Modal2D

The forward question starts with an object: choose its outline, element formulation, support, material and excitation, then compute its response. The inverse question starts with a musical requirement: find a geometry whose relevant partial reaches a target pitch while retaining a useful decay and strike response. The second task repeatedly calls the first; it does not replace the mechanics with a table of recorded sounds.

A mode frequency alone is an incomplete instrument specification. The mode also has a spatial shape, a damping rate and an excitation weight. Two objects can share a fundamental yet have different upper partials. Two strikes on one object can select very different mixtures of the same modes. Modal2D makes these quantities editable and inspectable before assembling them into a performance.

This separates three meaningful experiments: geometry changes alter the solved spectrum; strike changes alter the mixture of an existing spectrum; playback transposition scales that spectrum as a musical operation. The first two explore the mechanical model, while the third makes a finite instrument library cover a larger score.

The native system has two main layers. libmodal owns geometry, assembly, eigenmodes, excitation and modal synthesis behind opaque object/result handles. The viewer owns the interactive editor, OpenGL displays, tuning orchestration, score playback and the surrounding audio-processing chain. Read-only queries expose meshes, frequencies, mode shapes and amplitudes without making the interface duplicate the eigensolver.

A structural edit invalidates the relevant solve. The solve worker can assemble and compute a replacement result while the viewer remains available; the application installs that result when it is ready. A strike on an unchanged object instead updates modal excitation. Those two actions have very different costs, so responsive playback does not imply that every mesh rebuild finishes within one frame.

CUDA is used for the small dense eigensolve path and for evaluating audio samples. OpenGL serves a different role: geometry rendering, overlays and the exterior-field compute passes. The sparse eigensolver and the geometry-search logic run on the CPU. This division makes the data flow more informative than a single GPU-enabled label.

The workspace includes sandbox experimentation, instrument editing, testing, instrument sets, refinement, Music and import/export. Strike, repeated excitation, pluck, bow, sweep, region and sustained-interaction tools explore different driving conditions. Some later excitation and nonlinear voice options are synthesis or control extensions around the modal model; they are not automatically a nonlinear finite-element simulation.

• Build: outline and physical parameters → mesh → stiffness/mass → retained modal basis.
• Play: event location, direction and strength → modal weights → sound and displacement.
• Inspect: mesh, spectrum, mode envelopes and field diagnostics read the current result.
• Revise: a new geometry or material starts another solve; score and instrument metadata preserve the design context.

Other branches extend the representation: beams include rotation, strings use tension, thin plates model bending, and thicker plate or shell formulations introduce their own kinematics. Later solid experiments use tetrahedral meshes. These are different approximations to different mechanical regimes; switching between them is not simply changing the visual style of one universal solver.

The thin-plate route builds a curvature–displacement operator from transverse displacement and nodal rotations, with lumped translational mass and rotary inertia. Its DKT-style implementation fixes Poisson’s ratio to 0.3 inside the bending constitutive block. The thick-plate route adds transverse shear. The shallow-shell route composes membrane and bending blocks and adds height-gradient coupling: bending a curved surface can require stretching it, adding resistance to motion.

These choices determine which comparisons are meaningful. A string dominated by pre-tension can approach a harmonic series; a flexural bar or plate has different overtone spacing. Selecting a richer formulation is useful when it adds the missing motion, not simply because it has more degrees of freedom.

Planar outlines use constrained triangulation with quality and area controls. Hole contours change connectivity as well as removing material: they alter the available paths for stiffness and can localize motion. Porous presets therefore affect the mode structure, not just the object’s silhouette. The solid experiments use structured tetrahedral construction and can carve straight tunnel-like voids through the volume.

Mesh refinement should be judged by a sequence of solves: compare selected frequencies, mode shapes and strike response while decreasing the element area. A stable-looking animation at one resolution cannot reveal how far a partial has moved numerically. Extremely slender geometry is especially demanding because stiffness scales in different directions can become very different.

Disconnected parts introduce another issue. A global solver can return arbitrary mixtures of equal-frequency motions on independent components. The component solve isolates each structural block, embeds its modes into the full mesh, then merges the spectra. This keeps an instrument bar’s motion associated with that bar. Coupled cavity scenes use a shared structural basis because the air explicitly links the system.

