PAdicMIDI — offline applet

0. Before you start — instructions and references

What files can I upload?

Standard MIDI files (.mid or .midi), formats 0 or 1.
Recommended: 100 – 4 000 notes, duration 30 s – 10 min.
The in-browser engine treats all tracks as a quasi-monophonic stream; for rigorous polyphonic analysis use the Python package.
Not supported: SMPTE division, MIDI type 2, encrypted files, audio (WAV/MP3).

What does the applet do with your MIDI?

Parses the notes and builds a 12-D chromagram (pitch classes) on a beat-synchronous grid with bin Δ_b = 1/12.
Quantises windows of length pⁿ by k-means at each level n.
Builds the tower S₁ → … → S_{N_max} with forced inverse system π_n+1,n.
Computes the coherence invariant Coh_π(p,n) and compares it against the architectural floor 1/p.

What results will you see?

A table with Coh_π(p,n), the floor 1/p and a status badge.
Four musical views: temporal chromagram with prototype segmentation, prototype catalogue, inverse system π and the full p-adic tree.
CSV download of the values.

Privacy

Your MIDI file never leaves your browser. The applet makes no network requests, uses no cookies and sends no telemetry. You can save this HTML and run it fully offline.

What to expect as a diagnosis (with examples)

When a piece satisfies the structural hypotheses (SC) sibling-cover and (AI) ancestor-inclusion and one chooses r = p, Proposition 3.1 of paper [2] predicts Coh_π(p,n) = 1/p exactly for all n (the architectural null floor). Deviations above the floor are the quantitative fingerprint of polyphonic texture or non-canonical metric structure.

Reference piece	p	Expected outcome
BWV 1007 — prelude (monophonic, bundled demo)	2	`Coh_π(2,n) = 0.500000` exactly at every level. The badge should read on floor.
BWV 1007 — prelude	3	Range `[0.67, 0.999]` above the floor `1/3 ≈ 0.333`. Difference explained by r ≠ p: the binary content does not align with the ternary grid.
Polyphonic Bach (BWV 1049, 1050, 1079)	2	Measurable deviation from the floor: `Coh_π` typically between 0.55 and 0.95 depending on the movement. This is the texture discriminant reported in [2].
Any piece, high levels with a small corpus	—	If `\|S_n+1\|` is very small the value loses statistical stability — interpret with caution.

Key references

[1] Pérez-Buendía, J. R. (2026). Prime-power indexed multiscale graph diagnostics for symbolic temporal data: methodological exploration and delimitation via BWV 1007. Submitted to the Journal of Mathematics and Music (Taylor & Francis). Defines the descriptors β₀, D, β₀^spec and the diagnostic methodology that motivates this software.
[2] Pérez-Buendía, J. R. (2026). Profinite hierarchical patterns and prime-indexed multiscale invariants in symbolic music. Submitted. Introduces the p-adic tower D_p,n, the inverse system π_n+1,n, the invariant Coh_π(p,n) and proves Proposition 3.1 of the 1/p floor.
[3] Serre, J.-P. (1979). Local Fields. Springer GTM 67, ch. 1, §3. Theoretical support for the convergence of pushforward measures on ℤ_p.

Start with the demo: click Load BWV 1007 demo below, keep the default parameters (p=2, N_max=4) and run the analysis; you should see three rows with Coh_π = 0.500000. Then try p=3 on the same file to observe the contrast.

1. Load a MIDI file or a demo

Upload your own MIDI, drop it onto the box below, or start with one of the four bundled demos. Demos are useful to see what the theory predicts for known structures.

Expected results per demo (with K=8, N_max=4, seed 42)

Demo	Coh_π(2,·)	Coh_π(3,·)	Reading
BWV 1007 — prelude (mono Bach)	[0.500, 0.500, 0.500] floor 0.500	[1.00, 1.00, 0.54] floor 0.333	The paper's canonical case. Binary cello pulse puts p=2 exactly on the floor. With p=3 the invariant saturates: there is no ternary alignment.
Toy binary (synthetic, p=2)	[0.500, 0.500, 0.500] floor 0.500	[0.44, 0.56, 0.35] floor 0.333	Designed to satisfy (SC) and (AI) at p=2. Hits the floor for p=2; fluctuates near the ternary floor.
Toy ternary (synthetic, p=3)	[0.25, 0.50, 0.50] floor 0.500	[0.333, 0.41, 0.27] floor 0.333	Designed to satisfy (SC) and (AI) at p=3. The binary view is atypical (n=1 dips below). With p=3 level 1 hits the floor 1/3 exactly.
BWV 1079 — Crab canon (polyphonic)	[0.500, 0.500, 0.500] floor 0.500	[1.00, 0.96, 0.81] floor 0.333	Real polyphony with two voices in retrograde imitation. Strict binary structure. Same qualitative profile as BWV 1007.

To reproduce the values: load the demo, set p=2 (or 3), click Run analysis. Magnitudes can shift slightly with K and seed; the qualitative pattern (on-floor vs off-floor) does not. The column shows levels n=1, 2, 3.

