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acoustic coupling would be required for a physical sound-pressure prediction, while this release provides declared sonification only; people and cultures may make music.", "physicalChain": [ "geometry-topology-metric-boundary-material", "declared-operator", "eigenmodes", "excitation-and-damping", "structural-motion-and-mechanical-energy", "explicit-acoustic-coupling-or-declared-sonification", "audio-signal", "bounded-musical-perceptual-cultural-interpretation" ] }, "conversionEdges": [ { "doesNotPreserve": [ "unmodelled material detail", "real-world validity" ], "from": "geometry-topology-metric-boundary-material", "id": "specification-to-operator", "operation": "Choose a model class and its operator from all declared structural inputs.", "preserves": [ "declared units", "boundary assumptions" ], "status": "physical-model", "to": "declared-operator", "unitsIn": [ "metre", "kilogram", "newton-per-metre" ], "unitsOut": [] }, { "doesNotPreserve": [ "canonical basis inside a degenerate eigenspace" ], "from": "declared-operator", "id": "operator-to-eigenmodes", "operation": "Solve the finite or analytic eigenproblem under its assumptions.", "preserves": [ "operator spectrum", "multiplicity" ], "status": "exact-mathematical", "to": "eigenmodes", "unitsIn": [], "unitsOut": [ "dimensionless", "inverse-square-metre", "radian-per-second" ] }, { "doesNotPreserve": [ "units", "physical interpretation" ], "from": "dimensionless-graph-laplacian", "id": "bare-graph-to-hertz", "operation": "REFUSED unless mass and stiffness or another dimensional physical bridge are declared.", "preserves": [], "status": "no-canonical-conversion", "to": "hertz", "unitsIn": [ "dimensionless" ], "unitsOut": [] }, { "doesNotPreserve": [ "actual object validity" ], "from": "P4-plus-M-plus-K", "id": "graph-physical-bridge", "operation": "omega_i=sqrt((k/m)*lambda_i) for the equal model.", "preserves": [ "mode ordering under positive common k/m" ], "status": "physical-model", "to": "modal-angular-frequency", "unitsIn": [ "dimensionless", "kilogram", "newton-per-metre" ], "unitsOut": [ "radian-per-second" ] }, { "doesNotPreserve": [ "phase", "amplitude" ], "from": "radian-per-second", "id": "angular-to-cyclic-frequency", "operation": "f=omega/(2*pi).", "preserves": [ "time scale" ], "status": "exact-mathematical", "to": "hertz", "unitsIn": [ "radian-per-second" ], "unitsOut": [ "hertz" ] }, { "doesNotPreserve": [ "radiation efficiency", "spatial pressure", "hearing conditions" ], "from": "structural-motion", "id": "motion-to-pressure", "operation": "REFUSED: v0.1 supplies no acoustic coupling, propagation medium, receiver, or pressure calibration.", "preserves": [], "status": "no-canonical-conversion", "to": "sound-pressure", "unitsIn": [ "metre", "metre-per-second" ], "unitsOut": [] }, { "doesNotPreserve": [ "waveform shape", "physical amplitude", "mechanical energy", "acoustic pressure" ], "from": "selected-modal-frequency", "id": "mode-to-pcm", "operation": "Map selected frequencies to deterministic integer triangle phase accumulators.", "preserves": [ "declared target-frequency ordering" ], "status": "declared-sonification", "to": "PCM16", "unitsIn": [ "hertz" ], "unitsOut": [ "pcm16" ] }, { "doesNotPreserve": [ "SI energy", "loudness", "pressure" ], "from": "joule", "id": "mechanical-energy-to-amplitude", "operation": "REFUSED: no calibration maps mechanical energy to sample code.", "preserves": [], "status": "no-canonical-conversion", "to": "PCM16", "unitsIn": [ "joule" ], "unitsOut": [] }, { "doesNotPreserve": [ "culture", "preference", "meaning" ], "from": "frequency-relations", "id": "frequency-structure-to-consonance", "operation": "REFUSED in this artifact; any perceptual hypothesis would require declared stimuli, listeners, tasks, models, and contexts.", "preserves": [], "status": "no-canonical-conversion", "to": "perceived-consonance", "unitsIn": [ "hertz" ], "unitsOut": [] }, { "doesNotPreserve": [ "timbre", "style", "preference" ], "from": "12TET-registered-chords", "id": "pitch-to-voice-leading", "operation": "Minimize registered semitone motion under the declared equivalences and metric.", "preserves": [ "declared pitch-class membership" ], "status": "exact-mathematical", "to": "L1-voice-leading-distance", "unitsIn": [ "semitone-12tet" ], "unitsOut": [ "semitone-12tet" ] }, { "doesNotPreserve": [ "listener response", "culture", "aesthetic value" ], "from": "L1-voice-leading-distance", "id": "voice-leading-to-preference", "operation": "REFUSED without a contextual perceptual and cultural