Modal2D needs the vibration frequencies of a particular object, including its supports, material and thickness. FEM turns those choices into stiffness and mass; the retained eigenmodes then make playback inexpensive. Accuracy still depends on the mesh, boundary conditions and mechanical formulation. This is a different objective from Crucible2.5D’s evolving material histories and large deformations. FEM can simulate those too, while MPM also uses a weak-form grid discretization. The distinction comes from the model and what each project needs to compute.

The live mass–spring preview is a separate approximation, not a smaller copy of the FEM mesh. It solves a small lattice with lumped nodal masses, axial springs and diagonal links for shear response. Its simplified geometry and stiffness do not preserve arbitrary outlines, holes or support conditions. Use its patterns as qualitative editing feedback; use the full FEM result to judge the instrument’s pitch. A separate adjustable cavity bandpass is likewise a one-resonance sound proxy, not a solution of the cavity’s spatial pressure modes.

The geometry-sensitivity route saves work in a different way: it retains an already solved FEM eigenvector, perturbs the mesh width, reassembles stiffness and mass, and compares Rayleigh quotients without another eigensolve. This remains a FEM-derived estimate. The current helper targets the basic planar triangle route; it does not provide a general derivative for every formulation. A small perturbation along an isolated mode branch can guide a search, but a fresh solve must check the proposed geometry. Near crossings or after large shape changes, the old mode may cease to be a useful predictor.

The dispatcher currently tries the float32 cuSOLVER generalized symmetric eigensolver for systems of at most 512 degrees of freedom. Larger systems use the CPU sparse route; a dense-path exception also falls back to sparse. The threshold is an implementation policy, not a measured performance law for every GPU or every mesh.

Shift–invert makes the low structural eigenvalues easier to extract. The code factors K − σM once, repeatedly solves against M times a trial vector, and builds a Krylov basis with M-inner-product reorthogonalization. It solves a small projected symmetric eigenproblem and transforms the resulting Ritz values back into frequencies.

The CPU route uses a sparse shift-invert Lanczos construction with a projected solve. Its current implementation is a fixed-subspace method; an older thick-restart description should not be read as an implemented restart mechanism. A dense cuSOLVER route provides an additional GPU path for suitable problem sizes. The available routes have different memory and problem-size tradeoffs, rather than a demonstrated universal speed hierarchy.

Replacing a mode’s onset whenever another hit arrives would abruptly discard its previous ring. The synthesis state instead stores multiple amplitude/onset slots per mode and sums their responses. Rapid strikes then add new attacks while earlier hits continue to decay. The ring has finite capacity, so this is a bounded playback representation rather than unlimited event history.

The CUDA audio kernel assigns one thread to an output sample. That thread evaluates every retained mode and its active strike slots, applies the modal radiation coefficient, and writes a mono sample. The viewer’s audio chain then performs the selected voice and processing stages before stereo output.

Audio and visual time have separate purposes. Slow-motion displacement makes an oscillation readable, while audible playback runs on the audio clock. The native callback is configured for 44.1 kHz and 512-frame blocks, about 11.6 ms of samples per block. That block duration is not an end-to-end latency measurement; device buffering, synchronization and processing also contribute.

A single strike starts a free decay. With continuous contact enabled, Bow and Bow+ instead apply a force that depends on the bow’s speed relative to the same modal body at the contact. The audio-rate model uses implicit midpoint integration and a regularized friction law, so the vibrating body feeds back into the excitation. Rub uses a filtered stochastic force. Releasing a contact lets the accumulated modal state decay.

Continuous contact is enabled by default. Turning it off restores the earlier repeated-impact Bow/Bow+ and noise-driven Sustain routes; the latter uses band-pass biquads centered on solved modal frequencies. The current friction model uses bounded iterations, a force limit and an aggregate-work safeguard for simultaneous contacts. It approximates stick–slip behavior in normalized mechanical units; it does not model bow width, contact heating or a fluid-coated finger.

