dimensionless-graph-laplacian → hertz
REFUSED unless mass and stiffness or another dimensional physical bridge are declared.
Shape & Resonance Lab 01 · 0-static-synthetic · DRAFT
What can a shape sing?
Shape, vibration, energy, sound, and musical geometry without collapsed categories
Shape defines model-dependent possibilities and excitation selects motion; acoustic coupling would be required for a physical sound-pressure prediction, while this release provides declared sonification only; people and cultures may make music.
As of: 2026-08-26 · Next review due: 2027-08-26
Maintainer-authorized assisted construction; exact text has not been separately human-reviewed.
Mechanical energy balance
Every energy or power term is tied to the declared mechanical model and an SI unit; PCM amplitude, loudness, feeling, value, and metaphor are excluded.
| Term | Expression | Unit | Scope |
|---|---|---|---|
| kinetic-energy | T=(1/2) qdot^T M qdot | joule | Declared linear mechanical model only. |
| elastic-potential-energy | V=(1/2) q^T K q | joule | Declared linear mechanical model only. |
| total-mechanical-energy | E=T+V | joule | Conserved only when forcing and damping are absent under the declared model. |
| damping-power | P_d=qdot^T C qdot | watt | A nonnegative loss rate when C is positive semidefinite; it is not an energy store. |
| forcing-power | P_f(t)=eta_dot*Q(t)=eta_dot*Q0*cos(Omega*t) | watt | Signed instantaneous mechanical power supplied to the declared driven scalar mode; Q0 is the generalized-force amplitude. |
| pcm-amplitude-is-not-energy | PCM code != joule | pcm16 | Digital sample amplitude is a declared sonification control and carries no mechanical-energy unit. |
No canonical conversion
structural-motion → sound-pressure
REFUSED: v0.1 supplies no acoustic coupling, propagation medium, receiver, or pressure calibration.
joule → PCM16
REFUSED: no calibration maps mechanical energy to sample code.
frequency-relations → perceived-consonance
REFUSED in this artifact; any perceptual hypothesis would require declared stimuli, listeners, tasks, models, and contexts.
L1-voice-leading-distance → aesthetic-preference
REFUSED without a contextual perceptual and cultural study.
cyclic-gap-variance → musical-quality
REFUSED: arithmetic evenness is not a cultural or aesthetic ranking.
Eight finite exhibits
Exhibit 1 · spectral mechanics and forced linear response
A graph needs a physical bridge—and resonance needs a drive
When does a dimensionless graph mode acquire hertz, and when may a model call its response resonant?
Declared model
- System
- four-degree-of-freedom equal-mass equal-spring chain
- Geometry
- Four indexed displacement coordinates q0..q3; no ambient spatial embedding is required.
- Topology
- Path graph P4 with edges (0,1), (1,2), (2,3).
- Metric
- Graph Laplacian is dimensionless; displacement coordinates use metres after the physical-model declaration.
- Boundary
- Free rigid translation is retained as mode 0.
- No external endpoint anchoring is present.
- Material
- M=mI with m=0.01 kg; K=kL with k=6400 N/m.
- Operator
- K phi = omega^2 M phi; driven mode phi=[1,-1,-1,1]/2 has Laplacian eigenvalue 2.
- 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.
- 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.
- Acoustic coupling
- NONE. Acoustic radiation, medium impedance, receiver position, and sound pressure are not modelled; they would be required for a physical audio prediction.
- Sonification
- Three non-rigid modal frequencies control integer triangle oscillators in graph-modes.wav. This authored mapping is not radiated sound.