study.", "preserves": [], "status": "no-canonical-conversion", "to": "aesthetic-preference", "unitsIn": [ "semitone-12tet" ], "unitsOut": [] }, { "doesNotPreserve": [ "accent", "tempo", "performance" ], "from": "Z16-onset-set", "id": "onsets-to-cyclic-gaps", "operation": "Take sorted cyclic differences including wraparound.", "preserves": [ "onset count", "cycle length" ], "status": "exact-mathematical", "to": "cyclic-gap-sequence", "unitsIn": [ "cycle-index" ], "unitsOut": [ "cycle-index" ] }, { "doesNotPreserve": [ "meaning", "embodiment", "preference" ], "from": "cyclic-gap-variance", "id": "evenness-to-musical-rank", "operation": "REFUSED: arithmetic evenness is not a cultural or aesthetic ranking.", "preserves": [], "status": "no-canonical-conversion", "to": "musical-quality", "unitsIn": [ "dimensionless" ], "unitsOut": [] } ], "conversionStatuses": [ "exact-mathematical", "physical-model", "numerical-approximation", "declared-sonification", "perceptual-hypothesis", "cultural-interpretation", "no-canonical-conversion" ], "exhibits": [ { "accessibilitySummary": "The diagram and table expose the P4 topology, exact dimensionless spectrum, M and K bridge, zero rigid mode, modal frequencies, driven sweep, phase crossing, and sonification refusal boundary.", "audioAssetPaths": [ "audio/graph-modes.wav" ], "claims": { "claim": "A graph spectrum acquires angular-frequency units only through a declared M,K physical model. Under the separately declared force, coupling, damping, response observable, and sweep, the lambda=2 scalar model has an analytic displacement-response peak and a 90-degree phase crossing.", "falsifier": "Changing m, k, boundary anchoring, damping, drive shape, or response observable changes the modal frequencies or response peak. An orthogonal drive has Q=0 and produces no forced response in this mode even at omega_n; with c=0, exact-frequency forcing has no bounded steady-state amplitude.", "nonClaims": [ "The graph alone does not contain hertz.", "A natural frequency match alone is not called resonance.", "The synthetic peak is not a physical observation.", "The WAV is not an acoustic-pressure prediction.", "PCM amplitude is not mechanical energy." ], "scope": "Finite, linear, equal-mass, equal-spring synthetic model only." }, "derivations": [ { "assumptions": [ "equal masses", "equal springs", "small linear displacement" ], "caveat": "Without k and m, lambda has no hertz interpretation.", "expression": "K=kL and M=mI", "id": "laplacian-to-physical-operator", "result": "omega_i=sqrt((k/m)*lambda_i)", "status": "physical-model", "unitId": "radian-per-second" }, { "assumptions": [ "second is the time unit" ], "caveat": "A numerical decimal uses pi approximation only for display.", "expression": "f_i=omega_i/(2*pi)", "id": "angular-to-cyclic-frequency", "result": "modal cycles per second", "status": "exact-mathematical", "unitId": "hertz" }, { "assumptions": [ "linear time-invariant mode", "harmonic forcing Q(t)=Q0*cos(Omega*t)", "steady state" ], "caveat": "The sweep is calculated; no experiment or measurement occurred.", "expression": "A(Omega)=Q0/sqrt((k*lambda-m*Omega^2)^2+(c*Omega)^2)", "id": "driven-steady-state-response", "result": "displacement amplitude sweep", "status": "physical-model", "unitId": "metre" }, { "assumptions": [ "same driven scalar mode" ], "caveat": "Phase angle is displayed in degrees, not a unit of energy or frequency.", "expression": "delta(Omega)=atan2(c*Omega,k*lambda-m*Omega^2)", "id": "driven-phase", "result": "phase lag crosses 90 degrees at Omega=omega_n", "status": "physical-model", "unitId": "dimensionless" }, { "assumptions": [ "linear modal projection", "zero initial response in this mode" ], "caveat": "Frequency proximity without nonzero coupling is insufficient; this is why the aligned drive is called zero-net-spatial-force, not net-zero modal drive.", "expression": "Q=phi^T f0=0 for any drive vector orthogonal to phi", "id": "orthogonal-drive-falsifier", "result": "no forced response in the lambda=2 mode even when Omega=omega_n", "status": "exact-mathematical", "unitId": "newton" }, { "assumptions": [ "scalar lambda=2 modal equation", "instantaneous Q(t)=Q0*cos(Omega*t)", "c nonnegative" ], "caveat": "This is a model balance, not an observed ledger.", "expression": "d((1/2)m*eta_dot^2+(1/2)k*lambda*eta^2)/dt = eta_dot*Q0*cos(Omega*t)-c*eta_dot^2", "id": "mechanical-energy-balance", "result": "instantaneous forcing power minus damping power", "status": "physical-model", "unitId": "watt" }, { "assumptions": [ "no radiation or propagation model supplied" ], "caveat": "The release supplies declared