A real piano string is a fixed-fixed vibrating string that supports standing waves: when plucked or struck by a hammer, it settles into a superposition of the fundamental and its integer-multiple overtones (the harmonic series). Each mode n has n antinodes and n+1 nodes (including the two fixed ends), and its frequency is f_n = n × f_1, where f_1 is set by the string's length, tension, and mass per unit length via f_1 = (1/2L)√(T/μ). The relative strength of each harmonic determines the string's timbre — why a piano note sounds different from a flute playing the same pitch. Piano strings are under enormous tension, and bass strings are wound with copper to add mass without adding excessive stiffness, keeping their fundamental low while staying physically short enough to fit inside the case.

• Full piano range: A0 (27.5 Hz) to C8 (4186 Hz), just over 7 octaves.
• Typical string tension: roughly 700-900 N per string (some designs reach ~1000 N).
• Total tension of all ~220 strings in a grand piano: about 15-20 tons on the frame.
• Treble strings: plain high-carbon steel wire; bass strings: steel core with copper winding for added mass.
• Mode n has n antinodes and (n+1) nodes, including the two clamped endpoints.
• Harmonic frequencies follow f_n = n × f_1 (integer multiples of the fundamental).
• Struck strings decay over seconds; higher harmonics typically damp out faster than the fundamental.
• Middle C (C4) fundamental is 261.6 Hz; A4 (concert pitch) is 440 Hz.