Quantities
| Quantity | Value | Unit | Status | Meaning |
|---|---|---|---|---|
| Equal spring stiffness k | 6400 approx. 6400 | newton-per-metre | physical-model | Introduces force-per-displacement units. |
| Equal point mass m | 0.01 approx. 0.01 | kilogram | physical-model | Introduces inertia. |
| Driven mode natural angular frequency | 800*sqrt(2) approx. 1131.370849898 | radian-per-second | physical-model | For the normalized lambda=2 eigenvector. |
| Driven mode natural frequency | 400*sqrt(2)/pi approx. 180.063263231 | hertz | numerical-approximation | Cycles per second obtained by exact division by 2π before numerical display. |
| Aligned modal force amplitude F0=Q0 | 0.1 approx. 0.1 | newton | physical-model | The vector f0=F0*phi has zero component sum while generalized-force amplitude Q0=phi^T f0=F0 is nonzero. |
| Driven modal damping coefficient | 2*0.05*0.01*800*sqrt(2) approx. 1.13137085 | newton-second-per-metre | physical-model | Viscous modal damping c=2*zeta*m*omega_n. |
| Displacement-response peak ratio Omega_peak/omega_n | sqrt(1-2*zeta^2)=sqrt(0.995) approx. 0.997496867 | dimensionless | exact-mathematical | Valid for zeta<1/sqrt(2). |
| Steady-state modal displacement amplitude at analytic peak | 0.00007822284 approx. 0.00007822284 | metre | numerical-approximation | Synthetic model response, not a measurement. |
Derivations and refused bridges
| Bridge | Expression | Status | Assumptions | Result / unit | Caveat |
|---|---|---|---|---|---|
| laplacian-to-physical-operator | K=kL and M=mI | physical-model |
| omega_i=sqrt((k/m)*lambda_i) / radian-per-second | Without k and m, lambda has no hertz interpretation. |
| angular-to-cyclic-frequency | f_i=omega_i/(2*pi) | exact-mathematical |
| modal cycles per second / hertz | A numerical decimal uses pi approximation only for display. |
| driven-steady-state-response | A(Omega)=Q0/sqrt((k*lambda-m*Omega^2)^2+(c*Omega)^2) | physical-model |
| displacement amplitude sweep / metre | The sweep is calculated; no experiment or measurement occurred. |
| driven-phase | delta(Omega)=atan2(c*Omega,k*lambda-m*Omega^2) | physical-model |
| phase lag crosses 90 degrees at Omega=omega_n / dimensionless | Phase angle is displayed in degrees, not a unit of energy or frequency. |
| orthogonal-drive-falsifier | Q=phi^T f0=0 for any drive vector orthogonal to phi | exact-mathematical |
| no forced response in the lambda=2 mode even when Omega=omega_n / newton | Frequency proximity without nonzero coupling is insufficient; this is why the aligned drive is called zero-net-spatial-force, not net-zero modal drive. |
| mechanical-energy-balance | d((1/2)m*eta_dot^2+(1/2)k*lambda*eta^2)/dt = eta_dot*Q0*cos(Omega*t)-c*eta_dot^2 | physical-model |
| instantaneous forcing power minus damping power / watt | This is a model balance, not an observed ledger. |
| structure-to-sound-pressure | structural motion -> acoustic pressure | no-canonical-conversion |
| REFUSED / not-applicable | The release supplies declared sonification only. |
| mode | Laplacian eigenvalue | exact omega rad/s | approx omega rad/s | approx frequency Hz | classification |
|---|---|---|---|---|---|
| 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 |
| drive ratio Omega/omega_n | drive frequency Hz | displacement amplitude m | phase lag degrees | marker |
|---|---|---|---|---|
| 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 |
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.
Does not claim
- 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.
Falsifier / counterexample
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.
Scope: Finite, linear, equal-mass, equal-spring synthetic model only.
Driven-model resonance assessment
- Observable
- steady-state modal displacement amplitude and phase lag
- Peak
- sqrt(1-2*zeta^2); synthetic peak detected: true
- Phase criterion
- phase lag equals 90 degrees at Omega=omega_n and the analytic amplitude peak lies below omega_n for zeta=0.05
- Model criterion
- true
- Physical experiment / measured resonance
- false / false
The label applies only to the deterministic damped-forced model calculation; no physical system was driven or measured.