sonification only.", "expression": "structural motion -> acoustic pressure", "id": "structure-to-sound-pressure", "result": "REFUSED", "status": "no-canonical-conversion", "unitId": "not-applicable" } ], "domain": "spectral mechanics and forced linear response", "id": "graph-to-hz-driven-resonance", "model": { "acousticCoupling": "NONE. Acoustic radiation, medium impedance, receiver position, and sound pressure are not modelled; they would be required for a physical audio prediction.", "boundaryConditions": [ "Free rigid translation is retained as mode 0.", "No external endpoint anchoring is present." ], "damping": "Driven mode uses damping ratio zeta=0.05 and modal c=2*zeta*m*omega_n. The audio mix uses a separate integer fade envelope, not this physical damping.", "excitation": "Synthetic force f(t)=F0*phi*cos(Omega*t), F0=0.1 N; this has net spatial force zero and instantaneous modal force Q(t)=phi^T f(t)=Q0*cos(Omega*t), with nonzero amplitude Q0=F0.", "geometry": "Four indexed displacement coordinates q0..q3; no ambient spatial embedding is required.", "material": "M=mI with m=0.01 kg; K=kL with k=6400 N/m.", "metric": "Graph Laplacian is dimensionless; displacement coordinates use metres after the physical-model declaration.", "operator": "K phi = omega^2 M phi; driven mode phi=[1,-1,-1,1]/2 has Laplacian eigenvalue 2.", "sonification": "Three non-rigid modal frequencies control integer triangle oscillators in graph-modes.wav. This authored mapping is not radiated sound.", "systemKind": "four-degree-of-freedom equal-mass equal-spring chain", "topology": "Path graph P4 with edges (0,1), (1,2), (2,3)." }, "ordinal": 1, "quantities": [ { "approximateValue": 6400, "exactValue": "6400", "id": "spring-stiffness", "label": "Equal spring stiffness k", "note": "Introduces force-per-displacement units.", "status": "physical-model", "unitId": "newton-per-metre" }, { "approximateValue": 0.01, "exactValue": "0.01", "id": "point-mass", "label": "Equal point mass m", "note": "Introduces inertia.", "status": "physical-model", "unitId": "kilogram" }, { "approximateValue": 1131.370849898, "exactValue": "800*sqrt(2)", "id": "driven-natural-angular-frequency", "label": "Driven mode natural angular frequency", "note": "For the normalized lambda=2 eigenvector.", "status": "physical-model", "unitId": "radian-per-second" }, { "approximateValue": 180.063263231, "exactValue": "400*sqrt(2)/pi", "id": "driven-natural-frequency", "label": "Driven mode natural frequency", "note": "Cycles per second obtained by exact division by 2π before numerical display.", "status": "numerical-approximation", "unitId": "hertz" }, { "approximateValue": 0.1, "exactValue": "0.1", "id": "force-amplitude", "label": "Aligned modal force amplitude F0=Q0", "note": "The vector f0=F0*phi has zero component sum while generalized-force amplitude Q0=phi^T f0=F0 is nonzero.", "status": "physical-model", "unitId": "newton" }, { "approximateValue": 1.13137085, "exactValue": "2*0.05*0.01*800*sqrt(2)", "id": "modal-damping", "label": "Driven modal damping coefficient", "note": "Viscous modal damping c=2*zeta*m*omega_n.", "status": "physical-model", "unitId": "newton-second-per-metre" }, { "approximateValue": 0.997496867, "exactValue": "sqrt(1-2*zeta^2)=sqrt(0.995)", "id": "peak-drive-ratio", "label": "Displacement-response peak ratio Omega_peak/omega_n", "note": "Valid for zeta<1/sqrt(2).", "status": "exact-mathematical", "unitId": "dimensionless" }, { "approximateValue": 0.00007822284, "exactValue": null, "id": "peak-displacement", "label": "Steady-state modal displacement amplitude at analytic peak", "note": "Synthetic model response, not a measurement.", "status": "numerical-approximation", "unitId": "metre" } ], "question": "When does a dimensionless graph mode acquire hertz, and when may a model call its response resonant?", "resonanceAssessment": { "analyticPeakDriveRatio": "sqrt(1-2*zeta^2)", "couplingDeclared": true, "dampingDeclared": true, "explanation": "The label applies only to the deterministic damped-forced model calculation; no physical system was driven or measured.", "forcingDeclared": true, "measuredResonance": false, "modelResonanceCriterionSatisfied": true, "phaseCriterion": "phase lag equals 90 degrees at Omega=omega_n and the analytic amplitude peak lies below omega_n for zeta=0.05", "phaseCriterionSatisfied": true, "physicalExperimentPerformed": false, "responseObservable": "steady-state modal displacement amplitude and phase lag", "sweepDeclared": true, "syntheticResponsePeakDetected": true }, "sourceIds": [ "mit-modal-analysis" ], "tables": [ { "caption": "Natural modes