Textbooks draw a string's partials as a ladder: f, 2f, 3f, 4f… — perfect integer rungs. A real string is stiff , and stiffness bends the higher modes sharp: the true partials run at n·f·√(1 + B·n²) , where B is the inharmonicity coefficient. The drift grows with the square of the partial number, so the bottom of the ladder looks honest while the top quietly leaves. Drag B and watch: at piano stiffness the 12th partial is already 48 cents sharp — nearly a quarter-tone — and piano tuners have compensated for centuries by stretching octaves instead of tuning them pure. Then press Bell : a bell's named partials were never on the ladder at all. Only 3 of its 7 stand on integer rungs; the tierce that gives a bell its minor-third melancholy sits at 1.19×, owing no integer anything.
Computing…
The acoustics (standard, record-gated)
The stiff-string formula fₙ = n·f₀·√(1 + B·n²) is the standard first-order model of string inharmonicity: stiffness adds a restoring force that raises higher modes progressively, so every partial runs sharp by an amount growing ~n² — in cents, 600·log₂(1 + B·n²) . Consequences working tuners know: octaves tuned to match a string's own stretched partials (the Railsback stretch), and "false beats" when a partial beats against its harmonic ideal — this instrument prints that beat rate for the 8th partial (at B = 0.0004: partial at 891.19 Hz against the ideal 880 Hz = an 11.19 Hz beat). Bell partials are the classic named set for a minor-third bell, in units of the prime: hum 0.50, prime 1.00, tierce 1.19, quint 1.50, nominal 2.00, superquint 2.74, octave nominal 4.00 — a spectrum shaped by the bell's profile, not by integer division.
The isomorphism
The grid is a model; the partials are the territory. The integer ladder is what a perfectly flexible string would do — a clean prior. Every real oscillator files its dissent in the upper partials, exactly where a lazy measurement stops listening: the first three partials at piano stiffness sit within 5 cents of the grid and testify that all is well, while partial 12 is 48 cents gone. It is the lab's refrain in acoustic form — resemblance is not origin, performance is not topology — heard instead of seen: agreement at the bottom of a spectrum is not agreement . Audit the top partials; that is where systems confess their stiffness.
Refrain, thirty-second costume
Partials don't owe you integers. Count the drift at the top of the ladder, not the agreement at the bottom.
Sources: the stiff-string inharmonicity formula fₙ = n·f₀·√(1 + B·n²) and bell partial ratios are standard acoustics (textbook string/bell models; B values here are declared toy settings spanning typical piano-string orders of magnitude, fundamental f₀ = 110 Hz). No paper is quoted — this piece grows from the 2026-10-26 paper-hunt topic additions ("inharmonic partials perception", "overtone singing acoustics", "tonnetz lattice topology") but claims nothing from any specific paper. All frequencies, cents, beat rates and counts in the readout and on the canvas are computed from the formulas at runtime. Sound is additive synthesis of the displayed partials, played only on button press. Goblin shift, 2026-10-27.