Optional declared sonifications
No audio is requested on page load. Each ordinary same-origin WAV link makes a request only if a reader activates it; the browser may download, open, or play the file. The page has no embedded audio control. WAV access is never an acoustic prediction.
- P4 non-rigid modal sonification — declared sonification WAV, mono PCM16, 8000 Hz, 0.5 seconds; access on reader activation
A half-second integer-triangle mix controlled by the three non-rigid frequencies of the declared P4 mass-spring model.
- File byte receipt
sha256:7ce6187a0396e7108102e4dffa8a8ae26b090f8bd3d67de37312c96dd7c5f556- PCM byte receipt
sha256:bb90706034760906cc62527c8baa80eb4083cc25c6eb7438d5a9b8a365436634- Time scale
- 1 sample = 1/8000 second; 4000 frames = 0.5 second
- Amplitude scale
- Weighted PCM triangle codes 12:9:7 divided by 32 with a 320-frame integer fade; PCM code is dimensionless and not joules or pascals.
- Stored PCM min / max
- -26255 / 25649
- Pre-clamp peak / clipped samples
- 26255 / 0
- Normalization / stored PCM in range
- NONE / true
Carrier/sampling declaration: Physical modal targets f=ω/(2π): 97.449535840, 180.063263231, 235.263991073 Hz. Integer phase increments 52317821, 96670728, 126306393 produce actual digital carrier fundamentals 97.449535504, 180.063262582, 235.263990238 Hz, with errors -3.365e-7, -6.496e-7, -8.347e-7 Hz. This is a sonification, not radiated pressure.
Spectrum caveat: The three listed values are carrier fundamentals, not the complete WAV spectrum. Triangle oscillators add odd harmonics; finite duration and the 320-frame envelope add window leakage and sidebands.
Complete non-audio equivalent: The public snapshot contains the complete schedule, equations, units, tables, and relation that generated this file.
Exhibit 2 · one-dimensional wave equation
Same ideal string, different boundary family
Can equal length and wave speed still produce different allowed frequencies?
Declared model
- System
- ideal taut string
- Geometry
- Interval x in [0,1 m].
- Topology
- One connected one-dimensional interval.
- Metric
- Euclidean length L=1 m.
- Boundary
- Variant A: u(0,t)=u(L,t)=0.
- Variant B: u(0,t)=0 and spatial derivative ux(L,t)=0.
- Material
- Wave speed c=100 m/s is held equal; density and tension are represented only through c.
- Operator
- u_tt=c^2 u_xx
- Excitation
- Natural-mode eigenproblem only; no forcing.
- Damping
- No damping.
- Acoustic coupling
- NONE; transverse string motion is not converted to pressure.
- Sonification
- NONE.
Quantities
| Quantity | Value | Unit | Status | Meaning |
|---|---|---|---|---|
| Length L | 1 approx. 1 | metre | physical-model | Held equal. |
| Wave speed c | 100 approx. 100 | metre-per-second | physical-model | Held equal. |
Derivations and refused bridges
| Bridge | Expression | Status | Assumptions | Result / unit | Caveat |
|---|---|---|---|---|---|
| fixed-fixed-family | f_n=n*c/(2L) | exact-mathematical |
| integer harmonic family / hertz | Ideal linear string. |
| fixed-free-family | f_n=(2n-1)*c/(4L) | exact-mathematical |
| odd-quarter-wave family / hertz | Ideal linear string. |
| boundary | n | exact expression | frequency Hz |
|---|---|---|---|
| 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 |
Claim
Boundary conditions select different admissible mode families even when interval length and wave speed are equal.
Does not claim
- This is not a measured string.
- Equal outline and material summary do not erase boundary conditions.
- No acoustic output is predicted.
Falsifier / counterexample
Replacing the fixed-free derivative condition with a second fixed displacement condition returns the fixed-fixed family.
Scope: Ideal one-dimensional small-displacement wave equation.
Exhibit 3 · partial differential operators on a square
A membrane and a plate are not one square-shaped instrument
Does a shared square outline determine the frequency law?
