spectral-prosody

Spectral graph theory applied to metrical patterns across languages.

Poetry traditions have spectral fingerprints — the eigenvalues of graph Laplacians constructed from metrical structure reveal deep structural properties. Independent cultures converge on isospectral meters, suggesting universal constraints from breath and cognition.

Modules

Module	Description
`metrical_graph`	Construct weighted graphs from poetic corpora. Nodes = lines, edges = metrical similarity, weights = syllable distance.
`laplacian_scan`	Compute graph Laplacian eigenvalues for poetic traditions. Each tradition = point in spectral space.
`rhyme_adjacency`	Encode rhyme schemes as adjacency matrices. ABAB, AABB, ABBA, and free verse produce distinct spectral radii.
`tradition_embedding`	Low-dimensional spectral embeddings of poetic traditions. Visualize the "space of all poetry".
`iso_breath`	Test the conjecture that breath-constrained meters produce isospectral graphs regardless of language.
`dial_scan`	Connect to dial-theory: spectral distance between traditions as a dial dimension.
`linalg`	Gaussian elimination and Jacobi eigenvalue method from scratch. No external math dependencies.

Core Types

use spectral_prosody::*;

// A line of poetry with metrical information
let line = MetricalLine::new(
    vec![1.0; 10],                                        // syllable durations
    vec![false, true, false, true, false, true, false, true, false, true], // iambic stress
    "English".into(),
);

// Build a graph from multiple lines
let graph = MetricalGraph::from_lines(&lines);

// Extract spectral signature
let signature = SpectralSignature::from_graph(&graph, "English Iambic Pentameter");
println!("Fiedler value: {}", signature.fiedler_value());
println!("Spectral radius: {}", signature.spectral_radius());

// Classify rhyme schemes
let abab = RhymeScheme::from_str("ABAB");
println!("Spectral radius: {}", abab.spectral_radius());

// Compare traditions spectrally
let distance = sig_en.distance_to(&sig_fr);
let similarity = sig_en.cosine_similarity(&sig_fr);

Mathematical Foundation

Graph Construction

Lines of poetry become nodes. Edge weights encode metrical similarity:

w(i,j) = 0.6 × stress_similarity(i,j) + 0.4 × syllable_distance(i,j)

Laplacian Eigenvalues

The unnormalized graph Laplacian L = D - A encodes metrical structure in its spectrum:

λ₁ = 0 (always, for connected components)
λ₂ (Fiedler value) = algebraic connectivity, measures metrical coherence
λ_max = spectral radius, relates to rhythmic complexity

Jacobi Eigenvalue Method

Eigenvalues are computed via the Jacobi iterative method — rotations annihilate off-diagonal entries of the symmetric Laplacian matrix. Converges for all real symmetric matrices.

Cheeger Constant

The Cheeger constant h(G) measures the "bottleneck" of the metrical graph:

h²/2 ≤ λ₂ ≤ 2h

Approximated as h ≈ √(2λ₂).

Iso-Breath Conjecture

Meters constrained by human breath capacity (~1 breath unit ≈ 10-16 syllables) produce isospectral graph structures regardless of language family. Iambic pentameter (English, 10 syllables), Sanskrit sloka (16 syllables), and French alexandrine (12 syllables) all inhabit a narrow region of spectral space.

Dial Theory

Each eigenvalue index becomes a "dial dimension" along which traditions vary. The full dial-space distance between traditions captures structural divergence beyond simple syllable counting.

Testing

cargo test

65 tests covering:

Metrical graph construction and Laplacian properties
Jacobi eigenvalue computation (identity, 2×2, 3×3 path graph)
Gaussian elimination (identity, general, singular)
Rhyme scheme spectral classification (ABAB, AABB, ABBA, free verse)
Spectral clustering and tradition classification
Cheeger constant computation
Random walk traversal time (Kemeny's constant)
Tradition embedding and dimensionality reduction
Iso-breath conjecture validation
Dial-space construction and proximity ranking
Serde roundtrips for all public types

Dependencies

serde — serialization for all public types
serde_json — test-only JSON roundtrips

No external math libraries. All linear algebra implemented from scratch.

License

MIT

spectral-prosody 0.1.0