch, the walls in — stays under 4 at every semitone of MIDI 21–156 (power iteration, the sweep under a millisecond). A rejected patch returns false with the message on the log callback (the desktop prints it) and the live patch stands; voxo_suzu_coupling_bound bisects the PATCH’s bound (the swirl’s share reserved). FOUND: for every shipped preset within κ ≤ 4 the compensation’s condition binds first — the harmonic 16-mode chain’s bound is κ = 1.036 (total 1.536, where the SECOND partial’s own spring is gone) while its λ_max never leaves ~1.96 — so the sampling bound’s red control lives on the primitive: two cells at 23.04 kHz / 96 k, κ* = 0.567 — 5 % over (λ_max 4.106) blows up at once, 5 % under (3.894) stays bounded; the compensation’s red control through the ABI: the 1.05× patch with lattice_gate = 0 (the lab’s bypass, the table frozen at its last feasible row) plays C4’s fundamental 21 cents off — the partials cannot be placed. THE PRESETS (suzu::modal_table): harmonic string (r = m·sqrt(1 + B·m²), w = 1/m), stiff bar ((2m+1)²/9: 1, 2.78, 5.44, 9 …, w = 1/m), bell (0.5, 1, 1.2, 1.5, 2, 2.5, 2.667, 3 …, cast weights, the prime loudest), glass (1, 2.32, 4.25, 6.63 …, w = 1/m²) and the plucked string — Karplus–Strong in modal form, w = sin(mπp)/m² (at p = ½ the even partials are not fed: gated). The decays: γ_k = α + β·(r_k² − 1) with α = ln 1000 / decay_s — so the fundamental’s T60 IS the declared decay_s and an inharmonic preset’s highs die by their frequency; the spec writes α + βk² (FLAG: the shipped form keeps the fundamental exact and reads r, not k); decay_s = 0 declares no decay (the lab’s ledger). The ENERGY LEDGER: the bell, 8 modes, κ = 0.3, every decay zero, struck once, ten minutes through the ABI — RMS second 1 → second 600: −0.0026 dB (the envelope’s spread 0.034 dB, the inharmonic beats); decays restored (2 s, β 0.3), every mode’s T60 within 0.1 % of the declared, at κ = 0 and unchanged at κ = 0.05. The swirl (0xA0) adds up to 0.5 to κ per block through the 20 ms one-pole (the [ITERATE] binding, built as the spec’s thematic one; the author signs by ear). Patches ride the QoL preset file as a suzu object (presets/SCHEMA.md: the source and voxo_suzu_params_t field for field, kept when absent; the desktop writes and reads it). THE COST: the first cut computed a sine per mode per frame and 64 released voices cost 513 ms per second of audio (ten times step 56’s); the retune now runs only when the pitch moves (the block’s first frame, then each frame of a glide) — 60 ms/s for the 64-voice tail, sixteen voices 3.5 % of the callback (0.09 ms worst of 2.67). Voxo 0.9.0: voice_kind, modal_preset, modes, coupling, decay_s, decay_bright, stiffness, pluck, bow_onset_s, bow_position, breath_cc, lattice_gate; voxo_set_suzu_params returns bool; voxo_suzu_coupling_bound; voxo_stats_t.suzu_modes. The storm on the MacBook’s speakers, twice: 0 XRuns, 0 dropped, render max 0.83 ms, 96 fps. The profiles of the five presets and the bowed voice are the evidence’s and the drafts’.

  2. The bow is an energy servo per mode toward the breath’s target, bowed only while breath is held — and it can hold only the partials whose declared decay is slower than its own onset. SYNTH §2.6 as built: per sub-step each bowed mode’s state is scaled by 1 + g·u, u = clamp((E_t − E)/E_t, −4, +1), g = 1/(τ·rate) with τ = bow_onset_s (0.15 s default; 0 removes the bow) — anti-damped under the target, damped over it, the servo’s own rate bounded by g per step: a limit cycle by construction. E_t per mode is (A·b_k)² with A = level·(0.35 + 0.65·breath)·breath (the breath twice: a soft breath sings softly) and b_k = |sin(kπ·pos)| the bow’s profile (bow_position: 0 the fundamental alone, 1 every mode evenly — the profile-from-CC [ITERATE] stays fixed per patch). Breath is the patch’s CC (breath_cc, default 2) and CC 11 as its alias, per zone through the normalizer; a voice is bowed only while HELD with breath

    0, so no breath means no bow (silence is gated: the declared decays alone) and a released voice ends by amplitude as any other. A bowable mode at rest (E < 1e-10) is seeded at −20 dB of its target — the bow’s first grip on a resting string, a declared DRIVE. THE REACH, found: the servo’s injection tops at 1/τ (u ≤ 1), so a mode whose declared decay rate γ_k exceeds it cannot be held; the first cut re-seeded such modes each time they died (a −40 dB tick every 0.3 s on the eighth partial) — now they are not bowed at all: they ring from the strike and die. At the defaults (τ 0.15, decay 3 s, β 0.3) the first three partials of the harmonic string sing. The steady state sits UNDER the target by the decay’s share, E = E_t·(1 − γτ): measured A_ss 0.2022 against 0.2023 predicted under a 2 s declared decay at τ = 0.1 ([ITERATE]: the target corrected for the decay; μ from the breath’s attack rate, the spec’s). THE GATES (§5), on the cell: from silence (the seed) the orbit settles within 5 % in 4.9 τ, from 2× in 1.7 τ, and the two orbits agree to 0.00 %; zero breath — the servo off, the declared 2 s decay alone — falls monotonically to −66 dB in 2.2 s; over a second of steady state the injected and the extracted energy (the decay’s and the bow’s slips) balance within 0.000 % (the rotation itself injects nothing: symplectic); the RED control — the servo that gives and never takes, from 2× — is at 1000× the target’s energy in 0.43 s and 8e11× at 2 s. Through the ABI: breath 80/127 on a velocity-1 note sings at −39.4 dBFS, stable to 0.05 dB over the last second; the same tone from velocity 127 within 0.3 dB; the breath withdrawn, the held note is 84 dB down after 3 s. The bowed profile (harmonic string, breath 0.63 held under velocity-100 strikes) is a flat singing line across the keyboard (peak spread 1 dB; the second partial level with the first at bow position 0.3). The Brisa’s singing tone on the actual hardware is the author’s judgement.

  3. The binding table as built (SYNTH §3), for the author to sign by ear. Through the normalizer, every dimension one consumer:

    dimension (MPE)Suzu’s consumer
    strike (velocity)the kick across the modes by the preset’s profile — the plucked string’s pluck position in closed form, sin(kπp)/k²
    press (channel pressure)the output gain 0.35 + 0.65·p (step 56’s stage; §3’s “sustained small drive into damped modes” is NOT built — [ITERATE], the bow being the press consumer where there is breath)
    breath (CC 2, the patch’s breath_cc; CC 11 as alias)the bow’s E_target per mode (#11)
    glide (pitch bend)every mode’s ε retuned per frame, the orbits re-based, each partial on its ratio (the cell’s amplitude refreshed from its state first: the coupling moved energy the cell’s own amp did not see)
    slide (CC 74)the SVF’s cutoff, ±3 octaves about the patch’s
    swirl (0xA0, poly pressure)κ += 0.5·swirl per block (#10)
    lift (note-off)the release’s declared contraction, the tighter of the patch’s release and the mode’s own

    Not built, flagged: the strike-position profile from the layout’s cell (the [ITERATE] of §3 — a struck bar) and the bow profile from a CC. The desktop’s Sound section carries every knob when the source is Suzu and the voice the lattice (INI suzu_*); the roadmap’s “patch names” are the author’s — the five presets are named by their physics.

Step 58 — Strings & chaos (the Mac; Voxo 0.10.0)

Section titled “Step 58 — Strings & chaos (the Mac; Voxo 0.10.0)”

  1. The Verlet chain tunes itself under the CFL bound by shedding nodes, and the bound is a gate with a red control. SYNTH §2.8 as built (suzu::VerletString): M interior nodes between fixed ends, per sub-step every acceleration from the pre-update u, then v = v·d + k·Δu, u += v (symplectic Euler, a conformal per-node damping d from the patch’s string_decay_s as the fundamental’s T60; 0 = none). In dimensionless units the one number is s² = k·dt², and the m-th mode rotates by 2·asin(s·sin(mπ/(2(M+1)))) per step — the 1–2–1 Laplacian’s eigenvalues, the cell’s ε² = λ — so THE CFL BOUND s ≤ 1 keeps the top mode under Nyquist and past it the explicit scheme is unstable. Tuning couples M, k and dt: a note f₀ needs s = sin(πf₀/rate)/sin(π/(2(M+1))) ≤ 1, so the chain is REDUCED per note to floor(rate/(2f₀) − 1) nodes (48 hold to C6; 44 at C6, 21 at C7, 10 at C8 on the 96 kHz section; above rate/6 no chain is representable and the note is silent) and s re-derived — the pitch is exact by construction: the sweep MIDI 21–108 through the ABI reads 0.000 cent. The gate: voxo_set_suzu_params rejects a forced k·dt² over 1 (the lab’s string_cfl; the derived one never exceeds it) with its message; the red control — 1.05 with cfl_gate bypassed — goes non-finite within a second at C4 (25 samples on the primitive), 0.95 rings bounded. The pluck is a triangle ADDED to the state (re-plucking a ringing string, §2.8’s ask), released from rest; the pickup reads a node at pickup (0.25). A glide re-derives s per frame and saturates at the bound (the pitch stops rising where the chain cannot follow) — the chain has no re-base, so a glide ripples as a retensioned string does (not gated; the cell’s glide gate is the cell’s). The cost is honest: ten strings at 80 nodes with the SVF on take 20.8 % of the callback (0.55 ms worst of 2.67) — [ITERATE] the structure-of-arrays SIMD of §4 when the budget asks. The chain’s spectrum is the discrete string’s (its highs compress toward the top mode): the body the modal preset’s spectrum lacks.