Declared model
- System
- ideal membrane compared with simply supported thin plate
- Geometry
- Both domains are a one-metre square.
- Topology
- Both domains are simply connected.
- Metric
- Euclidean metric with side a=1 m.
- Boundary
- Membrane displacement zero on boundary.
- Plate fixture uses the simply supported sinusoidal eigenfamily.
- 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.
- Operator
- Membrane: -Delta. Plate: Delta^2 in the declared thin-plate model.
- Excitation
- Natural-mode comparison only.
- Damping
- No damping.
- Acoustic coupling
- NONE; no radiation or pressure model.
- Sonification
- NONE.
Quantities
| Quantity | Value | Unit | Status | Meaning |
|---|---|---|---|---|
| Square side a | 1 approx. 1 | metre | physical-model | Held equal as outline only. |
| Membrane c | 100 approx. 100 | metre-per-second | physical-model | Membrane parameter. |
| Plate sqrt(D/(rho*h)) | 10 approx. 10 | square-metre-per-second | physical-model | Thin-plate parameter. |
Derivations and refused bridges
| Bridge | Expression | Status | Assumptions | Result / unit | Caveat |
|---|---|---|---|---|---|
| membrane-frequency | f_mn=(c/(2a))*sqrt(m^2+n^2) | physical-model |
| square-root spectral scaling / hertz | Not a plate law. |
| plate-frequency | f_mn=(pi*beta/(2a^2))*(m^2+n^2) | physical-model |
| linear-in-eigenvalue scaling / hertz | Not a membrane law. |
| mode | m^2+n^2 | membrane exact Hz | membrane approx Hz | plate exact Hz | plate approx Hz |
|---|---|---|---|---|---|
| (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 |
Claim
The same outline and even the same sinusoidal labels can sit under different operators and frequency scalings.
Does not claim
- 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.
Falsifier / counterexample
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.
Scope: Comparison of two ideal linear PDE fixtures.
Exhibit 4 · symmetry and eigenspace geometry
Square symmetry fixes an eigenspace, not one privileged basis
What does a repeated eigenvalue leave undetermined?
Declared model
- System
- ideal fixed-boundary square membrane
- Geometry
- Square [0,a] by [0,a], a=1 m.
- Topology
- Simply connected domain.
- Metric
- Euclidean square metric.
- Boundary
- Dirichlet displacement boundary.
- Material
- Uniform ideal membrane represented by constant c; frequency value is not needed here.
- Operator
- -Delta with phi_mn=sin(m*pi*x/a) sin(n*pi*y/a)
- Excitation
- Natural eigenproblem only.
- Damping
- No damping.
- Acoustic coupling
- NONE.
- Sonification
- NONE.
Quantities
| Quantity | Value | Unit | Status | Meaning |
|---|---|---|---|---|
| Shared spatial eigenvalue lambda_12=lambda_21 | 5*pi^2/a^2 approx. 49.348022005 | inverse-square-metre | numerical-approximation | Multiplicity two on the perfect square. |
| Eigenspace dimension | 2 approx. 2 | dimensionless | exact-mathematical | Spanned by phi_12 and phi_21. |
Derivations and refused bridges
| Bridge | Expression | Status | Assumptions | Result / unit | Caveat |
|---|---|---|---|---|---|
| basis-rotation | psi_1=cos(theta)*phi_12+sin(theta)*phi_21; psi_2=-sin(theta)*phi_12+cos(theta)*phi_21 | exact-mathematical |
| another orthonormal eigenbasis for every theta / dimensionless | A perturbation or measurement convention may select a basis, but the unperturbed operator does not. |
| member | formula | eigenvalue | canonical? |
|---|---|---|---|
| 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 |
Claim
A repeated eigenvalue identifies a multidimensional invariant subspace; it does not canonically name individual basis vectors inside that subspace.
Does not claim
- 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.
