Wander through sound. Discover the mathematics behind each turn.
Each step finds a note. The luminous courier traces the Daedalian labyrinth and lets each turn change the phrase.
The stay-put chance is α in the probability controls below.
Cells map to E Dorian by row and column. This is a musical mapping we chose, not a physical property of the maze.
Press Play to follow the luminous courier and hear its path through the Daedalian labyrinth.
32 notes · E Dorian
Play starts sound. If you hear nothing, check your volume and audio output, then try Test audio. The same seed reproduces the same notes.
Play hears the pattern on screen. New variation creates and plays another. Previous variation lets you compare.
Change Pattern to hear different rules; change Tempo to slow down or speed up.
The other tabs are separate experiments. The maze journey has its own notes; its α slider does not change the Melody studio pattern. Graph-mode sonification has its own Play control. Fineberg analysis retains the full signal-to-spectrum-to-track pipeline.
Render what you hear, export note/timing data, or send structured state across the bounded Hg bridge.
WAV captures the current audible realization. Drop it directly into Ableton as audio.
If your browser supports a native Save As picker, the Lab will use it. Firefox may instead download immediately to your configured Downloads folder.
Playback uses the current room voice; WAV preserves that audible realization.
MIDI exports note/timing data for generated sequences or graph sonification using the current audition step timing. It does not capture timbre or Bridge provenance; graph-sonification Hz values are quantized to the nearest MIDI note.
Imported MIDI lives in an isolated working-sequence lane; it never overwrites the Lab’s saved experiments.
.hgstate is JSON underneath, but preserves Bridge provenance, losses, assumptions and mapping class. Use this for the Mercurial Synthesis Chamber.
The full transformation ledger remains available; each station also carries a contextual mini-ledger.
Bounded file/state exchange. Source drives export; receiver drives realization.
The source drives the export. The receiver drives the realization. This panel does not merge the Lab and Chamber, does not use live networking, and never overwrites the Lab’s saved state.
Choose an existing Lab experiment. The export captures the current working values and maps only fields the Bridge can actually carry. Chamber filter, envelope, drive, and saturation are never invented as source mathematics.
Load the exported .hgstate there and preview before Apply Import. Opening the Chamber does not create a live connection; the exported file remains the only transfer.
Load a Chamber bridge-state file. Validation and mapping happen before anything is applied. Apply creates a separate working analysis; the Lab controls and saved experiments stay unchanged.
Empty. An applied Chamber state will appear here without altering the Lab’s experiments.
Protocol notes, the parked formal-systems branch, and validation context live here without interrupting the instrument.
Scope: this instrument is bounded to the experiments shown here. Formal incompleteness remains a referenced boundary rather than a simulated proof system.
State Exchange: The exchange remains one-way and explicit. Importing or exporting state does not make the Labyrinth and Synthesis Chamber the same instrument.
Story boundary: nothing in this laboratory rewrites The Cadet Emerald Tablet or any story text.
Same mathematical object under the stated convention.
Invertible structure-preserving map.
Common formal skeleton, different interpretation.
Selected behavior retained with explicit losses.
Useful mapping with no claim of natural identity.
No mathematical or empirical inference permitted.
Every comparison below records the object/space, operator, constraints, modes, preserved information, changes, observations, failure boundary, and classification.
This laboratory concentrates on operator, graph, perceptual, tracking, music, and sonification experiments. Formal incompleteness is named here only as a boundary: this instrument does not simulate a proof system or claim a Gödel demonstration.
Added seeded reproducibility, ordinary-vs-lazy maze evolution, TV/parity diagnostics, exact hitting systems, graph Laplacian eigenmodes, an Operator Atlas, browser spectral synthesis, beating, a framewise FFT/peak/tracking pipeline, three Markov-music state models, graph-spectrum sonification, and an interactive invariants/losses view.
π_i∝deg(i) for undirected simple walks; bipartite period-2 behavior; P_lazy=αI+(1-α)P; harmonic hitting equations; L=D-A, L_rw=I-D^{-1}A, L_sym=I-D^{-1/2}AD^{-1/2}; similarity of L_rw and L_sym; P=I-L_rw; FFT decomposition; entropy and explicit sequence metrics.
See VALIDATION.md and source/validate_phase2.py for independent numerical checks. Browser functions also expose deterministic seeds for reproduction.
Graph-mode sonification is explicitly downgraded to ANALOGY / REPRESENTATIONAL MAPPING. The normalized-Laplacian/random-walk relation is sharpened to exact similarity/isomorphism at the linear-operator level under positive degree assumptions.
Browser audio is uncalibrated; psychoacoustic responses are not guaranteed. Jacobi eigenanalysis is appropriate for this small graph, not a production large-graph solver. Partial tracking is intentionally primitive. Music models are educational toys, not corpus-trained style models.