  2. The hybrid string’s junction is a scattering junction on the cell’s synchronized velocity, discretized by the midpoint rule so its balance is exact; the bridge’s reflection phase is solved in closed form and folded into the delay; the passive bound is a point. SYNTH §2.9 as built (suzu::HybridString): a ring of N samples read N behind the write (the history stays, so a glide’s integer steps read real samples) plus a first-order Thiran allpass for the fraction (|H| = 1), a one-zero loss y = (1 − a)x + a·x[n−1] with a = loop_loss/2 (the KS averager as a declared conformal loss, |H| ≤ 1) and a round-trip factor from string_decay_s (a T60 at the note’s period), into a BRIDGE of 1–3 magic-circle cells at bridge_hz × {1, 1.618, 2.618} with their declared decays. THE JUNCTION, three forms on the way: the spec’s explicit “kick the bridge by c·s, reflect −s + c_back·v” pumped energy through its c·F² term — the first probe went to infinity at any coupling; the physical scattering form (the incident wave’s force on the bridge is 2s − v_b, the −v_b share the bridge’s own motion radiating BACK into the string, the reflection v_b − s) with the midpoint rule on y also grew, because the staggered cell’s y sits half a step ahead of x and a kick on it changes the invariant by −ε·x·δ beyond the kinetic energy; on the synchronized velocity ỹ = y − ε·x/2 the balance holds EXACTLY per sample: F = (2s − Σỹ)/(1 + Σc/2), each mode kicked by c_m·F, the reflection s − g·v̄ with v̄ the midpoint velocity (the nut’s inversion folded in on the way back), g = 1 the conserving junction. Measured: every declared damping zeroed, the loop holds 60 s within 0.004 dB at any coupling, and the ten-minute soak through the ABI at A2 reads −0.018 dB (the spread 0.037: the allpass’s one sample of state). THE BRIDGE’S SCALE: c is the string’s impedance over the bridge’s mass per sample, and a real bridge is HEAVY — at c = 0.08 the reflection’s phase swung the loop by a semitone at every note (−120 cents saturating the search); shipped c = 0.002 (range 0..0.02), at which the pull is a few cents far from the bridge’s resonance and up to π at it (the wolf). THE PHASE, compensated: the junction is linear, so its per-sample map z’ = A·z + B·s, r = Cᵣ·z + D·s is read off by pushing unit vectors through one sample and its response H(ω) = Cᵣ(e^{iω}I − A)⁻¹B + D is a 2·nb complex solve at tune time; the loop resonates where its whole phase is 2π, so arg H/ω is folded into the fractional delay with the one-zero’s phase delay — the fundamental stays on the note (the sweep MIDI 21–108 within 0.144 cent; 0.07 on the primitive at c = 0.002, 0.58 at 0.008) while the partials keep the bridge’s pull: the body. The hybrid retunes once per block (a glide steps at most a few cents per block). THE GATE, twice (§5): the load-time probe — the closed loop at C6 for 300 ms (three hundred round trips), every declared damping zeroed, plucked; the energy’s peak over the start: 1.013 at g = 1 (the allpass’s wobble), 1.07 at 1.005, 1.15 at 1.01, 3.7 at 1.05 — rejects over 1.03 with its message; voxo_suzu_passive_bound searches above 1 and reads 1.0016; and the soak. The red control: g = 1.05 with the probe bypassed goes non-finite in 31 s at A2. FOUND: the conserving junction is a passive POINT, not a half-line — under 1 the bridge still takes the full force 2s − v̄ while the string sees less of v̄ come back, the cross term (1 − g)·v̄·(2s − (1 + g)v̄) is sign-indefinite, and the probe admits 0.95 only because it does not grow at C6 (the spec’s “coupling gain 1.05× the passive bound” reads, as built, “5 % over the conserving point”); a lossy bridge would need a resistor the junction has none of — the declared decays are the bridge cells’ own. The output is the string at the pickup (both directions, half) plus the bridge’s velocity in the string’s energy units (half): the body speaks — the profile shows its response, up to 11 dB louder around the bridge’s two modes with the wolves as dips exactly at 220 and 356 Hz, the modes ringing across every note. The [ITERATE] wave-digital escape hatch stays unused: no patch failed the probe on its merits.

  3. The Duffing cell is the cubic on the position; the rotor is the standard map with its momentum as the pitch on the torus. SYNTH §2.10: suzu::Duffing puts the hardening spring in the kick, y += ε·(x + βx³) — a shear of y by a function of x, symplectic across any swing (the spec writes “g(y) = −βy³·dt”, a shear of y by itself, which is not a map of det 1 — FLAG; step 56’s cubic shear on the drift is the same cell with the coordinates swapped, normalized and detune-calibrated, which is exactly what the Duffing cell must NOT be: the clang IS the detune). The settled pitch is exact with no compensation (the cubic vanishes with the orbit): C4 at velocity 127 with β 8 clangs +258 cents sharp and settles to +0.00 by 2.5 s under a 2 s declared decay (chart/duffing_clang.png, the pitch against time over the amplitude’s straight decibel line). The drive: a sinusoid at drive_ratio × the note, its amplitude drive × the press (channel pressure), added as a force in the kick; swept 0 → 1 over 24 s on A3 (β 8, a 0.3 s decay) the spectrum holds one partial and its harmonics until 0.87 and there bifurcates into a comb of new partials (chart/duffing_drive.png: the second chaos voice’s order → chaos, as a period-multiplying window rather than broadband at these settings — [ITERATE] the drive ratio and β for the Ueda-like storm). suzu::Rotor (§2.3): the cell carries the angle; once per NOMINAL cycle — a fixed clock at the note’s period, Chirikov’s kick period — the momentum takes p += K·sin θ with sin θ read from the cell’s quadrature, p is wrapped to (−π, π] (the map on its torus) and the cell is re-based onto the pitch f₀·(1 + p/2π): the momentum IS the pitch, within the octave about the note, and the kick is a re-based retune (phase- and amplitude-continuous), never a jump. K = rotor_k + (2.5 − rotor_k)·wheel (CC 1, per zone), delta-smoothed by the 20 ms one-pole per block; the chaotic modulator may add to it. Gated: K = 0 is the pure tone (0.000 cent); K = 0.3 undamped for ten minutes holds the orbit’s peak within 0.0000 dB and |p| ≤ 1.108 (the island’s libration, 2√K); past K_c the fundamental’s share of the power falls (97 % → 2 % → 20 % → 52 % → 36 % at K 0 / 0.47 / 0.98 / 1.5 / 2.5 — the small-K figure is the wide slow libration of the island, ±2√K in p: ±3 semitones at K 0.3, the map’s own scale). The K sweep 0 → 2.5 over 24 s on A3 is chart/rotor_sweep.png: the tone, its sidebands, the band widening toward the octave about the note — the visual Chirikov’s sibling, one theorem, two senses. FLAG for the author’s ear: whether K’s musical range wants a gentler mapping of the wheel (the island already swings three semitones at 0.3).

  4. The chaotic modulator is a double pendulum under RK4 with its energy projected, one per patch. SYNTH §2.10 says a leapfrog double pendulum; the double pendulum’s Hamiltonian is not separable, so a leapfrog (velocity Verlet) is not symplectic for it — measured over ten minutes at dt 0.02: E drifted from −2.0 to +1.4 and a hard kick went to NaN. Shipped (suzu::DoublePendulum): classical RK4 in double at a fixed sub-step of 0.005 (RK4 alone holds 6.0000 → 5.9998 over ten minutes), with the kinetic energy PROJECTED onto the trigger’s value after each control step — the velocities rescaled — a declared correction that measures a no-op (within 9e-8 of 1) and bounds by construction; the class-table row says so (FLAG: the spec’s word). The energy is set at the trigger from the velocity (a kick from rest at the bottom of 0.5 + 2.5·vel, E from −2.75 to 6: below the flip energy it swings, above it tumbles), the output sin θ₂ (bounded in [−1, 1]) is smoothed 20 ms per block, a unit of pendulum time is 50 ms / mod_rate, and the value goes to one smoothed target — the cutoff (±3 octaves × depth), the coupling (+0.5 × depth), the rotor’s K (+1 × depth) or the drive — the [ITERATE] mod-matrix corner left as one row. One pendulum per patch (the instance’s), re-energized by every strike; per voice is the [ITERATE].