Falsifier / counterexample
Any nontrivial orthogonal rotation of phi_12 and phi_21 gives a different basis with the same eigenvalue, directly refuting basis uniqueness.
Scope: Exact square symmetry under the declared membrane operator.
Exhibit 5 · finite inverse spectral counterexample
Different connected graphs can share one Laplacian spectrum
Does an exact spectrum uniquely identify every finite connected graph?
Declared model
- System
- pair of finite connected simple graphs
- Geometry
- Six labelled vertices in each drawing; drawing coordinates carry no metric meaning.
- Topology
- Graph A and Graph B use explicitly listed edge sets.
- Metric
- Combinatorial adjacency only; no edge lengths.
- Boundary
- Combinatorial Laplacian L=D-A.
- No continuum boundary.
- Material
- No mass, stiffness, or acoustic material is assigned.
- Operator
- Combinatorial Laplacian and det(tI-L).
- Excitation
- No excitation.
- Damping
- No damping.
- Acoustic coupling
- NONE.
- Sonification
- NONE.
Quantities
| Quantity | Value | Unit | Status | Meaning |
|---|---|---|---|---|
| Shared characteristic polynomial | t(t-2)(t-3)^2(t^2-6t+4) | dimensionless | exact-mathematical | Coefficient vector [1,-14,73,-176,192,-72,0]. |
| Shared sorted Laplacian spectrum | [0,3-sqrt(5),2,3,3,3+sqrt(5)] | dimensionless | exact-mathematical | Eigenvalue 3 has multiplicity two. |
| Graph A sorted degrees | [2,2,2,2,2,4] | dimensionless | exact-mathematical | Degree multiset is an isomorphism invariant. |
| Graph B sorted degrees | [1,2,2,3,3,3] | dimensionless | exact-mathematical | Different from Graph A. |
Derivations and refused bridges
| Bridge | Expression | Status | Assumptions | Result / unit | Caveat |
|---|---|---|---|---|---|
| exact-characteristic-polynomial | Faddeev-LeVerrier over BigInt traces of L^k | exact-mathematical |
| [1,-14,73,-176,192,-72,0] / dimensionless | The producer recomputes both graphs independently. |
| non-isomorphism-witness | degreeMultiset(A) != degreeMultiset(B) | exact-mathematical |
| graphs are non-isomorphic / dimensionless | This witness does not depend on drawing layout. |
| graph | edges | sorted degrees | connected | det(tI-L) coefficients |
|---|---|---|---|---|
| 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] |
Claim
The pair falsifies the universal proposition that a connected finite simple graph is uniquely determined by its combinatorial-Laplacian spectrum.
Does not claim
- 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.
Falsifier / counterexample
Equal exact characteristic polynomials establish equal spectra, while unequal degree multisets prove non-isomorphism.
Scope: Finite combinatorial Laplacian only; Kac and Gordon-Webb-Wolpert are separate continuum context links.
Exhibit 6 · finite discrete Fourier phase counterexample
Equal Fourier magnitude does not recover sample order
Can two different real finite waveforms have exactly equal DFT magnitudes?
Declared model
- System
- two finite mono PCM16 sequences
- Geometry
- A 4000-index discrete time cycle.
- Topology
- Cyclic group Z_4000 for the DFT theorem.
- Metric
- Uniform sample interval 1/8000 second.
- Boundary
- Cyclic reversal fixes n=0 and maps n to N-n for 1<=n<N.
- Material
- Synthetic integer sample codes only; no medium or transducer.
- Operator
- N-point discrete Fourier transform.
- Excitation
- Four deterministic integer-triangle tone bursts create x; y is its exact cyclic reversal.
- Damping
- The 80-frame edge envelope is authored digital amplitude shaping, not mechanical damping.
- Acoustic coupling
- NONE; PCM codes are not pressure.
- Sonification
- Both files are declared sonifications of a finite exact sequence relation.
