The Compass Limit of a Stretched String: The Frequency–Length Product Set by Wire Strength

For a plain stretched string sounding its fundamental, the product of frequency and vibrating length obeys f*L = 0.5*sqrt(sigma/rho), where sigma is the tensile stress and rho the material density. Because sigma cannot exceed the breaking stress, every string material has a hard ceiling C = 0.5*sqrt(sigma_break/rho) in hertz-metres that no gauge, tension, or workmanship can exceed. Music wire allows 287.8 Hz*m, natural gut 238.4 Hz*m, drawn nylon monofilament 268.2 Hz*m, and filament carbon fibre, aramid and UHMWPE 708.1-949.8 Hz*m on a filament basis. The top plain strings of the violin, guitar, classical guitar and piano all operate at f*L = 209.3-216.2 Hz*m (52.9-82.3% of breaking stress). A steel string at the violin's operating stress drifts flat by 1.68 cents per degree C between rigid supports; measured gut strings drift flat by up to about 2 cents per degree C. Any claim of a wire extending compass beyond the sqrt(sigma/rho) bound without changing material class is arithmetic, not engineering.