  5. The sampler and Suzu can sound together — the layered source — and the combined stress holds. The roadmap’s DONE asks for the Osmose storm + a heavy sampler preset + ten synth voices at once; Voxo’s source was exclusive (step 56: the switch ends every voice), so VOXO_SOURCE_LAYERED (2) is added: every note strikes a sampler voice AND a Suzu body in the same Voice, both release together, the voice ends when both have, the mixer sums them (SYNTH §4: the synth and the sampler share the mixer and the bus); switching to or from it still ends every voice. The desktop’s Source row gains “Both”; the preset file’s source carries 2. Gated: one note on the layered source is one voice whose RMS exceeds either body’s alone, and it ends when both have. THE COMBINED STRESS on the MacBook’s speakers (--voxo-storm 8 --voxo-source layered --voxo-preset <the author's Bösendorfer 280VC library> --voxo-suzu-voice 2): the fifteen-channel MPE storm with the sampler’s heavy library under the Verlet chain (48 nodes), twice — 0 XRuns, 0 dropped, render max 0.80 / 0.77 ms of 2.67, 96.1 / 94.5 fps, twelve layered voices at the end; and under the hybrid string — 0 XRuns, render max 0.39 ms, 94.7 fps. The boxes’ re-run of the suite (Windows, Linux) is the author’s fan-out, as at 55b. The bench’s --voxo-suzu-voice <kind> picks Suzu’s voice for a run (the storm’s and the profile’s) and --voxo-chart <dir> writes the chaos charts’ material (tools/chaos_chart.py draws it; the chart’s scripted run renders one settled frame first — the core’s Metal shutdown waits on a frame semaphore only a committed frame arms, and the first chart run hung there).

Step 58b — The bore & the jet: the flute (the Mac; Voxo 0.11.0)

Section titled “Step 58b — The bore & the jet: the flute (the Mac; Voxo 0.11.0)”

  1. The bore is the chain in acoustic variables on a staggered grid, its pitch its length, its ends declared ports whose radiation rises with frequency and whose reactance is an end correction the tuning counts. SYNTH §2.11 as built (suzu::Bore): Webster’s system on the Yee grid — p at the integer nodes, u (the volume velocity) at the half nodes, both in the units where the bore’s characteristic impedance at its mouth is 1, S(x) a per-node weight normalized at the mouth — u −= λ·S·Δp then p −= λ·Δu/S, symplectic Euler on the wave equation; profiles cylinder, cone S ∝ (x₀ + x)² (closed at the truncated apex) and the Bessel flare S ∝ (1 − x/x₁)^{−γ} (58c’s). THE CFL: λ²·μ_max(L_S) < 4 with μ_max the weighted Laplacian’s top eigenvalue by power iteration (3.996 on the uniform grid, 4.015 on the cone’s apex cells) — the bound λ ≤ 2/√μ_max, derived, never trusted; the lab’s bore_cfl forces a multiple of it and the gate rejects over 1 with its message; the red control (1.05×) goes non-finite in 57 sub-steps on the primitive and within a second through the ABI, 0.95× rings bounded. PITCH IS BORE LENGTH: n = ⌊λ_max·rate/(2f₀) − ends⌋ cells (open–open or the cone; a quarter-wave closed–open takes half) and λ = 2f₀(n + ends)/rate ≤ λ_max absorbs the fraction — the KS-delay tuning move — so a note’s dispersion stays under a cell’s worth; above the cell cap λ falls instead (A0 at 128 cells runs at λ 0.07, stable and near enough to harmonic for m ≪ n). THE SERIES, measured on the primitive at A4: the closed–open cylinder’s first four peaks on the odd series within 0.0 cent, the open–open cylinder’s on the integers within 0.0, the cone’s (apex 5 % of its length, counted to the apex — without it the whole series read 84 cents flat, exactly 1/1.05) on the integers within 8.7 cent, stretched +0.7 / +2.5 / +5.1 / +8.7 as a truncated cone is (the mouthpiece’s compensation is 58c’s to find). THE ENERGY, in the staggered form the leapfrog conserves exactly — ½Σ S·p_n·p_{n+1} + ½Σ u²/S — holds the closed lossless bore within 0.0003 dB for ten minutes (the symmetric p² form wobbled a quarter of a decibel at λ ≈ 1, and a first staggered form paired the wrong levels). THE ENDS are declared ports: closed is the mirror (a half cell), open is p = ∓z·(u − lp(u)) with lp a one-pole at bore_corner_hz — the radiation resistance of an unflanged pipe grows as (ka)², and this port’s response z·iω/(ω_c + iω) has a non-negative real part at every frequency, so it only absorbs: the fundamental sees little of z, the upper modes all of it (with z 0.3 and the corner at 1500 Hz the first mode’s T60 at A4 is 0.19 s, the third’s 0.077) — the bore’s own selectivity between its registers, without which the jet’s flat gain locked the sixth mode. FOUND: the port’s reactance is an inertance z·ω_c/(ω_c² + ω²), i.e. λ·that many cells of extra bore — 2.8 per end at A4 — and the flute read 81 cents flat until the tuning counted it (end_correction): the pinned-end theory is not the bore’s. A glide changes the cells and λ per block; the wave keeps circulating (the cells past the new end are dropped). Cost: cells × rate × voices, as the chain.

  2. The jet is a bandpass amplifier whose centre rises with the breath; its drive is the labium’s dipole, power-limited by the mouth per sample; the flute overblows by itself and flattens when blown softly. SYNTH §2.13 as built (suzu::Jet), the loop found in five probes: (1) the acoustic DISPLACEMENT at the flue — a leaky integral of the bore’s velocity there, DC-blocked at 10 Hz — not the velocity, carries the jet: with the velocity the phase condition landed a quarter period off; (2) the jet’s travel time τ = jet_tau·T·√(P_ref/P_mouth) — d/(αU₀) with the embouchure following the note (the delay in periods of the note at the reference breath, a v1 choice: the physical jet with a fixed distance, where the player blows harder for high notes, is the [ITERATE]), Hermite-interpolated in a 4096-sample ring (A0’s period and a half: a 1024 ring left the low half of the keyboard silent); (3) the RECEPTIVITY BAND: the sinuous instability grows fastest at one Strouhal number f·d/U₀ = f·τ, so a second-order bandpass (the Chamberlin core, jet_q) on the displacement centred at ½/τ amplifies best around the fundamental at the reference breath and around the octave at four times it — the register jump IS the band crossing the modes, and without it a full-period delay locked every frequency at once into a broadband saturation (the very soft end); (4) the labium’s tanh partition Q_in = (Q₀/2)(1 − tanh(η − y₀)), Q₀ = jet_area·U₀, η = −G·(band)(t − τ) + the stochastic vector σ·U₀·white (chiff, filtered by the bore), an inward displacement carrying the jet in (more flow); (5) THE DRIVE: a flow alone does no work at a pinned open end, so the source is the jet drive’s dipole across the labium, a pressure port at node 0 ∝ dQ_in/dt — p_src = −jet_drive·(Q − Q_prev) per sub-step, O(1): a first form carried an arbitrary ×T/2π that let the acoustic injection exceed the mouth’s work sixfold (the ledger caught it). The jet’s gain G = e^{μd} at A4 (jet_gain 560) follows the note as f² because the bore’s radiation loss does (the embouchure follows the note again). THE POWER-LIMITED PORT: the dipole never does more work on the bore in a sample than the mouth does on the jet, P_mouth·Q_in — a declared limiter (it acts on 0.5 % of the samples of the test’s phrase), so §1’s self-excited row for the winds holds by arithmetic: over a 4 s phrase (a swell, a note change A4 → C5, a release) the stored energy never exceeds 0.06 % of ∫P_mouth·Q_in. THE BREATH: P_mouth = P_ref·r with r from 1/√range to √range across the breath about breath_ref (0.44 → the reference, the note in tune; breath_range 12); no breath, no tone. THE EMBOUCHURE FOLLOWS THE NOTE, three laws in v1: the jet’s delay in periods (above), its gain rising as f/440 (the bore’s losses rise with the pitch) and its area — the flow — falling as √(440/f): the drive is a derivative, so its saturated amplitude climbs 6 dB an octave otherwise (C2 read 27 dB under A4; with the laws C2–C7 sit within 8 dB, the sub-contra octave below the flute’s range quieter still); the physical jet with one distance and one width, the player blowing harder for height, is the [ITERATE]. THE WALL: the radiation alone left the low notes nearly lossless and seconds slow to speak, so the bore carries a declared per-node damping, bore_wall_s (a T60 of 1 s) — the class table’s “string’s per-node damping” row in acoustic clothes. THE OUTPUT is the mouth end’s volume velocity (the standing wave’s amplitude there; the port’s own pressure falls as f² toward the bass and read 60 dB down at C2), scaled to sit near the cell’s −24 dBFS at A4; the labium’s offset y₀ (0.3 jet widths) is the partition’s asymmetry, where the even harmonics come from (the profile still reads the odd ones 20 dB ahead — the jet’s asymmetry is an [ITERATE] for the ear). THE PRESS BLOWS: the author, at the desk, heard nothing — the winds and the bow sound only under breath (CC 2 or 11), and the controller in hand had none. So press_blows (default on; the desktop’s “the press blows” switch): the mouth is the larger of the channel’s breath and the voice’s pressure, for the flute and the bow alike — an Osmose or an aftertouch keyboard plays them; a pure breath player switches it off. FLAG: the press then has two consumers on those voices (the gain stage and the mouth), SYNTH §3’s one-consumer rule bent where the alternative was silence; a keyboard without pressure or breath still has nothing to blow with (a velocity floor is the [ITERATE]). Gated: A4 under channel pressure alone sounds; with the switch off it stays silent. MEASURED, the delay scan at A4: the first register sustains for τ between 0.35 and 0.6 periods — +39 cents at 0.3, +3 at 0.5, −37 at 0.7 — the octave below 0.3, silence past 0.8 (the second hydrodynamic mode a whisper near 1.2); the breath ramp through the ABI, A4, 0 → 127 over 12 s, no other change: −283, −147, −59, −23, −3, +9, +21, +37, +77, +157, +265 cents second by second, then +1193 — THE OVERBLOW, the octave within 7 cents, autonomous; soft blowing flattens 32 cents at breath 32/127 against 53/127 — the τ-phase lag, free (chart/flute_ramp.png, the spectacle). The bore’s own arithmetic with centred port products does not close (injected 0.049 against radiated 0.057 on the phrase): the ports’ discrete power at the half step is an [ITERATE]; the exact conservation is the closed bore’s gate. The Brisa’s breathing flute is the author’s judgement.