Quantities
| Quantity | Value | Unit | Status | Meaning |
|---|---|---|---|---|
| Sample rate | 8000 approx. 8000 | hertz | declared-sonification | Digital sampling convention. |
| Frames per file | 4000 approx. 4000 | sample | exact-mathematical | Duration 0.5 second. |
| Real cyclic reversal | y[0]=x[0]; y[n]=x[N-n] | dimensionless | exact-mathematical | Ensures Y[k]=conj(X[k]). |
Derivations and refused bridges
| Bridge | Expression | Status | Assumptions | Result / unit | Caveat |
|---|---|---|---|---|---|
| reversal-dft | Y[k]=sum_n x[-n] exp(-i2*pi*k*n/N)=conj(X[k]) for real x | exact-mathematical |
| |Y[k]|=|X[k]| for every k / dimensionless | The theorem concerns the finite PCM sequence, not an analogue signal outside the file. |
| pcm-to-pressure | PCM16 code -> pascal | no-canonical-conversion |
| REFUSED / not-applicable | Playback hardware may produce sound, but the artifact predicts no pressure field. |
| start sample | length samples | triangle period samples | target peak PCM code | phase offset samples | carrier Hz |
|---|---|---|---|---|---|
| 200 | 400 | 80 | 10000 | 0 | 100 |
| 900 | 550 | 50 | 7500 | 7 | 160 |
| 1700 | 300 | 40 | 12000 | 13 | 200 |
| 2800 | 600 | 32 | 9000 | 3 | 250 |
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.
Does not claim
- 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.
Falsifier / counterexample
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.
Scope: Finite N=4000 real PCM sequence and its cyclic DFT.
Optional declared sonifications
No audio is requested on page load. Each ordinary same-origin WAV link makes a request only if a reader activates it; the browser may download, open, or play the file. The page has no embedded audio control. WAV access is never an acoustic prediction.
- Forward finite phase fixture — declared sonification WAV, mono PCM16, 8000 Hz, 0.5 seconds; access on reader activation
A half-second schedule of four integer-triangle tone bursts.
- File byte receipt
sha256:157389bd7321f81c47a870d4895ddf4ae8c824f87baee8af03bfa769428f57b5- PCM byte receipt
sha256:1df5ac6f7375c16bccf1a1fa416a6cebde450927bce88b1534dbd5f316c5f227- Time scale
- 1 sample = 1/8000 second; 4000 frames = 0.5 second
- Amplitude scale
- Each burst declares an integer peak PCM code and an 80-frame linear edge envelope; no normalization.
- Stored PCM min / max
- -12000 / 12000
- Pre-clamp peak / clipped samples
- 12000 / 0
- Normalization / stored PCM in range
- NONE / true
Carrier/sampling declaration: Each integer-triangle burst period maps to sampleRate/period: 100 Hz, 160 Hz, 200 Hz, 250 Hz. These are authored digital carriers, not physical modes.
Spectrum caveat: The listed rates are carrier fundamentals, not complete spectra. Triangle carriers contain odd harmonics and each finite 80-frame edge envelope produces sidebands and finite-window leakage.
Complete non-audio equivalent: The public snapshot contains the complete schedule, equations, units, tables, and relation that generated this file.
- Cyclically reversed phase fixture — declared sonification WAV, mono PCM16, 8000 Hz, 0.5 seconds; access on reader activation
The exact cyclic sample reversal of phase-forward.wav; its DFT magnitudes are equal and phases conjugated.
- File byte receipt
sha256:db2a43b42f37c6feb3876bb085e2868f68d976e904fc4110a82ef0e4199638f4- PCM byte receipt
sha256:89609bbd464e1a0934275bc0b86bef66e91aec422a954b5561e86877aede29dd- Time scale
- 1 sample = 1/8000 second; 4000 frames = 0.5 second
- Amplitude scale
- Exact cyclic sample reversal of phase-forward.wav; no normalization. DFT phases conjugate while magnitudes remain equal.