A frequency sweep provides another experiment: the largest forced response occurs near a retained resonance, with damping setting peak width. A driven nodal pattern can be compared with a free mode, but a displayed Chladni-style field is not itself a simulated bed of moving grains.

Tuning is a substantive inverse-design problem. A geometry may satisfy a frequency target while exciting that mode weakly, decaying too quickly or producing competing partials. Finding the right note is central, but it is not universally the only difficult part: amplitudes, strike location and the desired spectrum matter too.

Bharaj et al., Computational Design of Metallophone Contact Sounds (SIGGRAPH Asia 2015), optimize object geometry through deformation and perforation to match target frequencies and amplitudes. Their method explores a complex optimization landscape using Latin Complement Sampling. It goes beyond finding a sweet spot on an already tuned object.

Modal2D takes a more interactive, guided route with its present search tools: inspect the modes, audition the object, adjust the material or base design, then search a constrained parameter range. Multiple search methods are available, but our most dependable workflow so far remains a 1D search supported by manual decisions. This states what the tool currently does well without claiming the paper’s more general shape-optimization pipeline.

A tuner first needs a scalar observation. It may use the lowest elastic mode, a manually pinned mode, the strongest audible mode or a mode within a decay window. Those policies can return different answers on the same spectrum. Pinning and auditioning the intended partial makes the target reproducible.

The shared tuning cycle builds a candidate, waits for its solve, applies or reuses the intended strike, reads the selected response and proposes another geometry. The visible cents error is a logarithmic frequency ratio, so octave-spaced targets have comparable tolerances. The search should also preserve its best measured candidate when a later proposal worsens the result.

A small error certifies the selected pitch condition at the measured geometry. Timbre, decay, excitation and support remain separate requirements. A useful tuned note stores its shape and relative strike point together, so performance excites the same motion that was inspected during tuning.

Bisection is effective when a continuous mode branch is bracketed across the target. A preliminary sweep can find several such brackets instead of trusting one global trend. Alternating width and height handles two parameters, but can waste work on an axis to which the selected mode barely responds.

Fine-Grad measures local width and height elasticities with probe solves. It takes a damped minimum-norm step in log dimensions, bounded by the search region and a maximum change. This gives an interpretable direction: move the dimension whose local influence best reduces the measured error.

Experimental Bayesian width or width/height searches model the response and choose informative next evaluations. Calibration tools fit a local size-to-frequency relationship from actual solves, predict a useful next geometry and then refine it with the physical solver. A surrogate prediction is a proposal, not a completed instrument: the next eigensolve supplies the actual frequencies.

The current range driver begins from the chosen family, height, shell setting and material. It samples widths across its supported range, solves each candidate and records usable low frequencies. For each requested note it picks the nearest measured candidate in cents, then delegates to the existing refinement process. Targets far from all sampled candidates are reported as missed.

This differs from the earlier two-solve calibration design, whose power-law fitting helper remains in the source and tests. A size law can be a good local approximation, but a mixed or poorly conditioned family may change mode ordering or lose a low mode as geometry changes. A measured table avoids assuming global monotonicity; its finite sampling still cannot prove that every achievable note was found.

The polish step remains the meaningful completion path. The optional unpolished branch seeds a library entry with target metadata, so its displayed zero error should not be treated as an independently verified solve. The brochure’s demonstrated workflow uses refinement and a pitch check after material or geometry edits.

• Choose the base geometry and material; select the score’s needed range.
• Measure usable candidate widths and inspect the attainable frequency span.
• Seed each target from the closest observation, then refine the actual geometry.
• Keep missed targets visible; check a saved note’s pitch before building the ensemble.

The score supplies a pitch, onset, duration and velocity. The physical source supplies solved modal frequencies, damping, mode shapes and the excitation weights at a playable contact. Imported instruments, manually tuned instruments and hybrid ensembles reuse these ingredients to fill notes that were never physically tuned in the source library.

Selection comes first. Fit score pitches prefers an exact note label, then the same note name in the closest available octave, then a nearby chromatic note. Among equally distant contacts, the lower source note wins. Thus C4 is preferred over B5 for a requested C6 when octave-first fitting is enabled. Imported libraries also respect their configured shift limits; manually tuned instrument and hybrid fitting can extend a single octave across a score.