Step 58c — The reed & the lips: the saxophone and the trumpet (the Mac; Voxo 0.12.0)

Section titled “Step 58c — The reed & the lips: the saxophone and the trumpet (the Mac; Voxo 0.12.0)”

  1. The valve is one kick–drift cell with a pressure force and a stop; its junction with the bore is solved implicitly, the Bernoulli quadratic against the bore’s one-step impedance, and both the valve’s swept volume and its energy are counted — the mouth-power ledger holds by arithmetic, and the naive junction is the red control. SYNTH §2.12 as built (suzu::Valve): a 1-DOF mass–spring, v −= ω²y, v += ±g·Δp, y += v, v ×= damp (Q declared), inward-striking (the reed, closing with Δp = P_mouth − p[0]) or outward (the lips, opening with it); the opening h = h₀ + y clamped at 0 and the aperture A·h; the force gain g = h₀ω²/P_close so the rest opening closes at the declared pressure. THE JUNCTION: the flow through the aperture is Bernoulli, q = A·h·√|Δp|·sgn — but Δp is the pressure AFTER the flow has entered the end node, and the bore’s closed-end update (58b’s mirror, p[0] −= 2λ(u₀ − q_b)/S₀) makes that a quadratic: q = A h √(P_mouth − p_ahead − Z q) with p_ahead the end pressure the step would produce without the flow and Z = 2λ/S₀ its one-step impedance (end_pressure_ahead, end_impedance, flow_implicit — the closed form). The naive form (q from the previous step’s Δp) is the spec’s “explicit coupling that can blow up though the continuous system is passive”: on the primitives it goes non-finite within the test’s phrase (a swell, a note change, a release) and stays finite on a steady blow at every pressure — its instability is kicked by transients — so the load-time probe blows twice (soft, 0.6 P_ref, where the reed’s local gain is highest; hard, 2.5) with the bore’s declared losses zeroed AND a retune a fourth up halfway (the state kept), and asserts E_bore + E_valve ≤ 1.05·∫P_mouth·Q after 30 ms (the two staggered energy forms wobble against a per-step work sum; the reed beating hard reads 1.02 with the losses at zero). The naive form reads 1e300× and is rejected with the message; the ABI’s red control (the gate bypassed, a bend of four semitones) goes non-finite within the second. FOUND, three things the ledger caught before the ear could: (1) the valve’s displacement moves air — the swept volume ±S_r·v into the bore in the force’s direction (the reed closing pushes air INTO the chamber: −S_r·v; a first form carried the wrong sign and the ledger read the valve doing work on the mouth); its mass is S_r/g so the valve’s energy is in the flow’s units; (2) the valve’s energy in the kick–drift’s own conserved form, (S_r/g)·½(v² + ω²·y·y_prev) — the symmetric ½ω²y² wobbles at the valve’s frequency and tripped the probe; (3) an undamped explicit valve step diverges even where the continuous loop is passive, so the probe zeroes the BORE’s losses, not the valve’s damping — the lip on the reed is structural. THE STOP (the reed’s only): the reed meets the lay and rests closed at h = 0 — an inelastic collision, the velocity dies, the potential falls (a loss: the ledger only gains margin). Without it the reed swung on through into a closed phase clocked by its own resonance and the first register let go above C4. The lips keep their swing-through: they meet each other softly, and the trumpet’s registers were tuned on it (with the stop the lips reopened at once and the tuning went). THE NOISE: the breath’s turbulence is σ·P_mouth·noise on Δp, the perturbation a static reed grows from (a reed at rest is a fixed point); the reed’s is low-passed at 2 kHz (white noise at the sub-rate seeded its upper registers), the lips’ white. ζ: with A 0.6 the reed’s coupling ζ = Z_c·A·h₀·√(2/P_M) was 4.3 — ten times a real reed’s — and the bore’s returning wave ran to eight times the mouth pressure; at 0.14 it is 0.8 and the mouthpiece pressure swings about ±P_mouth as the textbook’s does.

  2. The saxophone is a cone truncated at a quarter of its length with a mouthpiece holding the missing apex’s volume, its bell and its wall loss scaled with the note, a quasi-static reed, and its intonation calibrated at four notes when the patch loads. THE CONE (SYNTH §2.11’s S ∝ x², closed at the truncation, open at the mouth) is normalized at the REED end (normalized at the mouth its throat impedance read 880 and the junction saturated). THE MOUTHPIECE: a bare truncated cone’s series is stretched (+36 cents at the fourth peak at apex 0.1); Benade’s rule — a chamber holding the missing apex’s volume — brings it back: a cylinder of the throat’s area and a third of the cut cone’s length (cone_mouthpiece), and the pinned cone’s first four peaks sit within 0.1 cent of the integers, the ported one’s within 31 (the radiation’s end correction shrinks with frequency). THE TRUNCATION decides the register, and it took a direct impedance measurement to see it (a sinusoidal flow at the throat, |p[0]|/|q| scanned about each harmonic — an impulse’s Goertzel had listed the peaks in the wrong order): at apex 0.1 the cone’s first peak at A3 is 3.5 throat impedances against 7.1, 9.5 and 10.3 for the next three — the FIRST PEAK IS THE WEAKEST of a deeply truncated cone (the cylinder’s is 71 against 18, 9, 6) — and the reed took the strongest, the third register with a slow modulation at the note’s period, which every estimator read as a wandering fundamental; at C6 (0.9 against 5.8 for the eighth) nothing held. At apex 0.2 the peaks are 11, 21, 21, 16; at 0.25 (the default: an alto’s ratio, the missing cone a fifth of the body) the first is within a factor two of the second and the first register holds; at 0.3 the fourth peak stretches 3 %; at 0.5 the chamber swallows the third and fourth. THE SAX SCALES WITH THE NOTE (as the flute’s embouchure does, #19): its bell’s radiation corner and its wall loss are declared at A3 and follow the pitch — corner·f/220, T60·220/f — so every note is the A3 cone in miniature (measured: A3 11/21/21, C5 18/27/24, C6 27/29/25); with the corner fixed the high notes’ peaks sat under the bell’s cutoff and the reed took the octave from F#4. THE REED is quasi-static: reed_hz 12000 with Q 0.7 (its response settles in two sub-steps; at 2500 the fifth harmonic of C5 met the resonance and the register hopped; past 20 kHz the kick–drift’s ω nears 2 and the reed misbehaves), its stiffness the closing pressure 3 P_ref, its rest opening 0.5, its flow factor 0.14. THE ONSET ISLAND, measured through the ABI over 104 onsets (C3–C6, breath 60–127): apex 0.25 with A between 0.12 and 0.15 starts every note in its first register; 0.22 starts 17 in the third, A 0.18 starts 10, apex 0.3 with A 0.1 starts 20 — the defaults sit in the island’s middle (0.25, 0.14), and the mouth pressure rises through an 80 ms one-pole so a hard blow on C3 does not seed the third. THE BREATH: P_mouth = P_ref·(1.1 + 0.55·breath) — from just under the reed’s threshold (P_M/3 in theory, 1.15 P_ref with the losses) to 0.55 of the closing pressure at full breath (at 0.6 two notes thinned and read +23 cents at the top; at 0.7 some squealed to the fourth register). FOUND BY THE AUTHOR: a first map started at 0.7 P_ref, so the sax said nothing until the breath passed 50 of 127 and then jumped to nearly full level — a pedal or a light press never reached it (“I hear nothing”), while the trumpet whispers from 5. Now a note held under a slow ramp speaks at 4–10 of 127 (measured through the ABI at 0.3, 2 and 6 s ramps, by CC 11 and by pressure). THE DYNAMICS, declared: a beating reed’s bore amplitude grows 1.7 dB from the threshold to full breath where the lips’ grows 14, so the sax’s level follows the breath itself (0.32 + 0.68·breath, −10 dB at the threshold, smoothed as the pressure is) — the player’s crescendo, the physics adding the brightening on top; the press adds the gain stage’s own (SYNTH §3’s two consumers, #19). THE RELEASED VOICE’S MOUTH GOES TO ZERO: the pressure is what smooths, and a released voice’s target is 0 — a first form smoothed the breath and mapped it, so a released sax kept blowing at half the reference and its tail read as the next note’s fundamental a second and a half later (the test heard the previous note, 300 cents down, within 15 dB). THE INTONATION: the reed’s pull is measured at load — C3, C4, C5, C6 blown 0.45 s at 1.4 P_ref, the period of the last 0.2 s against the note (−13, −18, −20, −22 cents at the defaults) — and the bore is cut to cancel it, lerped in octaves between the four (held flat outside); the calibration is keyed by the embouchure’s fields and the kind, ~150 ms, once per change, logged. THE PERIOD, not the lowest partial: a sax’s first register carries its fundamental under the second harmonic, and “the lowest partial within 15 dB” read it as the octave and mis-cut the bore; the period is the autocorrelation’s first maximum within 0.08 of its top (the mean removed — a DC offset flattens it; 0.005 read a jittery cycle as chaos), refined by a 4-cent Goertzel scan — in the calibration and the gates alike. MEASURED through the ABI at breath 70/127: C3–C6 every third semitone within 5 cents (the worst C6 at −5); the cylinder with the same reed, for the record: the textbook square wave at γ 0.47 but period-doubled (r(2T) 1.000 against r(T) 0.962), periodic at γ 0.37, 0.4 and 0.6 — Maganza’s cascade on the near-lossless Raman model, a T60 of 0.1 s suppresses it; the cone at the defaults shows no doubling. THE LEVEL: the mouth end’s velocity ×1.0 sits A3 at −24 dBFS at full breath (−30 at 0.55, the profile’s); C2–C7 within 6 dB, the fundamental and the second harmonic within a few dB of each other at half breath and the fundamental ahead from C4 up; below C2 the cell cap (128) thins it — the sax’s range. The ramp chart at A3: the tone from a light breath, swelling 10 dB and brightening as the reed closes further — no register change, the sax’s own spectacle is its steadiness.