- Stored PCM min / max
- -12000 / 12000
- Pre-clamp peak / clipped samples
- 12000 / 0
- Normalization / stored PCM in range
- NONE / true
Carrier/sampling declaration: Each integer-triangle burst period maps to sampleRate/period: 100 Hz, 160 Hz, 200 Hz, 250 Hz. These are authored digital carriers, not physical modes.
Spectrum caveat: The listed rates are carrier fundamentals, not complete spectra. Triangle carriers contain odd harmonics and each finite 80-frame edge envelope produces sidebands and finite-window leakage.
Complete non-audio equivalent: The public snapshot contains the complete schedule, equations, units, tables, and relation that generated this file.
Exhibit 7 · bounded mathematical music theory
Chord geometry starts by declaring equivalences and a metric
What does a one-semitone path mean, and what does it not mean?
Declared model
- System
- 12-tone equal-temperament pitch-class and registered voice-leading fixture
- Geometry
- Pitch classes form Z_12; registered pitches are integer semitone coordinates.
- Topology
- Octave equivalence maps p to p mod 12; chord order is quotiented by permutation only for pitch-class sets.
- Metric
- Voice-leading distance is the minimum L1 sum of absolute registered semitone moves under declared bijections and octave representatives.
- Boundary
- 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].
- Material
- No instrument, tuning drift, timbre, listener, or culture is represented beyond the named Western 12-TET convention.
- Operator
- Finite quotient and assignment search.
- Excitation
- No physical excitation.
- Damping
- No damping.
- Acoustic coupling
- NONE.
- Sonification
- NONE; the geometry is shown, not sounded.
Quantities
| Quantity | Value | Unit | Status | Meaning |
|---|---|---|---|---|
| Octave size in this convention | 12 approx. 12 | semitone-12tet | exact-mathematical | Defines p~p+12. |
| Declared minimum L1 path | |0-(-1)|+|4-4|+|7-7|=1 approx. 1 | semitone-12tet | exact-mathematical | C major [0,4,7] to E minor [-1,4,7]. |
Derivations and refused bridges
| Bridge | Expression | Status | Assumptions | Result / unit | Caveat |
|---|---|---|---|---|---|
| pitch-to-pitch-class | pc(p)=p mod 12 | exact-mathematical |
| cyclic pitch-class coordinate / cycle-index | Other tuning systems require another space. |
| voice-leading-distance | min over declared bijections and octave representatives of sum_i |p_i-q_sigma(i)| | exact-mathematical |
| distance 1 for the fixture / semitone-12tet | Another equivalence relation or metric can change the result. |
| distance-to-preference | short voice-leading distance -> preferred or consonant | no-canonical-conversion |
| REFUSED / not-applicable | Mathematical proximity is not an aesthetic verdict. |
| source voice | target representative | signed move st | absolute move st |
|---|---|---|---|
| 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 |
Claim
Chord and voice-leading geometry is well-defined only after equivalence relations, register choices, and a metric are declared.
Does not claim
- The L1 distance is not consonance, quality, emotion, or preference.
- 12-TET is not a universal tuning system.
- A short path does not make a progression culturally neutral.
Falsifier / counterexample
Changing the metric, tuning lattice, chord equivalence, cardinality rule, or register window can change the nearest path.
Scope: One explicit Western 12-TET equal-cardinality L1 convention.
Exhibit 8 · finite cyclic rhythm geometry
Equal onset count does not imply equal spacing—or musical rank
How can five onsets be compared on a sixteen-step cycle?
Declared model
- System
- two onset subsets of Z_16
- Geometry
- Sixteen equally spaced positions on a displayed circle.
- Topology
- Cyclic residue classes modulo 16.
- Metric
- Clockwise cyclic gap length in integer steps.
- Boundary
- Rotation begins at onset 0 for display.
- Wraparound gap is included.
- Material
- No performer, tempo, accent, duration, timbre, movement, or musical culture is encoded.
- Operator
- Cyclic difference and population variance of five gap lengths.
- Excitation
- No excitation.
- Damping
- No damping.