of the declared P4 mass-spring model", "columns": [ "mode", "Laplacian eigenvalue", "exact omega rad/s", "approx omega rad/s", "approx frequency Hz", "classification" ], "rows": [ [ "0", "0", "0", "0.000000000", "0.000000000", "rigid translation; excluded from audio" ], [ "1", "2-sqrt(2)", "800*sqrt(2-sqrt(2))", "612.293491784", "97.449535840", "elastic mode; included in declared sonification" ], [ "2", "2", "800*sqrt(2)", "1131.370849898", "180.063263231", "elastic mode; included in declared sonification" ], [ "3", "2+sqrt(2)", "800*sqrt(2+sqrt(2))", "1478.207252018", "235.263991073", "elastic mode; included in declared sonification" ] ] }, { "caption": "Forced damped lambda=2 steady-state displacement response", "columns": [ "drive ratio Omega/omega_n", "drive frequency Hz", "displacement amplitude m", "phase lag degrees", "marker" ], "rows": [ [ "0.500000", "90.031631616", "1.039359539e-5", "3.814075", "sweep point" ], [ "0.800000", "144.050610585", "2.118461502e-5", "12.528808", "sweep point" ], [ "0.950000", "171.060100070", "5.739019136e-5", "44.255941", "sweep point" ], [ "sqrt(1-2*zeta^2)", "179.612540964", "7.822283974e-5", "87.130424", "analytic displacement peak" ], [ "1.000000", "180.063263231", "7.812500000e-5", "90.000000", "natural-frequency phase crossing" ], [ "1.050000", "189.066426393", "5.324212841e-5", "134.309723", "sweep point" ], [ "1.200000", "216.075915878", "1.713003944e-5", "164.744881", "sweep point" ], [ "1.500000", "270.094894847", "6.205480241e-6", "173.157227", "sweep point" ] ] } ], "title": "A graph needs a physical bridge—and resonance needs a drive", "visualAssetPath": "assets/graph-to-hz.svg" }, { "accessibilitySummary": "Two text-labelled frequency families and four exact table rows per boundary provide the nonvisual equivalent.", "audioAssetPaths": [], "claims": { "claim": "Boundary conditions select different admissible mode families even when interval length and wave speed are equal.", "falsifier": "Replacing the fixed-free derivative condition with a second fixed displacement condition returns the fixed-fixed family.", "nonClaims": [ "This is not a measured string.", "Equal outline and material summary do not erase boundary conditions.", "No acoustic output is predicted." ], "scope": "Ideal one-dimensional small-displacement wave equation." }, "derivations": [ { "assumptions": [ "Dirichlet at both ends" ], "caveat": "Ideal linear string.", "expression": "f_n=n*c/(2L)", "id": "fixed-fixed-family", "result": "integer harmonic family", "status": "exact-mathematical", "unitId": "hertz" }, { "assumptions": [ "Dirichlet at x=0", "Neumann at x=L" ], "caveat": "Ideal linear string.", "expression": "f_n=(2n-1)*c/(4L)", "id": "fixed-free-family", "result": "odd-quarter-wave family", "status": "exact-mathematical", "unitId": "hertz" } ], "domain": "one-dimensional wave equation", "id": "same-string-different-boundary", "model": { "acousticCoupling": "NONE; transverse string motion is not converted to pressure.", "boundaryConditions": [ "Variant A: u(0,t)=u(L,t)=0.", "Variant B: u(0,t)=0 and spatial derivative ux(L,t)=0." ], "damping": "No damping.", "excitation": "Natural-mode eigenproblem only; no forcing.", "geometry": "Interval x in [0,1 m].", "material": "Wave speed c=100 m/s is held equal; density and tension are represented only through c.", "metric": "Euclidean length L=1 m.", "operator": "u_tt=c^2 u_xx", "sonification": "NONE.", "systemKind": "ideal taut string", "topology": "One connected one-dimensional interval." }, "ordinal": 2, "quantities": [ { "approximateValue": 1, "exactValue": "1", "id": "string-length", "label": "Length L", "note": "Held equal.", "status": "physical-model", "unitId": "metre" }, { "approximateValue": 100, "exactValue": "100", "id": "wave-speed", "label": "Wave speed c", "note": "Held equal.", "status": "physical-model", "unitId": "metre-per-second" } ], "question": "Can equal length and wave speed still produce different allowed frequencies?", "resonanceAssessment": null, "sourceIds": [ "mit-wave-equation", "ucsb-energy-method" ], "tables": [ { "caption": "First four allowed frequencies", "columns": [ "boundary", "n", "exact expression", "frequency Hz" ], "rows": [ [ "fixed-fixed", "1", "1*c/(2L)", "50" ], [ "fixed-fixed", "2", "2*c/(2L)", "100" ], [ "fixed-fixed", "3", "3*c/(2L)", "150" ], [ "fixed-fixed", "4", "4*c/(2L)", "200" ], [ "fixed-free", "1", "(2*1-1)*c/(4L)", "25" ], [ "fixed-free", "2", "(2*2-1)*c/(4L)", "75" ], [ "fixed-free", "3", "(2*3-1)*c/(4L)", "125" ], [ "fixed-free", "4", "(2*4-1)*c/(4L)", "175" ] ] } ], "title": "Same ideal string, different