  3. The trumpet is a cylinder with a Bessel flare, its pitch on the bore’s third peak, the lips’ resonance an outward valve that CC 74 bends an octave either way — the registers — with the intonation calibrated as the sax’s and a declared bell shear for the brass. THE BORE (BORE_TRUMPET): S = 1 along the tube, the flare S = (1 − t)^{−γ} from bell_start (0.6) with t reaching 1/(1 + ε), ε 0.1, γ 0.7, to the mouth; the radiation port scaled by the mouth’s area (in tube units the flare blew up); the bell’s brassiness a §2.4 shear in the last cell’s (p, u), p −= brass·u³ — amplitude-driven, det 1, declared (the spec’s v1 stand-in for the shock). THE PEAKS of a flared bore are not a harmonic series from its quarter-wave fundamental, so they are MEASURED at load (bore_peaks: a 200-cell reference bore, a 0.6 s pulse, a 2 % Goertzel scan, prominence ×3, threshold 0.2 % — a first threshold found only the first) and the note is placed on the partialth (the third: the trumpet’s written middle register), the bore cut so that peak lands there. THE LIPS: outward-striking, Q 3, rest opening 0.05, closing at P_ref, area 0.5, at CC 74 centre their resonance 0.95 of the note (an outward valve plays above its resonance, on the bore’s peak: the pull reads +89, +87, +86, +92 cents at C3–C6 and the calibration cuts the bore to cancel it — before it the trumpet played 151 cents sharp); CC 74 bends the resonance ±lip_range octaves, so at 0 the lips sit on the peak below and at 127 on the peak above — the register key is the embouchure, as the spec’s [ITERATE] said it would be. FOUND: at lip_ratio 0.9 the lips at full CC 74 sat between the fifth and sixth peaks and notes from F#3 fell to the pedal; at 0.95 they land on the sixth — the octave — for every note C3–C6 (measured through the ABI: CC 74 at 0 → −772 cents on all thirteen, at 64 → +14…+25, at 127 → +1276…+1296). THE BREATH: P_mouth = P_ref·(0.5 + 2.5·breath), the attack a 20 ms one-pole (the lips choose their register at once). THE LEVEL: ×4.0 sits A3 at −17 dBFS; C3–C6 within 1 dB, C7 at −26 — the trumpet’s range; below C3 the bore wants more cells than the cap (its fundamental is the note’s third) and the bass falls away. The embouchure chart at A3, CC 74 swept 0 → 127 over 16 s: the registers as a staircase — the second peak, the third (the note), the fourth, fifth, sixth — each bending within itself as the lips pull it, the pedal showing through where the lips cross between peaks. The trumpet held its gates from its first build; it was the sax that took the step.

  4. Flags for the author (the spec folds). SYNTH §2.12: the reed’s stop at the lay (an inelastic collision, a loss); the swept volume ±S_r·v and the valve’s energy in the kick–drift’s form; the implicit Bernoulli junction against the one-step impedance (the naive form the red control); the probe’s design (two blows with a retune, the bore’s losses zeroed, 5 % after 30 ms); the breath noise low-passed for the reed; ζ in a real reed’s range (0.8). §2.11: the cone normalized at the reed end; the mouthpiece as the missing apex’s volume (Benade); the truncation ratio decides the register — the first peak of a deeply truncated cone is its weakest, and an impulse’s peak listing is not an impedance measurement; the sax’s bell corner and wall loss scaled with the note (the flute’s law again); the trumpet’s flare with ε and the port scaled by the mouth’s area; the peaks measured at load; the bell shear. §3: the intonation calibration at four notes per embouchure (the pull lerped in octaves), logged; the period as the pitch measure; the released mouth at zero; the sax’s breath map 1.1 → 1.65 P_ref (from its threshold) with an 80 ms attack and its level following the breath (a declared 10 dB, where the lips’ physics gives 14), the trumpet’s 0.5 → 3 with 20 ms; CC 74 as the trumpet’s register key (the [ITERATE] resolved: lip tension); the press blows the reed and the lips as it does the jet and the bow (#19). Carried [ITERATE]s: the reed channel’s flow inertia (a lowpass on the flow that would favour the first register physically, where the truncation does it geometrically); toneholes and a register vent; the two-mass lips; the physical reed at 2–3 kHz with the lip’s damping (the quasi-static reed is v1’s); the ports’ discrete power at the half step (#19); a sax below C2 and a trumpet below C3 (the cell cap — 256 cells or a sub-rate for the bass). The steadiness measure (the period per 50 ms window, its spread) is in the scratch probes, not the suite: the suite’s gates are the pitch, the register and the ledger.

Step 59 — The orbit trace & the phase close (the Mac; Voxo 0.13.0)

Section titled “Step 59 — The orbit trace & the phase close (the Mac; Voxo 0.13.0)”

  1. The synth draws itself from its own state: each traced voice keeps the last 85 ms of its orbit in a lock-free ring, and a poll from any thread takes what is new, decimates it curvature-weighted to a handful of segments and hands it over unit-normalized with its amplitude. SYNTH §2.7 as built, Voxo’s side (voxo_set_trace, voxo_trace_poll): the rendering thread writes one point in eight sub-steps (12 kHz) into a 1024-point ring per voice when the voice’s kind is in the mask — one shared word, the write index, and the reader keeps its own cursor per slot, re-synced by the voice’s serial (a new voice in the slot starts a quarter ring back at most) — so the audio path is untouched: the rendering with the trace on is BIT-IDENTICAL to the rendering without it (gated for the cell, the rotor, the hybrid string and the sax), and a mask of 0 costs nothing. THE PAIR per voice kind: the single cell’s own (x, y); the lattice’s (Σx, Σy) over its modes with the mix (the output and its quadrature); the Duffing cell’s and the kicked rotor’s (x, y) — the rotor’s momentum on its torus is what scribbles; the strings and the winds, whose state is a chain, trace their output against its scaled derivative, (s, ṡ/ω) with ω the note’s — the phase plane of the sound itself, honest where no single cell exists. THE DECIMATION: of n points keep at most K + 1 — the first, the last, and the points where the running weight (the turning angle at each point plus half the step length in orbit radii) crosses each K-th of its total, so the bends get the vertices and a straight run gets few; the polyline comes back divided by its peak radius over the window, the radius beside it, so the shell scales by amplitude × its trace scale (the spec’s law). MEASURED: the single cell at A3, eight segments asked — nine points, all on the unit circle within 0.4 % (the magic circle’s orbit is a circle; the amplitude reported 0.175, the level’s); ten rotor voices polled at four segments answer with five points each, finite; the hybrid string’s phase plane comes back with its note’s amplitude.