- Acoustic coupling
- NONE.
- Sonification
- NONE.
Quantities
| Quantity | Value | Unit | Status | Meaning |
|---|---|---|---|---|
| Onsets in each pattern | 5 approx. 5 | dimensionless | exact-mathematical | Cardinality held equal. |
| Population variance of gaps [3,3,3,3,4] | 4/25 approx. 0.16 | dimensionless | exact-mathematical | Mean gap 16/5. |
| Population variance of gaps [1,1,1,1,12] | 484/25 approx. 19.36 | dimensionless | exact-mathematical | Mean gap 16/5. |
Derivations and refused bridges
| Bridge | Expression | Status | Assumptions | Result / unit | Caveat |
|---|---|---|---|---|---|
| onsets-to-gaps | sorted cyclic differences including wraparound | exact-mathematical |
| E gaps [3,3,3,3,4]; cluster gaps [1,1,1,1,12] / cycle-index | Rotation changes the sequence start but not the gap multiset. |
| evenness-to-aesthetic | lower gap variance -> musically better or universal | no-canonical-conversion |
| REFUSED / not-applicable | Even spacing is a combinatorial property only. |
| pattern | onset set | cyclic gaps | gap variance | interpretation |
|---|---|---|---|---|
| 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 |
Claim
A cyclic onset set can be compared by exact gap structure under a declared convention.
Does not claim
- 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.
Falsifier / counterexample
Both patterns have five onsets yet have unequal exact gap variances, while neither arithmetic result supplies an aesthetic ordering.
Scope: Finite Z_16 onset-only comparison.
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- Lecture 24: Modal Analysis: Orthogonality, Mass, Stiffness, Damping Matrix — MIT OpenCourseWare; teaching-orientation. Orientation for generalized eigenmodes and mass, stiffness, and damping matrices.
- The Wave Equation — MIT OpenCourseWare; teaching-orientation. Orientation for wave-equation boundary-value models.
- The energy method — University of California, Santa Barbara; teaching-orientation. Orientation for kinetic and elastic energy in the ideal string wave equation.
- Vibration of plates — NASA Technical Reports Server; technical-monograph-orientation. Orientation for plate-vibration theory and boundary-condition dependence.
- Can One Hear the Shape of a Drum? — American Mathematical Monthly author-course copy; mathematical-primary-paper. Historical formulation of inverse spectral questions.
- One cannot hear the shape of a drum — arXiv; mathematical-primary-paper. Context for noncongruent isospectral planar domains; not a source for the finite graph fixture.
- Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects — Journal of the Optical Society of America A author copy; signal-processing-primary-paper. Orientation for the Fourier phase problem; the finite reversal theorem is proved directly in the fixture.
- The Geometry of Musical Chords — Science; music-theory-primary-paper. Orientation for pitch-class quotient spaces and geometrical voice-leading models.
- Generalized voice-leading spaces — PubMed record for Science; music-theory-primary-paper. Orientation for explicitly chosen chord equivalence relations and quotient spaces.
- The Euclidean Algorithm Generates Traditional Musical Rhythms — Bridges Archive; rhythm-primary-paper. Orientation for Euclidean distribution of onsets; not evidence of aesthetic superiority or universality.
- Simultaneous Consonance in Music Perception and Composition — Psychological Review via PubMed Central; psychoacoustics-review. Orientation for multiple accounts of simultaneous consonance and their limits.
- Indifference to dissonance in native Amazonians reveals cultural variation in music perception — Nature author copy; psychoacoustics-primary-paper. Evidence against treating consonance preference as a context-free universal aesthetic.
- On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces — Philosophical Transactions of the Royal Society historical copy; historical-primary-paper. Historical context for particle patterns on vibrating surfaces and their material dependence.
- Tonal consonance and critical bandwidth — PubMed record for the Journal of the Acoustical Society of America; psychoacoustics-primary-paper. Historical psychoacoustic orientation for roughness and critical bandwidth; not a universal preference rule.