boundary family", "visualAssetPath": "assets/boundary-variants.svg" }, { "accessibilitySummary": "The table states both exact laws and selected numerical values; the diagram labels the operators and shared outline.", "audioAssetPaths": [], "claims": { "claim": "The same outline and even the same sinusoidal labels can sit under different operators and frequency scalings.", "falsifier": "Changing the operator from -Delta to Delta^2 changes sqrt(m^2+n^2) scaling to (m^2+n^2) scaling under the declared models.", "nonClaims": [ "The chosen parameters do not assert identical material.", "A square silhouette does not identify an object as a membrane or plate.", "No sound-pressure field is produced." ], "scope": "Comparison of two ideal linear PDE fixtures." }, "derivations": [ { "assumptions": [ "ideal membrane", "fixed displacement boundary" ], "caveat": "Not a plate law.", "expression": "f_mn=(c/(2a))*sqrt(m^2+n^2)", "id": "membrane-frequency", "result": "square-root spectral scaling", "status": "physical-model", "unitId": "hertz" }, { "assumptions": [ "Kirchhoff-Love thin plate", "simply supported fixture" ], "caveat": "Not a membrane law.", "expression": "f_mn=(pi*beta/(2a^2))*(m^2+n^2)", "id": "plate-frequency", "result": "linear-in-eigenvalue scaling", "status": "physical-model", "unitId": "hertz" } ], "domain": "partial differential operators on a square", "id": "same-square-different-operator", "model": { "acousticCoupling": "NONE; no radiation or pressure model.", "boundaryConditions": [ "Membrane displacement zero on boundary.", "Plate fixture uses the simply supported sinusoidal eigenfamily." ], "damping": "No damping.", "excitation": "Natural-mode comparison only.", "geometry": "Both domains are a one-metre square.", "material": "Membrane wave speed c=100 m/s; plate beta=sqrt(D/(rho*h))=10 m^2/s. These illustrative parameters are not claimed to describe one material.", "metric": "Euclidean metric with side a=1 m.", "operator": "Membrane: -Delta. Plate: Delta^2 in the declared thin-plate model.", "sonification": "NONE.", "systemKind": "ideal membrane compared with simply supported thin plate", "topology": "Both domains are simply connected." }, "ordinal": 3, "quantities": [ { "approximateValue": 1, "exactValue": "1", "id": "square-side", "label": "Square side a", "note": "Held equal as outline only.", "status": "physical-model", "unitId": "metre" }, { "approximateValue": 100, "exactValue": "100", "id": "membrane-wave-speed", "label": "Membrane c", "note": "Membrane parameter.", "status": "physical-model", "unitId": "metre-per-second" }, { "approximateValue": 10, "exactValue": "10", "id": "plate-beta", "label": "Plate sqrt(D/(rho*h))", "note": "Thin-plate parameter.", "status": "physical-model", "unitId": "square-metre-per-second" } ], "question": "Does a shared square outline determine the frequency law?", "resonanceAssessment": null, "sourceIds": [ "mit-wave-equation", "nasa-vibration-plates" ], "tables": [ { "caption": "Selected square mode labels under two operators", "columns": [ "mode", "m^2+n^2", "membrane exact Hz", "membrane approx Hz", "plate exact Hz", "plate approx Hz" ], "rows": [ [ "(1,1)", "2", "50*sqrt(2)", "70.710678119", "5*pi*2", "31.415926536" ], [ "(1,2)", "5", "50*sqrt(5)", "111.803398875", "5*pi*5", "78.539816340" ], [ "(2,2)", "8", "50*sqrt(8)", "141.421356237", "5*pi*8", "125.663706144" ] ] } ], "title": "A membrane and a plate are not one square-shaped instrument", "visualAssetPath": "assets/membrane-plate.svg" }, { "accessibilitySummary": "Three labelled panels show two conventional basis members and one rotated basis; equations and noncanonical status are tabulated.", "audioAssetPaths": [], "claims": { "claim": "A repeated eigenvalue identifies a multidimensional invariant subspace; it does not canonically name individual basis vectors inside that subspace.", "falsifier": "Any nontrivial orthogonal rotation of phi_12 and phi_21 gives a different basis with the same eigenvalue, directly refuting basis uniqueness.", "nonClaims": [ "The perfect square is an ideal symmetry.", "A rendered nodal pattern is not the only possible observed combination.", "Degeneracy does not imply mystical balance or aesthetic value." ], "scope": "Exact square symmetry under the declared membrane operator." }, "derivations": [ { "assumptions": [ "orthonormal starting basis", "same repeated eigenvalue" ], "caveat": "A perturbation or measurement convention may select a basis, but the unperturbed operator does not.", "expression": "psi_1=cos(theta)*phi_12+sin(theta)*phi_21; psi_2=-sin(theta)*phi_12+cos(theta)*phi_21", "id": "basis-rotation", "result": "another