  2. The gesture route is the shell’s bridge, Voxo → libsumi at frame rate, no core change: the orbit lands at the note’s cell centre from the layout probe’s own table, as tine or wake segments, budgeted at twenty-four a frame over all voices with the overflow merged within each voice; the scope lives in the settings window as a miniature of the canvas. THE PLACEMENT: the shell scans the probe once per layout (a 96 × 54 grid of sumi_layout_probe, the first cell centre each note answers with — a multi-echo layout’s first echo; rebuilt when the layout, the aspect or the params change), so no core query was needed; the rolls answer nothing and their notes stay unplaced (a scrolling sheet has no home for a trace: the scope draws them at the centre, the ink skips them). THE EMISSION: each kept vertex pair is one sumi_add_tine (the mouse’s convention — alpha 0.035, the magnitude the segment’s length in canvas heights; an exact pass) or one sumi_add_wake (tip a quarter of the trace radius, clamped 0.005–0.08; sub-stepped — class by inheritance under the strictest-member rule), placed at cx + x·r/aspect, cy + y·r with r = amplitude × scale. THE BUDGET: the shell’s calls bypass the mapper’s per-frame budget (they go straight to the deform queue, which holds 4096), so the bridge budgets itself — at most 24 segments a frame over all voices; over it, every voice keeps the same share of its polyline, at least one segment, its vertices re-sampled evenly (merged within the voice, never one voice culled while another draws — the echo rule’s spirit); the mapper’s own 64 stay untouched by construction. MEASURED on the bench (--trace-test, ten rotor voices under a held press for 240 frames, K swept by the wheel): peak 20 segments a frame of the 24 (ten voices × ⌊6 × 0.4⌋), 4448 inked over the run, 9600 merged by the budget, none unplaced; the bridge’s own cost 0.17 ms a frame (the poll, the placement, the emission). THE TOGGLE IS CLEAN: the field after the run with the trace OFF is bit-identical to the run with no trace object at all (2 097 152 bytes equal); with the trace ON it differs in 1.97 million — the ink lands. FOUND, a core matter for the author (the core is frozen this phase): the same script run twice UNTRACED reads bitwise in Sumi (0 bytes differ) but not in Anod (1.1 million of 2 097 152 differ) — the strikes’ episodes are not bit-reproducible run to run through the gesture ABI either, with a 600-frame quiet start and the dip between runs; the gate therefore runs in Sumi and Anod’s reproducibility is logged here for Phase 10’s core reopening. THE SCOPE: the spec’s live composite overlay at the voice positions would be a core pass — the canvas window is the core’s swapchain on every backend (Metal, GL, D3D11) and the desktop’s ImGui lives in a second window — and the phase never touches libsumi, so the scope route ships as the Sound section’s miniature of the canvas: the same polylines at their cells, held voices bright, released ones dim, unplaced ones in the centre in blue, with the frame’s counts beneath (voices, polled, inked, merged, the peak against the budget); non-destructive, and outside the dip by construction. The on-canvas overlay is the [ITERATE] for a core reopening. THE BINDINGS: per voice KIND (bit k of trace_kinds), the kicked rotor on by default — the chaos voice scribbling its own noise into the water is the demo the spec asked for; trace_scale 0.25 canvas heights per unit amplitude (a cell’s amplitude is the level, 0.25 at velocity 127: a full-velocity note traces a 0.06-height orbit, a soft one a fifth of that); 6 segments per voice per frame (4–8); the stroke tine by default (exact, and the comb’s look), wake as the alternative; the two routes each their own switch; all six in the Sound section, the INI and the preset file. Voxo’s mask is 0 when neither route is on — no capture at all. The trace scale and the rotor’s default ON are the author’s taste sign-off (the step’s inputs).

  3. The demo, and the phase’s close. --trace-demo <dir>: Anod on the chroma grid, a chord of four rotor voices struck in turn and held under the press while the wheel sweeps K from 0 to 2.5 over twelve seconds, then released — the orbits inked as tines at their cells, eight segments a voice, scale 0.35 — the composite exported at 720 × 720 whenever the last export has landed, the frames encoded with ffmpeg: docs/evidence/step59/rotor_anod.mp4, the rotor scribbling its chaos into Anod, the synth drawing its own phase portrait in ink (a still beside it). Phase 8’s spec folds queue for the author in #23 and here: the trace’s per-kind pairs (§2.7’s “the voice’s cell” for the chain voices is their phase plane), the poll and the decimation as the ABI, the shell-side budget (the gesture ABI bypasses the mapper’s), the scope’s home, Anod’s run-to-run reproducibility. The phase end is the author’s: the tag v2.0.0-alpha.3 on the desktops with the synth played on every device, the tablets regression-checked (they compile Voxo 0.13.0 unchanged and stay on the sampler), and the fold of this file into DECISIONS.md Part VII.

  4. The scope view is a live-composite pass in libsumi (1.3.0, additive): the shell hands the composite up to 128 segments a frame and it draws them screen-locked, over the medium or instead of it — the spec’s on-canvas scope route, with the author’s ask on top: the water hidden and the polylines alone in the main window. The author, playing #25’s scope in the settings window: “can we add a checkbox to replace the medium with the scope in the main window”. The main window is the core’s swapchain on every backend, so the ask is a core addition — the phase’s rule kept libsumi untouched, and this entry records the one exception, chosen over a platform overlay per shell (an NSView on Metal, GL draws on Linux, nothing on D3D11 — rule 5: the core identical on every platform). THE PASS: the composite shader gained the scope’s uniforms in the plate guide’s own pattern (dbg_cells, DECISIONS_5): a mode, a count and 128 segments (x0, y0, x1, y1, normalized), drawn at the end of the fragment as the distance to the nearest segment in texels — a line a texel and a half wide with a faint halo, amber — SCOPE_OVER on the medium, SCOPE_REPLACE on the scope’s dark glass with the medium not drawn (the water underneath keeps marbling and the ink route keeps inking; only the view changes). LIVE PATH ONLY: the renderer passes the scope to the swapchain composite alone — like the live ripple, the print, the export and the bloom’s source never see it (gated: with the scope replacing the water for 240 frames of ten rotor voices, the field and the dip’s print are bit-identical to the run without it, 2 097 152 and 1 048 576 bytes). THE FIXTURES HOLD: mode 0 is the shipped composite — the composite gate reads bitwise on Metal (0 of 1 048 576 channel samples differ, the negative control red as required) and the field gate holds the phase invariant (no field pass changed). THE ABI: sumi_set_scope(inst, points_xy, strip_lengths, strips, mode), the segments copied, mode 0 or no strips clearing it; sumi_version 1.3.0 (the Step-33 minor-bump pattern: additive, nothing moved); the C compile test takes its address. THE SHELL: the bridge packs its placed polylines (sixteen voices at eight segments fill the 128) and sets the scope each frame the trace runs, clears it once when the trace stops; the Sound section’s “Suzu trace on the canvas”: off / over the water / the scope alone (the water hidden) — the INI’s and the preset’s trace_canvas; off by default (the author’s taste). The composite’s cost with the scope on: 128 segment distances per fragment at most, on the live frame only; off, one uniform compare. The shader regenerated for the four dialects (metal_macos, hlsl5, glsl410, glsl300es); the tablets compile the core unchanged in behaviour (mode 0).

Step 59b — Suzu on the web: the engine in a worklet, the flute panel (the Mac; Voxo 0.14.0) — added by the author, 2026-10-04

Section titled “Step 59b — Suzu on the web: the engine in a worklet, the flute panel (the Mac; Voxo 0.14.0) — added by the author, 2026-10-04”

  1. Suzu runs in the browser as itself: all of Voxo compiled to a standalone wasm with no audio device, driven by an AudioWorklet that owns it, talking to the page by messages — and one additive Voxo call reads a voice’s internal state for the pages. The author, with six standalone visual drafts (visuals/, untracked): “we can do better using directly a wasm version of suzu.” A spike answered first: all of Voxo compiles to wasm UNCHANGED with a stub for the device — 110 KB, the sampler’s parsers dropped at link because nothing reaches them — and renders every voice kind. The drafts each rewrote the physics in JavaScript, and some showed the opposite of the engine: the flute’s register picked by pressure thresholds (Suzu’s overblow is emergent, #19), the reed coupled explicitly (the form Suzu’s load gate rejects, #20), strings coupled across voices (Suzu has no coupling between voices). The author added the steps (59b the host and the flute, 59c every family) and moved the phase end to 59c. AS BUILT: a none backend (voxo/src/backend_none.cpp: voxo_start answers false, the host renders — the web, node, and the gate’s native reference); Voxo’s CMake builds one source list with either backend, the web branch fetching pugixml, miniz and dr_libs (dead code in the lab’s link) and no miniaudio. The lab’s wasm (web/suzu/suzu_web.c → build-web/suzu- dist/suzu.wasm): STANDALONE (no emscripten JavaScript glue: an AudioWorkletGlobalScope has no fetch, timers or TextDecoder), fixed 32 MB memory (the worklet’s views never detach; Voxo’s instance is 3.5 MB), 122 KB, ZERO imports; a flat surface — the parameters by NAME through a table generated from voxo.h’s struct field for field (75), MIDI bytes in, interleaved stereo out, the trace and the inspection as documented float records, Voxo’s log lines kept for the host — so no JavaScript depends on a C struct’s layout. ONE ENGINE, TWO HOSTS: web/suzu/site/suzu-engine.js wraps it with nothing DOM-bound, and both the worklet (suzu-worklet.js: renders each quantum, takes MIDI and parameters between quanta, posts a snapshot ~60 times a second) and the node gate import it. Messages, no SharedArrayBuffer: GitHub Pages serves it without cross-origin isolation headers. THE INSPECTION (voxo_suzu_inspect, Voxo 0.14.0, additive): a copy of each sounding voice’s state — per kind, documented in voxo.h: the cell’s and the rotor’s (x, y), the lattice’s modes with their ratios and bow targets, the Verlet chain’s shape and velocity, the hybrid’s loop read around the ring with its bridge cells, the bores’ pressure, flow and area along their length, the flute’s jet read from its delay line at x·τ (the jet’s shape from the flue to the labium, the engine’s gain at the labium) with the block’s mouth pressure, U₀, τ, gain and area, the valves’ opening, displacement, velocity and flow. Read between renders (the rendering thread, or no device): it copies live state without a lock, and the rendering is bit-identical with or without it (gated). The flute’s per-block values are written to five floats in its voice for it; nothing in the render reads them. The patch calibration of the sax and the trumpet runs on the worklet’s thread when their parameters change (0.3 to 0.8 s in wasm): the worklet mutes for the quanta after a slow apply — a 59c concern, the flute calibrates nothing.