Transposition comes next. Every retained modal frequency is multiplied by the same ratio: borrowing C4 for C5 doubles the spectrum; borrowing C4 for C♯4 multiplies it by about 1.05946. Relative partial spacing survives, including inharmonicity, but this does not guarantee unchanged timbre, decay or perceived tuning. Large shifts are especially different from redesigning the instrument. An optional measured-pitch correction uses the selected contact's measured modal pole instead of trusting its nominal note label; that pole is not a guarantee of the perceptually dominant pitch.

There is no new finite-element solve for a fitted note: the mesh, material and spatial mode shapes stay as they were. In manually tuned instrument and hybrid playback, events using the same source and frequency ratio share a state and can excite its resonance. Different ratios use independent modal banks. Their combined visualization does not imply mechanical coupling between those transposed copies. Manual contact playback still excites the original, unshifted body.

Preserve fitted-note level compensates the extra amplitude scaling introduced by transposing the bank. It keeps high borrowed notes from becoming artificially faint; it does not normalize every note to the same loudness. The sound still comes from the model, with no recorded instrument sample, but the frequency and level fitting are musical playback operations. Exact-only and legacy nearest-contact options remain available for comparison.

A simple size law can work for a restricted shape family. Once width, height, cavities, supports or materials change independently, the audible partial need not move monotonically. Modes can exchange order or character near crossings, and a mode that reaches the target may barely respond at the chosen contact. Searching only for the nearest eigenfrequency can therefore produce the right number and the wrong instrument.

Physical auto-tuning needs repeated solves, mode tracking and a check that the target partial can actually be excited and heard. The tuning and exploration tools investigate that problem. Score fitting intentionally takes a cheaper route: reuse a solved response and transpose it so the music can play now. It is one of the clearest departures from the project's physical construction, kept for convenience rather than presented as a substitute for solving the inverse problem.

In Hybrid ensemble, click a pad to hear its routed sound and contacts; Shift-click selects silently. Auditioning uses a separate buffer, keeping the rendered full score ready to play again. The live score, illuminated pads, layer colors and contact links follow whichever performance is currently heard, including every activated set when ranges are layered.

Manually tuned instruments remain visible as complete moving bodies. Selecting an imported source also opens an enlarged gallery view while keeping its source-pad thumbnail. Borrowed notes connect to the actual mesh node where that source is excited.

Instruments and coverage exposes the selected source's Body material and imported Strike X/Y controls. Material changes rebuild its physical model; moving a strike reprojects the existing modes without another eigensolve. The material selector distinguishes known recipes from imported or custom values.

Body vibration, cavity pressure and exterior medium have independent display switches. Motion magnification spans 0–8×; cavity and medium exposure each span 0.125–16×. Reset display reference clears the held visual normalization, and setup files retain the gains and visibility choices. These controls change the display, not the sound: cavity pressure comes from the coupled solve, while the exterior medium remains a radiation proxy.

Each newly imported instrument file becomes one named set containing its note bodies. Copying the current custom-instrument library also creates one set. Lowest note and Highest note include both endpoints; allowed note names filter target pitches within that range. Member limits intersect the parent set's coverage, so a one-note material variant stays local. A manually tuned instrument acts as an independent set.

A set's configured range can exceed its physical notes: with pitch fitting enabled, one octave may cover MIDI 0–127. Use physical note span selects its authored endpoints; Full MIDI range permits 0–127 and all note names. Show coverage on pads highlights the configured allowed range even when another source wins some notes. Fit grid to this range changes the visible octaves, not the routing limits.

Layer overlapping instrument sets is off by default. One source then wins: exact note first; with octave-first matching enabled, the same note name precedes another pitch class; nearest pitch and source order resolve the remaining choices. Turn layering on and one best contact is selected within every eligible set, then those sets play together. Thirteen imported notes from one file therefore supply one suitable contact, rather than thirteen simultaneous copies.