The laboratory retains the crossing-partials reference audio, three comparison MIDI files, deterministic media-generation notes, numerical validation code, and the written validation record.
The nearest-neighbor partial tracker can swap track identity at crossings by design. Large maze sizes would require a better eigensolver. Formal-system mini-lab remains parked. No perceptual calibration or listener study is included.
Stationarity and convergence separate cleanly in the same maze; normalized graph spectra connect exactly to random-walk dynamics while combinatorial Laplacian modes remain distinct; and sonification makes visible how much “meaning” can be introduced by an arbitrary display mapping.
The most reusable tool is not a universal operator theory but an invariants-and-losses ledger: every representation change should state what survives, what is destroyed, and whether the map is invertible.
Probability moves through a finite graph. Stationarity, periodicity, laziness, parity and hitting structure stay visible together.
Gold ring: one walker. Bubbles: the probability of being in each cell. Green and amber mark the two hitting targets; white marks the start.
Top: total-variation distance to the same stationary distribution. Bottom: probability mass on the start cell's bipartition. Nonlazy parity oscillates; laziness damps the oscillation and makes the chain aperiodic.
3-D height is a display coordinate for probability mass only. Exact probabilities remain available numerically.
For target A versus B, the exact finite linear systems are solved on the generated graph.
Preserved: graph, communicating class, degree-proportional stationary distribution, hitting probabilities between fixed absorbing targets when self-loops are inserted uniformly.
Changed: temporal spectrum, period, expected clock steps to hit, TV convergence profile.
Break: if self-loop probability varies by state, even hitting-time and reversible-measure relationships can change.
stationary law and hitting probabilities in the stated experiment
period-2 parity locking / original clock
No: added self-loop changes stepwise trajectory
self-loops; changed clock
nonuniform/state-dependent laziness needs separate analysis
SHARED MATHEMATICAL STRUCTURE / exact stated relations
Graph structure is inspected through distinct operators and their eigenstructure without collapsing their domains or normalizations.
Thus L_rw and L_sym are similar and share eigenvalues; their eigenvectors are related by degree weighting. L generally has different eigenvalues and must not be silently substituted.
For the combinatorial Laplacian, the second eigenvector is the Fiedler vector when the graph is connected. Low modes often vary slowly across strongly connected regions and can reveal bottlenecks, but any narrative or perceptual meaning must be tested rather than assigned.
No generic “spectrum” is allowed to erase the operator's domain, normalization, constraints, or interpretation.
| Object / space | Operator | Constraints | Symmetry / normality | Eigenstructure / modes | Interpretation |
|---|---|---|---|---|---|
| Probability row vectors on finite states | P | P_ij≥0, row sums 1 | Not generally symmetric/normal; reversible chains are similar to symmetric form | λ=1 stationary left mode; other modes govern evolution under hypotheses | One-step stochastic evolution |
| Continuous-time probabilities | Q | off-diagonal ≥0; row sums 0 | Not generally symmetric | 0 stationary mode; nonzero eigenvalues control e^{tQ} | Infinitesimal transition rates |
| Functions on graph vertices | A | Encodes edges/weights | Symmetric for undirected graph | Adjacency modes | Connectivity operator, not itself diffusion |
| Functions on graph vertices | D | Diagonal degrees | Symmetric diagonal | Coordinate axes | Local degree weighting |
| Functions on vertices | L=D-A | PSD for undirected graph | Symmetric | Orthogonal graph modes; 0-mode constant | Combinatorial smoothness / Dirichlet energy |
| Functions on vertices | L_rw=I-D^{-1}A | Requires positive degrees | Usually nonsymmetric in Euclidean inner product | Same eigenvalues as L_sym | Directly related to random-walk P |
| Degree-weighted vertex space | L_sym=I-D^{-1/2}AD^{-1/2} | Requires positive degrees | Symmetric PSD | Orthogonal normalized graph modes | Symmetric representative of random-walk geometry |
| Periodic sampled signals | Translation / circular convolution | Translation invariance / cyclic boundary | Circulant operators are normal | DFT modes diagonalize all circulant operators | Ordinary discrete Fourier structure |
| E-Dorian note states | P_music | Explicit row-stochastic note-transition model | No guaranteed symmetry | Statistical modes of that chosen state model | Local sequence probabilities, not “tonality” by itself |
| Comparison | Object / space | Operator / constraints | Modes | Preserved | Changes / observed | Breaks | Class |
|---|---|---|---|---|---|---|---|