orthonormal eigenbasis for every theta", "status": "exact-mathematical", "unitId": "dimensionless" } ], "domain": "symmetry and eigenspace geometry", "id": "square-symmetry-degeneracy", "model": { "acousticCoupling": "NONE.", "boundaryConditions": [ "Dirichlet displacement boundary." ], "damping": "No damping.", "excitation": "Natural eigenproblem only.", "geometry": "Square [0,a] by [0,a], a=1 m.", "material": "Uniform ideal membrane represented by constant c; frequency value is not needed here.", "metric": "Euclidean square metric.", "operator": "-Delta with phi_mn=sin(m*pi*x/a) sin(n*pi*y/a)", "sonification": "NONE.", "systemKind": "ideal fixed-boundary square membrane", "topology": "Simply connected domain." }, "ordinal": 4, "quantities": [ { "approximateValue": 49.348022005, "exactValue": "5*pi^2/a^2", "id": "degenerate-spatial-eigenvalue", "label": "Shared spatial eigenvalue lambda_12=lambda_21", "note": "Multiplicity two on the perfect square.", "status": "numerical-approximation", "unitId": "inverse-square-metre" }, { "approximateValue": 2, "exactValue": "2", "id": "eigenspace-dimension", "label": "Eigenspace dimension", "note": "Spanned by phi_12 and phi_21.", "status": "exact-mathematical", "unitId": "dimensionless" } ], "question": "What does a repeated eigenvalue leave undetermined?", "resonanceAssessment": null, "sourceIds": [ "mit-wave-equation" ], "tables": [ { "caption": "One repeated square-membrane eigenspace", "columns": [ "member", "formula", "eigenvalue", "canonical?" ], "rows": [ [ "phi_12", "sin(pi*x/a) sin(2*pi*y/a)", "5*pi^2/a^2", "no; basis member" ], [ "phi_21", "sin(2*pi*x/a) sin(pi*y/a)", "5*pi^2/a^2", "no; basis member" ], [ "rotated pair", "orthogonal theta rotation", "same", "no; equally valid basis" ] ] } ], "title": "Square symmetry fixes an eigenspace, not one privileged basis", "visualAssetPath": "assets/square-degeneracy.svg" }, { "accessibilitySummary": "Both edge lists, degree witnesses, coefficient vectors, factorization, and spectrum are provided as text and machine data.", "audioAssetPaths": [], "claims": { "claim": "The pair falsifies the universal proposition that a connected finite simple graph is uniquely determined by its combinatorial-Laplacian spectrum.", "falsifier": "Equal exact characteristic polynomials establish equal spectra, while unequal degree multisets prove non-isomorphism.", "nonClaims": [ "It does not claim these graphs are planar drums.", "It does not assign hertz or sound.", "It does not prove every inverse spectral problem is non-unique." ], "scope": "Finite combinatorial Laplacian only; Kac and Gordon-Webb-Wolpert are separate continuum context links." }, "derivations": [ { "assumptions": [ "finite simple graph", "integer combinatorial Laplacian" ], "caveat": "The producer recomputes both graphs independently.", "expression": "Faddeev-LeVerrier over BigInt traces of L^k", "id": "exact-characteristic-polynomial", "result": "[1,-14,73,-176,192,-72,0]", "status": "exact-mathematical", "unitId": "dimensionless" }, { "assumptions": [ "graph isomorphisms preserve degrees" ], "caveat": "This witness does not depend on drawing layout.", "expression": "degreeMultiset(A) != degreeMultiset(B)", "id": "non-isomorphism-witness", "result": "graphs are non-isomorphic", "status": "exact-mathematical", "unitId": "dimensionless" } ], "domain": "finite inverse spectral counterexample", "id": "finite-laplacian-cospectral-counterexample", "model": { "acousticCoupling": "NONE.", "boundaryConditions": [ "Combinatorial Laplacian L=D-A.", "No continuum boundary." ], "damping": "No damping.", "excitation": "No excitation.", "geometry": "Six labelled vertices in each drawing; drawing coordinates carry no metric meaning.", "material": "No mass, stiffness, or acoustic material is assigned.", "metric": "Combinatorial adjacency only; no edge lengths.", "operator": "Combinatorial Laplacian and det(tI-L).", "sonification": "NONE.", "systemKind": "pair of finite connected simple graphs", "topology": "Graph A and Graph B use explicitly listed edge sets." }, "ordinal": 5, "quantities": [ { "approximateValue": null, "exactValue": "t(t-2)(t-3)^2(t^2-6t+4)", "id": "shared-polynomial", "label": "Shared characteristic polynomial", "note": "Coefficient vector [1,-14,73,-176,192,-72,0].", "status": "exact-mathematical", "unitId": "dimensionless" }, { "approximateValue": null, "exactValue": "[0,3-sqrt(5),2,3,3,3+sqrt(5)]", "id": "shared-spectrum", "label": "Shared sorted Laplacian spectrum", "note": "Eigenvalue 3 has multiplicity