Inharmonische

Inharmonische, in music theory often referred to as inharmonicity, describe partial frequencies that do not form an exact harmonic series relative to the fundamental. In a purely harmonic system, the n-th overtone has frequency n times the fundamental. In real instruments, overtones deviate from this ideal, producing inharmonische Obertöne that enrich or alter the instrument’s timbre.

Causes of inharmonische include physical properties of vibrating bodies. String stiffness, nonuniform density, geometry, boundary conditions, and air loading all contribute to deviations from exact integer multiples. Percussion instruments, bells, and metallic bodies frequently exhibit stronger inharmonicity due to their shape and stiffness. The effect is especially noticeable in instruments with short, stiff strings or complex resonators.

Impact and implications include shaping timbre, perceived brightness, and the perception of pitch. Inharmonische complicate tuning and intonation, particularly across wide pitch ranges. On the piano, string stiffness causes higher partials to be steeper than the ideal harmonics, which leads tuners to apply stretch tuning to compensate, tuning octaves slightly wider than pure octaves.

Measurement and modeling of inharmonische often use an inharmonicity coefficient, a parameter in models such as f_n ≈ f_1 n sqrt(1 + B n^2), where f_n is the frequency of the n-th partial and B reflects stiffness and construction. Larger B indicates stronger deviation from the harmonic series, influenced by material, geometry, and tension.

Applications of understanding inharmonische span instrument design, tuning systems, psychoacoustics, and digital sound synthesis, where accurate modeling of partials improves realism and perceptual quality.