For example, a harp allowed C3–C6 and a bell set allowed C4–C5 both play C4 when layering is on; only the harp plays C3. A manual override changes the contact in its own set and may bypass its range, while other eligible sets still participate. Disabled sets or members remain excluded, and muting a pitch suppresses every layer for it. Pad colors and live connections expose the selected layers and their actual contacts.

Older imports without file-group provenance remain independent. Group legacy imported notes as one set is an explicit migration that preserves their member limits. New groups, ranges, overlap policy and coverage highlights persist with the Hybrid setup. The original import mode stays unchanged. Audio layers are summed before one final normalization; the mix does not create mechanical coupling between independent sets, nor assign different timbres to MIDI tracks sharing the same pitch.

The Music grid displays the instruments that cover the score, with strikes linked back to their source objects. Per-instrument material assignment allows a set to mix different mechanical responses. A material change can invalidate a stored pitch: the instrument needs another solve and check before the old tuning label can be trusted.

The ensemble file carries more than note names. It retains instrument geometry, material metadata, strike coordinates and arrangement settings so a session can be reconstructed. Older files without per-instrument material fields use compatibility defaults. Importing an additional library and replacing the current one are deliberately different operations.

Offline audio rendering and WAV export turn the design into an output artifact. The visual performance remains useful for inspecting which source plays each note; an exported phrase supplies the listening comparison. A complete evidence capture pairs the note mapping with the generated audio and records whether reverb, nonlinear voice or pitch shifts are enabled.

The selected-note inspector reconnects the performance to its mechanism: geometry, material, strike location and modal information remain visible. Ensemble persistence and audio export make the result reusable. The MIDI importer handles the supported note, timing and part data, but does not claim a complete General MIDI synthesizer with every controller, program and pitch-bend behavior.

• Inspect one saved note: geometry, actual pitch, material and strike location.
• Build the grid and review missing or borrowed assignments before playback.
• Change one material or contact parameter, recheck the relevant pitch, then replay the same phrase.
• Save the ensemble and export WAV with the processing state recorded.

Sound Analysis separates two sources. Solved modes builds a modeled line spectrum from modal envelopes and damping widths; it is useful for inspecting a single object's mechanics. Heard output analyzes the final stereo samples sent to playback, including rendered scores, fitted pitches and downstream effects. Auto selects heard output for Music and the continuous instrument voices, so the waterfall continues to follow the sound when playback comes from a rendered buffer.

With heard output selected, the spectrogram and waterfall use windowed FFT analysis; cochleagram and note-band views reorganize that spectrum. Stereo powers are combined without cancelling opposite-phase channels. The callback's sample clock supplies fresh data, and pause holds the spectral history. Restarting or seeking clears the old segment. The CQT-like view remains a chromatic band display rather than a separate constant-Q transform. Per-mode fields and decay diagnostics still require solved-mode data: an audio mixture alone does not identify the physical mode shapes of its instruments.

The audio chain includes optional sustained and nonlinear voices, gating, filtering, soft limiting and stereo reverb. Reverb offers a Schroeder–Moorer-style comb/all-pass route and partitioned convolution with an impulse response. Comparing a modal envelope with a post-effect waveform distinguishes what came from the object from what came from the listening-space and voice processing.

The primary guides span several generations of the application. The technical details here follow the inspected implementation where those accounts differ: the sparse solver has one fixed Krylov build, the plate kernel’s concrete choices include lumped mass and a fixed internal Poisson ratio, and Music’s active range driver samples candidate widths. The long source notes remain available as design history and deeper mathematical context.

A useful next validation campaign would connect element convergence, eigenpair residuals, shape perturbations and audible partial selection. For acoustic coupling, the next model improvement is signed interface quadrature and explicit open-boundary treatment. For nonlinear instruments, a reduced nonlinear mechanical model would connect amplitude-dependent pitch and energy exchange to structural parameters.

The source includes focused tests for mechanics and later workflows, including tuning scale behavior, music rendering, MIDI handling, persistence and acoustic extensions. This brochure's source audit did not run a fresh native build or those tests, so their presence is reported as implementation evidence rather than a new pass claim. The screenshots are genuine desktop captures.