two.", "status": "exact-mathematical", "unitId": "dimensionless" }, { "approximateValue": null, "exactValue": "[2,2,2,2,2,4]", "id": "degree-witness-a", "label": "Graph A sorted degrees", "note": "Degree multiset is an isomorphism invariant.", "status": "exact-mathematical", "unitId": "dimensionless" }, { "approximateValue": null, "exactValue": "[1,2,2,3,3,3]", "id": "degree-witness-b", "label": "Graph B sorted degrees", "note": "Different from Graph A.", "status": "exact-mathematical", "unitId": "dimensionless" } ], "question": "Does an exact spectrum uniquely identify every finite connected graph?", "resonanceAssessment": null, "sourceIds": [ "kac-shape-drum", "gww-isospectral-drums" ], "tables": [ { "caption": "Exact finite counterexample data", "columns": [ "graph", "edges", "sorted degrees", "connected", "det(tI-L) coefficients" ], "rows": [ [ "A", "[[0,2],[0,3],[0,4],[0,5],[1,4],[1,5],[2,3]]", "[2,2,2,2,2,4]", "true", "[1,-14,73,-176,192,-72,0]" ], [ "B", "[[0,2],[0,4],[0,5],[1,2],[1,4],[1,5],[2,3]]", "[1,2,2,3,3,3]", "true", "[1,-14,73,-176,192,-72,0]" ] ] } ], "title": "Different connected graphs can share one Laplacian spectrum", "visualAssetPath": "assets/cospectral-graphs.svg" }, { "accessibilitySummary": "The full event schedule, reversal equation, WAV format metadata, PCM receipts, and waveform-order distinction remain available without playback.", "audioAssetPaths": [ "audio/phase-forward.wav", "audio/phase-reversed.wav" ], "claims": { "claim": "For a real sequence, cyclic time reversal conjugates every DFT coefficient, preserving every magnitude while changing phases and, for this nonsymmetric fixture, sample order.", "falsifier": "The producer verifies the two PCM blocks are unequal and satisfy y[n]=x[-n mod N] sample by sample; the displayed algebra then forces magnitude equality.", "nonClaims": [ "Magnitude equality does not imply equal waveform.", "Different waveform does not guarantee a universal perceptual difference.", "The files are not recordings or physical-acoustic predictions." ], "scope": "Finite N=4000 real PCM sequence and its cyclic DFT." }, "derivations": [ { "assumptions": [ "finite real sequence", "cyclic indexing" ], "caveat": "The theorem concerns the finite PCM sequence, not an analogue signal outside the file.", "expression": "Y[k]=sum_n x[-n] exp(-i2*pi*k*n/N)=conj(X[k]) for real x", "id": "reversal-dft", "result": "|Y[k]|=|X[k]| for every k", "status": "exact-mathematical", "unitId": "dimensionless" }, { "assumptions": [ "no calibration, transducer, gain, or acoustic model" ], "caveat": "Playback hardware may produce sound, but the artifact predicts no pressure field.", "expression": "PCM16 code -> pascal", "id": "pcm-to-pressure", "result": "REFUSED", "status": "no-canonical-conversion", "unitId": "not-applicable" } ], "domain": "finite discrete Fourier phase counterexample", "id": "same-magnitude-different-phase", "model": { "acousticCoupling": "NONE; PCM codes are not pressure.", "boundaryConditions": [ "Cyclic reversal fixes n=0 and maps n to N-n for 1<=n preferred or consonant", "id": "distance-to-preference", "result": "REFUSED", "status": "no-canonical-conversion", "unitId": "not-applicable" } ], "domain": "bounded mathematical music theory", "id": "twelve-tet-voice-leading-geometry", "model": { "acousticCoupling": "NONE.", "boundaryConditions": [ "Source chord C major uses registered [0,4,7].", "Target E minor may use registered [-1,4,7], whose pitch classes are [11,4,7]." ], "damping": "No damping.", "excitation": "No physical excitation.", "geometry": "Pitch classes form Z_12; registered pitches are integer semitone coordinates.", "material": "No instrument, tuning drift, timbre, listener, or culture is represented beyond the named Western 12-TET convention.", "metric": "Voice-leading distance is the minimum L1 sum of absolute registered semitone moves under declared bijections and octave representatives.", "operator": "Finite quotient and assignment search.", "sonification": "NONE; the geometry is shown, not sounded.", "systemKind": "12-tone equal-temperament pitch-class and registered voice-leading fixture", "topology": "Octave equivalence maps p to p mod 12; chord order is quotiented by permutation only for pitch-class sets." }, "ordinal": 7, "quantities": [ { "approximateValue": 12, "exactValue": "12", "id": "octave-size", "label": "Octave size in this convention", "note": "Defines p~p+12.", "status": "exact-mathematical", "unitId": "semitone-12tet" }, { "approximateValue": 1, "exactValue": "|0-(-1)|+|4-4|+|7-7|=1", "id": "voice-leading-distance", "label": "Declared minimum L1 path", "note": "C major [0,4,7] to E minor [-1,4,7].", "status": "exact-mathematical", "unitId": "semitone-12tet" } ], "question": "What does a one-semitone path mean, and what does it not mean?", "resonanceAssessment": null, "sourceIds": [ "tymoczko-chord-geometry", "callender-generalized-voice-leading", "consonance-models-review", "mcdermott-cultural-variation", "plomp-levelt-critical-bandwidth" ], "tables": [ { "caption": "One declared minimal voice leading", "columns": [ "source voice", "target representative", "signed move st", "absolute move st" ], "rows": [ [ "C4 / 0", "B3 / -1", "-1", "1" ], [ "E4 / 4", "E4 / 4", "0", "0" ], [ "G4 / 7", "G4 / 7", "0", "0" ], [ "total", "under declared bijection", "vector [-1,0,0]", "1" ] ] } ], "title": "Chord geometry starts by declaring equivalences and a metric", "visualAssetPath": "assets/musical-geometry.svg" }, { "accessibilitySummary": "Both onset sets, all cyclic gaps, exact variances, and the no-aesthetic-conversion boundary are provided in text.", "audioAssetPaths": [], "claims": { "claim": "A cyclic onset set can be compared by exact gap structure under a declared convention.", "falsifier": "Both patterns have five onsets yet have unequal exact gap variances, while neither arithmetic result supplies an aesthetic ordering.", "nonClaims": [ "Lower gap variance is not a universal quality score.", "This fixture does not identify one pattern as traditional to a specific culture.", "Onset geometry omits accent, tempo, embodiment, and interpretation." ], "scope": "Finite Z_16 onset-only comparison." }, "derivations": [ { "assumptions": [ "Z_16 cycle", "start at onset 0" ], "caveat": "Rotation changes the sequence start but not the gap multiset.", "expression": "sorted cyclic differences including wraparound", "id": "onsets-to-gaps", "result": "E gaps [3,3,3,3,4]; cluster gaps [1,1,1,1,12]", "status": "exact-mathematical", "unitId": "cycle-index" }, { "assumptions": [ "no listening, culture, metre, accent, or task model" ], "caveat": "Even spacing is a combinatorial property only.", "expression": "lower gap variance -> musically better or universal", "id": "evenness-to-aesthetic", "result": "REFUSED", "status": "no-canonical-conversion", "unitId": "not-applicable" } ], "domain": "finite cyclic rhythm geometry", "id": "cyclic-rhythm-spacing", "model": { "acousticCoupling": "NONE.", "boundaryConditions": [ "Rotation begins at onset 0 for display.", "Wraparound gap is included." ], "damping": "No damping.", "excitation": "No excitation.", "geometry": "Sixteen equally spaced positions on a displayed circle.", "material": "No performer, tempo, accent, duration, timbre, movement, or musical culture is encoded.", "metric": "Clockwise cyclic gap length in integer steps.", "operator": "Cyclic difference and population variance of five gap lengths.", "sonification": "NONE.", "systemKind": "two onset subsets of Z_16", "topology": "Cyclic residue classes modulo 16." }, "ordinal": 8, "quantities": [ { "approximateValue": 5, "exactValue": "5", "id": "onset-count", "label": "Onsets in each pattern", "note": "Cardinality held equal.", "status": "exact-mathematical", "unitId": "dimensionless" }, { "approximateValue": 0.16, "exactValue": "4/25", "id": "euclidean-gap-variance", "label": "Population variance of gaps [3,3,3,3,4]", "note": "Mean gap 16/5.", "status": "exact-mathematical", "unitId": "dimensionless" }, { "approximateValue": 19.36, "exactValue": "484/25", "id": "cluster-gap-variance", "label": "Population variance of gaps [1,1,1,1,12]", "note": "Mean gap 16/5.", "status": "exact-mathematical", "unitId": "dimensionless" } ], "question": "How can five onsets be compared on a sixteen-step cycle?", "resonanceAssessment": null, "sourceIds": [ "toussaint-euclidean-rhythms", "mcdermott-cultural-variation" ], "tables": [ { "caption": "Two five-onset subsets of Z_16", "columns": [ "pattern", "onset set", "cyclic gaps", "gap variance", "interpretation" ], "rows": [ [ "Euclidean fixture E(5,16)", "[0,3,6,9,12]", "[3,3,3,3,4]", "4/25", "more even under declared gap variance" ], [ "clustered contrast", "[0,1,2,3,4]", "[1,1,1,1,12]", "484/25", "less even under declared gap variance" ] ] } ], "title": "Equal onset count does not imply equal spacing—or musical rank", "visualAssetPath": "assets/rhythm-cycle.svg" } ], "labId": "shape-resonance-lab-01", "limitations": [ "All fixtures are finite, deterministic, synthetic, idealized, and Phase 0.", "No acoustic coupling or sound-pressure model is present; every WAV is declared sonification.", "The producer and page load make no network request and embed no third-party resource. 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