Drop a .mid file here.

2. Parameters

Prime p N_max K K_child seed

Defaults reproduce the gold-standard runs of the companion papers (J. R. Pérez-Buendía, 2026) with the proviso that the in-browser implementation uses a simplified MIDI parser limited to monophonic flows; for full reproducibility use the Python package.

3. Coherence table

n	Coh_π(p,n)	1/p	\|S_n\|	\|S_n+1\|	status

4. Musical visualisations

These views show what the method is actually looking at inside the piece, not just the abstract tree. Use the buttons below to pick a level n and inspect how the piece is segmented and which prototypes dominate.

4.0 Player — listen to what is being analysed

Basic WebAudio synthesiser (triangle wave + decay envelope) that plays the loaded MIDI fully offline. Use it to relate what you hear with what you see on the chromogram in panel 4.1: pressing Play draws a red cursor that scrolls over the chromogram. Clicking directly on the chromogram seeks the player to that position.

0.00 / 0.00 s Vol

4.1 The piece coloured by prototype

Chromagram across time (width = number of beat bins of size 1/12; height = 12 pitch classes, C at the bottom, B at the top). Blue intensity encodes chromatic activation. The bottom band assigns each window of length pⁿ the colour of the prototype that quantises it (this is the temporal fingerprint of level n). Colour changes mark musically meaningful transitions.

4.2 Catalogue of prototypes at this level

Each card is a prototype: a typical chromatic pattern of length pⁿ extracted as the centroid of the windows assigned to it. Rows = 12 pitch classes (C bottom, B top); columns = time positions inside the prototype. Intensity encodes chromatic activation. The card colour matches the temporal band in 4.1, so you can read where each pattern occurs in the piece.

4.3 Inverse system π — flow of windows between levels

Diagram of the forced inverse system π_{n+1, n}: S_n+1 → S_n. Block heights and edge widths are proportional to the number of real windows falling on each prototype (label Pi·k: prototype i, k windows). Barely visible blocks = prototypes that are almost unused; thick bands = branches that concentrate traffic. When the piece satisfies (SC)+(AI) of Paper 2, the parent quantiser classifies consistently with this inheritance and Coh_π(p,n) = 1/p exactly; the visual imbalances explain why the invariant deviates from the floor.

4.4 p-adic tree — concentration of data per prototype

The full tower

S₁ → S₂ → … →
        S_{N_max}

with the real mass of the piece on top. Each node radius is proportional to √(number of windows assigned), and edge thickness/opacity reflect the flow of the corresponding child; almost-transparent nodes are practically empty prototypes. Hover over a node for the exact count. To the right of each level you see how many of the pⁿ formal prototypes are actually occupied: imbalances reveal the piece's internal repertoire (e.g. a head heavily repeated + many unique windows, vs. a more uniform distribution).
Technical note. Combinatorial branching is controlled by K_child in panel 2, not by the prime p: by default they are kept in sync (K_child = p) for an honest p-adic tower.

4.5 Heatmap of the transition matrix π

Matrix M_i,j = number of windows at level n+1 mapped to child j hanging off parent i at level n. Empty rows = parents with no occupied descendants; concentrated columns = branches absorbing most of the piece. This is the cleanest visualisation of how unbalanced the tower is in the data: the tree and the Sankey are regular by construction, but this matrix exposes the effective arithmetic of the piece. If the top bar looks roughly uniform and the matrix looks like a diagonal band you are close to the theoretical floor 1/p; if it is concentrated in a few cells, the piece has a restricted internal repertoire (lots of repetition).

Parent level n

What does this all mean?

The method quantises the piece into windows of length pⁿ (the p-adic balls) for each level n. At each level, the windows are clustered into K · pⁿ⁻¹ prototypes by k-means. The inverse system π forces each prototype at level n+1 to explicitly inherit a parent at level n.

The invariant Coh_π(p,n) measures the fraction of windows that are consistent with this inheritance. Proposition 3.1 proves that whenever the structural hypotheses (SC) and (AI) hold, the invariant equals 1/p exactly — the architectural null floor. Pieces that deviate from the floor reveal polyphonic texture or non-canonical metric structure.

In the in-browser applet, the engine is restricted to monophonic flows and small primes. For exhaustive analyses on full polyphonic corpora and comparison against null models, use the Python package with padicmidi run-suite.

About

PAdicMIDI · A Python toolkit for hierarchical, ultrametric, and p-adic analysis of symbolic music data. Jesús Rogelio Pérez Buendía (publication name: J. Rogelio Pérez-Buendía), researcher, CIMAT-Mérida. ORCID 0000-0002-7739-4779 · Web www.cimat.mx/~rogelio.perez · Email rogelio.perez@cimat.mx. Funding: SECIHTI grant CF-2019/217367. Source code: MIT licence. Companion papers: Prime-power indexed multiscale graph diagnostics (JMM, 2026) and Profinite hierarchical patterns and prime-indexed multiscale invariants in symbolic music (2026).