  2. The web gate: the wasm against a native build of the SAME flat surface over a Voxo built as wasm computes — bit for bit where the math library agrees, a declared −80 dB where it does not; the page itself proven in headless Chrome. tools/suzu_web_gate.mjs renders tests/fixtures/suzu_web_script.txt (every voice kind: two notes on two MPE channels under breath, a bend, a release, the trace and the inspection snapped three times) through the engine module in node, and tests/suzu_web_reference.c renders it natively through web/suzu/suzu_web.c — the lab’s own surface — linked against voxo_nofma: the same sources with no device, -ffp-contract=off (wasm has no fused multiply-add) and -O3 whatever the tree’s build type. FOUND, twice: (1) with the desktop’s default contraction the output differs from the first millisecond; without it seven of nine kinds are BIT-IDENTICAL to the wasm, the chaotic rotor included; (2) a Debug reference read the lattice differently — at -O1 and up the compiler rewrites some math-library calls (pow(2, x) as exp2(x)), so the reference is optimized as the wasm is. The remaining differences are the math libraries’ last bits — Apple’s libm against emscripten’s musl — where a voice’s path calls them on arguments that differ: the lattice’s sinf per sample under a glide (its first difference falls on the script’s bend, block 150), the flute’s tanhf from the first block, the trumpet’s powf. THE VERDICT: bit for bit on the audio, the trace and the inspection for the cell, Verlet, the hybrid, Duffing, the rotor and the sax; within −80 dB of each record’s peak for the lattice (measured −113), the flute (−92) and the trumpet (−728: one ulp); the negative control (one reference sample moved by 0.25) red. THE REPORT against the SHIPPING desktop build (contraction on): −68 to −111 dB for every kind but the rotor, which diverges as chaos does with a last-bit difference — the statistics, not the samples. Making the web and the desktops render bit-identically (Voxo without contraction everywhere, Suzu’s own tanh and pow) is the author’s input on the step; not done (the default). THE PAGE GATE (tools/suzu_lab_gate.mjs): the page’s ?gate=overblow renders a breath ramp — A4 held, the breath 0 → 127 over 12 s, then 3 s at the top — OFFLINE through the same worklet (OfflineAudioContext: the script rides in processorOptions, deterministic), reads the pitch with its own estimator every quarter second and posts it; GREEN in Chrome 154: the audio finite; mid-ramp on the note (closest 0.4 cents); the jump at breath 116/127 with nothing programmed; the last second on the octave (+4.4 cents); soft blowing flattens (−46 cents at breath 32 against +3 at 56) — the 58b chart’s spectacle, in a browser. The cost: 15 s rendered offline in 227 ms; one flute voice 38–61 µs a quantum in the live worklet (1.3–2.1 % of its time), 72 µs for two voices in node; the marble’s own web gate stays green on the rebuilt tree.

  3. The flute panel: the bore’s standing wave and its envelope, the jet swinging across the labium, the phase plane, the spectrum with the note’s harmonics, and the sounding pitch read from the audio — the page holds no physics. web/suzu/site/ (index.html, lab.js, lab.css; the site’s paper in the light, a scope’s glass in the dark; a phone stacks the panels). THE BORE: the engine’s pressure node by node inside the tube drawn from its area, the flow dashed, and the ENVELOPE — an RMS per node the worklet accumulates on every quantum over a quarter second, drawn as √2·RMS: one half-sine on the first register, two humps with a node in the middle on the octave, the overblow made visible. FOUND by the drawing, measured in node: (1) a single snapshot is jagged though the field is not chaotic — the odd harmonics sit 13 dB under the fundamental (harmonics 1–8 hold 88 % of the bore’s pressure energy), so a peak-hold envelope caught every crest; the RMS at 375 samples a second shows the mode; (2) the grid’s own shortest waves — near the staggered grid’s cutoff, slow by its dispersion — hold 6 % of the bore’s pressure energy at breath 70 and 21 % at full breath, driven by the jet’s sharp switching (the same with the jet’s noise at zero), inaudible at the mouth end, loud in a snapshot: the drawing averages along the bore ([1, 2, 1]/4 twice: the 8th harmonic of a 94-cell bore kept at 96 %) and its caption says so; the engine is unchanged (a gentle grid-scale damping in the bore is an [ITERATE]). THE JET: the delay line read along it, scaled to its displacement at the labium (where the engine applies the gain), compressed past three jet widths so the swing stays on the page; the labium’s edge at its offset y₀; the share entering the bore, ½(1 − tanh(η − y₀)). THE PHASE PLANE: the sound against its slope, (s, ṡ/ω), the last two periods read from the audio at the device’s rate — the flute’s orbit trace is this very pair (#24), but its poll decimates ~16 ms (seven periods of A4) to 16 segments, which aliases the orbit into a streak: the trace serves the ink route, the scope reads the sound. THE PITCH: the autocorrelation’s first lag within 0.08 of its maximum, refined by a parabola (the desktop calibration’s measure, #21), the register as the nearest multiple of the note, the last 12 s strip- charted against the breath. PLAY: a keyboard C4–C7 (the flute’s range; the computer’s A…K row from C5), the breath slider on CC 2, the breath ramp (12 s, the overblow), four embouchure knobs (the jet’s delay and gain, the breath range, the wall loss) sent as parameters, Web MIDI in (a breath controller plays it). Served locally by tools/web_serve.py --dist build-web/suzu-dist (localhost is a secure context); deployed at the docs step with the rest (the documentation-timing rule). The author’s drafts are superseded by the engine for the flute; their other ideas — the A/B switches, the bore beside the reed’s portrait — are 59c’s.

Step 59c — The Suzu lab: every voice family (the Mac; Voxo 0.15.0) — added by the author, 2026-10-04

Section titled “Step 59c — The Suzu lab: every voice family (the Mac; Voxo 0.15.0) — added by the author, 2026-10-04”

  1. One host, six panels: the pages share a core and hold no physics; the red controls are switches on the engine’s own lab parameters, and the worklet, not the page, keeps them off the speakers. web/suzu/site/: lab-core.js (the audio graph and the MIDI a controller would send, the keyboard, the pitch read from the sound — 59b’s estimator — the spectrum, a portrait with persistence, a strip chart, the bore view, the red controls’ banner, the navigation, and the offline runner every page’s browser check renders through), a page per family (cell, modal, strings, chaos, flute — 59b’s page moved onto the core — and winds) and a front page (index.html: a card per panel, how faithful the lab is, what the red controls are). The panels: THE CELL — its orbit from the ring at full density, the conserved form over whole turns, the shears combing harmonics in; red: the naive update, the plain retune. THE MODAL VOICE — a bar per mode at its ratio with the bow’s target and where the servo settles (#33), the servo strip, the lattice’s orbit, the presets, the coupling against the load gate (the refusal shown as the engine words it); red: the gate bypassed. THE STRINGS — the Verlet chain’s shape, the hybrid’s ring and bridge, the modal pluck, A/B by one switch (the author’s drafts asked for it); red: the CFL number forced over with the gate bypassed. THE CHAOS VOICES — the rotor’s Chirikov section (θ, p) at each kick from the ring’s aux channels, K on the mod wheel, the Duffing cell’s stroboscopic section and its clang; the page links the canvas’s Chirikov operator page and the operator draft links back (one theorem, two senses; DOCS_ROOT in lab-core.js — the lab’s address is the docs step’s, step 67). THE WINDS — the bore with the valve drawn at its blown end, the valve’s portrait (its displacement against its velocity, from the aux channels), the ledger as a live meter and strip (#32), CC 74’s register staircase for the trumpet; red: the naive junction on a lossless bore, the valve gate bypassed, a bend to trip it. THE SAFETY (the worklet): FOUND with the naive cell — Voxo clips its mix at full scale, so a runaway voice never leaves the range: it sits at ±1, a full-scale wave, while its state reaches 10¹² and beyond. So a quantum that reaches |x| ≥ 0.98 (the lab’s voices sit under 0.5) goes silent at once, and a non-finite sample or twelve such quanta in a row is a BLOW-UP: the page is told (the banner says which gate’s reason it was) and the engine restarts from the staged parameters. Nothing a page offers reaches the speakers at full scale. THE CHECKS: each page’s ?gate=<name> renders a scripted demonstration offline through the same worklet and measures it with the page’s own code (tools/suzu_lab_gate.mjs runs them all; --shots page[:arg][:light][:phone] captures the live pages through the DevTools protocol, the colour scheme and a phone emulated). GREEN in Chrome 154, every page (the evidence’s suzu_lab_gate.txt).

  2. Voxo 0.15.0 (additive): the trace’s density and its recent points with two aux channels per kind; the winds’ mouth-power ledger in the inspection. The 59b poll merges ~16 ms into segments for the ink route; a portrait wants every point. voxo_set_trace_decimation (1..64 sub-steps a point, default 8) and voxo_trace_recent (the last ≤ 1023 points of one sounding voice at full density, oldest first: x, y and the kind’s aux pair — the lattice’s mode 0, the hybrid’s bridge cell, Duffing’s drive phase and amplitude for a stroboscopic section, the rotor’s momentum and K, the flute’s η at the labium and the flow in, the winds’ valve displacement and velocity; voxo.h documents each). THE LEDGER (SYNTH §5): the winds’ inspection carries k[8], the mouth’s work ∫P_mouth·Q since the strike (accumulated in the render per sub-step in double, only while the mouth blows), and k[9], the energy the bore and the valve hold — k[9] ≤ k[8] always, the losses take the difference. Gate 26 in voxo_suzu_tests: the cell’s 1023 recent points keep x² + y² − εxy within 7e-5 of itself (the orbit undecimated); the rotor’s aux shows one kick a period (4 expected, 4 seen) at the wheel’s K; the sax’s and the trumpet’s ledgers read every block for a second hold (the held energy at worst 0.52 and 0.14 of the work); a render read at density 2 every block is bit-identical to the unread one. The node gate’s script now sets density 2 and its snapshots carry the recent points: every kind’s recent trace bit for bit or within the kind’s declared −80 dB (the lattice −106.5, the flute −106.0). suzu.wasm 124 024 bytes.

