ScoreCompute

Design a drum for a chord

shape_the_sound

Read-onlyIdempotent

Kac's question in reverse: design a polygonal drumhead whose lowest tones play requested frequency ratios (a fifth, a triad, a pentatonic scale), each accepted within ±tolerance_cents. proven_impossible is returned only when proven theorems exclude every ratio combination inside those tolerance windows for a connected uniform membrane with a fixed edge: simple fundamental, f2/f1 ≤ j11/j01 ≈ 1.5933 with equality for the disk (Ashbaugh–Benguria 1992), and Yang's universal inequality minimized exactly over the window box; an exact-target exclusion is reported separately. Otherwise multi-start L-BFGS with exact finite-element shape derivatives designs a 5–16-vertex polygon, then reanalyses it on three refined meshes with the shared hear_the_shape solver: achieved ratios, errors in cents with estimated uncertainty (not rigorous bounds), resolution warnings, optional numerical tuning of the fundamental in Hz. not_achieved is a bounded-search result, never an impossibility proof. Ideal membrane only.

Input Schema

Name	Required	Description
`seed`	No	Default 1. Same seed, same drum.
`vertices`	No	Polygon vertices, default 10.
`wave_speed`	No	m/s, √(tension/areal density).
`target_ratios`	Yes	Frequency ratios f2/f1, f3/f1, … of the lowest modes, ascending (e.g. [1.25, 1.5] for a just major triad).
`fundamental_hz`	No	With wave_speed: scale the drum so its fundamental sounds at this frequency.
`max_evaluations`	No	Search budget in finite-element evaluations, default 300.
`tolerance_cents`	No	Success when \|error\| + uncertainty ≤ tolerance for every ratio; default 5 cents.
`include_mode_shapes`	No	Adds computed mode shapes from the shared forward solver (≤1024 triangles, full-precision coordinates) for visualization.

Schema Changelog

Changes observed during successful MCP inspections.

Changed1 schema field changedNov 9, 2026

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AddedOct 4, 2026

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