| P ↔ L_rw | same vertex-state coordinates | P=D⁻¹A; L_rw=I−P | same right eigenvectors; eigenvalues shift λ↔1−μ | modal directions and complete operator information | stochastic evolution is recentered as a Laplacian operator | formula changes if a different walk/normalization is chosen | SHARED MATHEMATICAL STRUCTURE exact invertible affine relation |
| L_rw ↔ L_sym | degree-weighted coordinate spaces | L_rw=D⁻¹/² L_sym D¹/² | same eigenvalues; eigenvectors degree-rescaled | spectrum and similarity invariants | Euclidean symmetry/orthogonality appears in L_sym coordinates | requires positive degrees | ISOMORPHISM / SIMILARITY |
| L ↔ L_sym | vertex functions | different degree normalizations | generally different eigenvalues/eigenvectors | zero modes encode connected components; both measure edgewise variation | degree weighting changes modal geometry | cannot substitute spectra on irregular graphs | SHARED MATHEMATICAL STRUCTURE |
| circulant walk ↔ DFT | cycle-state basis ↔ Fourier basis | circulant P diagonalized by F | DFT exponentials are exact eigenvectors | all linear information under invertible basis change | coordinates only | fails as ordinary DFT diagonalization when translation symmetry is absent | ISOMORPHISM |
| graph eigenvalue ↔ audible pitch | dimensionless graph mode ↔ acoustic Hz | user-chosen display map | rank may be preserved by monotone maps | only deliberately chosen invariants | intervals/register/perceptual salience change with mapping | no canonical physical conversion | ANALOGY / REPRESENTATIONAL MAPPING |
vertex set; operator-specific algebraic structure
operator normalization may change eigenvectors and spectrum
Only for declared complete basis transforms
visual height/color encodings
operator families are not interchangeable
IDENTITY / ISOMORPHISM / SHARED STRUCTURE only where stated
Frequency, amplitude, phase, detuning and perceptual approximation become one linked analysis field.
For equal-amplitude sinusoids, the trigonometric envelope varies at |f₂−f₁|. Whether the listener hears beats, roughness, or two resolved tones depends on frequency region, separation, level, and auditory filtering.
explicit component frequencies and amplitudes
magnitude-only views lose phase; perceptual model omits listener context
Full complex representation yes; perceptual mapping no
MERCURIAL AUDITION timbre/spatial coloration
uncalibrated playback cannot establish thresholds
ISOMORPHISM for full DFT; APPROXIMATION / ANALOGY downstream
A synthetic signal is transformed, reduced and tracked so every loss in the analysis chain can be inspected.
This is a controlled synthetic case designed to make the analysis assumptions visible. It is not a reconstruction of a Fineberg composition. The synthetic signal contains two crossing chirps, one disappearing partial, and one late-entering partial.
| Arrow | Input → transformation | Lost / changed | Reversible? | Class |
|---|---|---|---|---|
| 1 | waveform → framed windowed FFT | finite-window resolution; edge/time localization tradeoff | Not from displayed magnitudes alone | APPROXIMATION |
| 2 | spectrum → selected peaks | nonselected bins, broadband energy, phase | No | APPROXIMATION |
| 3 | peaks → nearest-neighbor tracks | identity becomes model-dependent at crossings/disappearances | No unique inverse | APPROXIMATION |
| 4 | tracked frequencies → 12-TET pitches | microtonal deviation and continuous trajectory | No | APPROXIMATION |
REFERENCE DOWNLOAD — PREBUILT. crossing/disappearing partials — reference WAV. This is a fixed reference asset, not a recording of the current controls.
selected time-frequency components
unselected spectrum, window context, track identity at crossings, pitch deviation after quantization
Only full complex FFT stage; later reductions no
peak-selection/tracking/model assumptions
greedy tracks can exchange identity
ISOMORPHISM → APPROXIMATION as reductions accumulate
Stochastic generation and graph sonification remain separate mappings with explicit metrics and failure boundaries.
The original 32-note generator. Independent of the maze journey above.
Technical shorthand: CONSTRAINT SATISFACTION ≠ STYLE MODEL ≠ BACH.
The same 32-note E-Dorian task is generated three ways using reproducible randomness. The expanded-history model remains Markov after the state is redefined as (previous,current); its transition weights are hand-specified, not learned from Bach or any corpus.
REFERENCE DOWNLOADS — PREBUILT, NOT YOUR CURRENT GENERATED RESULT. These three fixed reference files remain available so the same examples can always be inspected again: reference first-order MIDI · reference expanded-state MIDI · reference hard-constraint MIDI. To save what you just generated, use the SAVE buttons directly above.
D→E at an 8-note phrase boundary.The graph eigenvalues are unchanged. Only the display mapping from ordered eigenvalues to audible frequency changes.
declared transition constraints or eigenvalue ordering
corpus/form beyond model state; interval geometry under arbitrary pitch mapping
No for reduced models / sonification
mapping scale, register, instrument realization
constraint satisfaction is not style; pitch map is not natural law
APPROXIMATION / ANALOGY / REPRESENTATIONAL MAPPING