  3. What the panels measured, each page’s check: the engine as shipped behaves as its gates say, and four drawings had to learn the physics first. THE BOW is a proportional servo, not an integral one: each mode settles where the bow’s push balances its own decay, E/E_t = 1 − γ_k·τ (γ_k = ln 1000 / T60 + the stiffness term, τ the bow’s onset 0.15 s) — the first check against the target failed by 2.6 dB, the check against this balance holds within 0.38 dB mode by mode (the harmonic string, A3, breath 80, T60 3 s); with no decay and no coupling the fundamental sits ON its target (0.031 dB). The bar view draws both ticks. The bell at κ 0.4 stays in tune (its first four peaks at +0 cents of 0.5, 1, 1.2, 1.5); at κ 1 the load gate refuses it, saying why. THE HYBRID’s body rings loud beside its string (the bridge modes at 341 and 472 Hz, −3 and +2 dB against the note at A3 — #14’s design), so a period estimator mixes them: the page reads the spectral peak at the note (+0.0 cents). The Verlet chain −1.3 cents, the modal pluck +0.1; the CFL gate refuses k·dt² = 1.05 and the bypassed chain blows up in its first quantum. THE ROTOR below K_c: the tori confine the momentum to the primary island (|p| ≤ 1.07 rad at K ≈ 0.3, the island’s half-width 2√K ≈ 1.1); past it (K = 2.5) the momentum spreads into the sea (1.57 rad against a uniform sea’s π/√3 = 1.81 and 0.62 confined) — a first check asking for the whole circle failed because the period-1 island about θ = π stays stable to K = 4, and the sea surrounds it. DUFFING struck hard at β 8: +226 cents in the first 80 ms, +0.0 at 2 s. THE NAIVE CELL grows 26 dB in 0.38 s at A3 before the lab catches it; the leapfrog’s conserved form drifts 7e-6 dB over 2.5 s. THE NAIVE JUNCTION: the valve gate’s probe measures the bore + valve holding 5.6e7× the mouth’s work and refuses it; bypassed, the bend trips it at 0.30 s. THE CONE’S DRAWING: along a cone the pressure grows toward the apex as one over the distance, so the narrow end took the scale and the sax’s bore looked still; it is p·r that stands as a sine, and the bore view draws the cone’s pressure times the local radius (the caption says so; the trumpet and the flute are drawn as they are). THE VALVE’S OPENING is averaged over the snapshots (one catches the lips shut). FOUND, an engine [ITERATE] for the author (58c, not changed here): THE SAX at A3, breath 70, sounds +18 to +19 cents by the period estimator, natively and in wasm alike, at every block size 64–512; its spectrum is QUASI-PERIODIC — the fundamental on the note (0 cents), the second harmonic split into 440 Hz (−16 dB) and +18 cents (0 dB), the third into +12 and +24 cents: a ~4.6 Hz sideband family. The sweep is bistable near the calibration’s pressure — breath 50: +1, 60 … 110: +19 … +14; the calibration’s step blow lands in the lower regime, the voice’s 80 ms onset in the upper (59b’s suite log read −3 cents for A3 within its own sequence: the regime follows the history). A reading, not measured on the bore: the cone’s second resonance sits sharp of twice the first and the reed’s oscillation shares itself between them. The winds check’s bound (the first register within 40 cents) holds; the cone’s resonances (the truncation’s correction) are where a fix would go.

  4. Flags for the author. (1) SYNTH §7 still lists the web build among the deferrals; steps 59b–59c brought it forward at the author’s word (the roadmap says so) — the spec’s line is the author’s to strike. (2) The roadmap’s “the mode splitting” on the modal panel: the shipped voice has no unison pair (every preset places distinct ratios; the splitting was measured at 57 with two bare cells in the suite, #12), so the panel shows the coupling’s exchange of energy and keeps the splitting chart as the draft’s figure; a live splitting needs a unison preset — an engine change, not this step’s. (3) Which red controls the public page exposes, and the panels’ taste: the author’s sign-off (the roadmap’s author input). (4) The older Suzu drafts say “for the docs, step 63”; the roadmap’s docs step is 67. (5) Safari and Firefox were not run (the browser gate is Chrome’s). (6) The lab’s address: the drafts and the Chirikov operator draft link /suzu/…, beside /marble/; the docs step decides.

Phase close — 2026-10-04

Section titled “Phase close — 2026-10-04”
  1. Phase 8 closes: Voxo has a synth, the synth draws itself, and the synth runs in the browser as itself; the fold, and what carries. The author, with the Suzu lab’s six panels green on the Mac (“I think we are good”): fold the phase. Shipped across steps 55b, 56–59 and 59b–59c (Voxo 0.8.0 → 0.15.0, libsumi 1.2.0 → 1.3.0): the field stored as a displacement (#1–#7); Suzu, a source beside the sampler — the symplectic cell with its shears and its sound profile (#8–#9), the modal lattice with its compensated coupling, its load gate and the breath bow (#10–#12), the Verlet chain under the CFL bound, the hybrid string’s scattering junction, the Duffing cell, the kicked rotor and the chaotic modulator, the layered source (#13–#17), the acoustic bore and the jet — the flute that overblows by itself (#18–#19) — the valve, the cone and the flare — the saxophone and the trumpet with the mouth-power ledger as their gate (#20–#23); the orbit trace through the gesture ABI and the scope view, the phase’s one core touch (#24–#27); Suzu in a standalone wasm under an AudioWorklet, the web gate bit for bit against a native build, the lab’s panel for every family with its red controls (#28–#34). The tablets compile every Voxo and stay on the sampler. THE FOLD: this file merges into DECISIONS.md as Part VII (continuing step 55b’s #1–#7; this entry its last), the Phase-8 section of ROADMAP_5.md moves to ROADMAP.md as Part 7, the evidence of steps 56–59b leaves the tree with the changelog’s step entries as its condensation (git history keeps it at df110c9: the profiles, the charts, the rotor’s video step59/rotor_anod.mp4, the lab’s screenshots; the figures the docs need are already in site/drafts/suzu/), step 59c’s evidence rode its own commit (66c0a30, the fold on top at 5111748) and left the tree the day after; the chart scripts already live in tools/ (sound_profile.py, chaos_chart.py, mode_splitting_chart.py, the two Suzu gates) — nothing else outlived the folders. specs/SYNTH_SPEC.md is folded into specs/TO_PROJECT_SPEC.md §13 corrected to what shipped (its sections in order, every [ITERATE] resolved by the entry named — #12’s binding table, #23’s and #26’s spec folds, #28–#34’s lab, §7’s web-build line struck) and leaves the tree, as SOUND_SPEC did at the Phase-7 close (git history keeps it: git show 5111748:specs/SYNTH_SPEC.md; the author, the day after the fold: “fold the synth spec into the project specs”); the transcription found the jet’s gain law written as f² in #19 and in voxo.h’s comment where the code scales it as f/440 — the comment corrected to the code’s law on the author’s word, §13.2 states it, #19’s wording stands. The author’s six standalone visual drafts (visuals/, #28 — superseded by the lab) left the tree the same day (git history: 5111748). CARRIED to the author: (1) the tag v2.0.0-alpha.3 (the phase’s pre-release; its notes the v2.0.0 section) and the devices played — the synth on the desktops, the lab in the browsers, the tablets regression-checked; (2) the sax at A3, quasi-periodic about 18 cents sharp at the playing breath (#33, an engine [ITERATE] from 58c); (3) the bore grid’s shortest waves under the jet’s switching — a gentle grid-scale damping (#30); (4) the D3D11 bench’s variable dip frame count (#7, from 55b); (5) Anod’s strikes not bit-reproducible run to run through the gesture ABI (#25, logged for Phase 10’s core reopening); (6) the [ITERATE]s the entries carry: the press as a sustained drive into damped modes and the strike-position profile from the layout’s cell (#12 — Phase 9’s, where the layouts know the cell), the chain’s SIMD when the budget asks (#13), the hybrid’s wave-digital escape hatch (#14), the Duffing drive for the Ueda-like storm and the wheel’s mapping of K (#15, the author’s ear), the modulator per voice and its mod matrix (#16), the physical jet with one distance, the jet’s asymmetry, the ports’ discrete power at the half step and a velocity floor for keyboards with neither pressure nor breath (#19), the reed’s flow inertia, toneholes, the two-mass lips, the physical reed and the bass below the cell cap (#23); (7) the web and the desktops rendering bit-identically (#29, not done, the default), Safari and Firefox unrun, the lab’s address and which red controls the public page exposes (#34); (8) a unison preset for live mode splitting (#34); (9) the older Suzu drafts and the operator drafts say “step 63” where the open roadmap’s docs step is 67 (#34). Phase 9 (Instruments) opens DECISIONS_8.md at step 60, its #1 the fingering confirmation the roadmap names.

midi-sink 2.0.0-rc.5 · AGPL-3.0 · source