Inharmonicity of Coupled Vibration of a Piano String and a Soundboard
Takeshi Yanagisawa, Mikio MORIOKA, Kijuro Nakamura · Journal of the Japan Society of Precision Engineering · 1967
This paper describes inharmonicities of a piano string vibration. Inharmonicities δon are first measured with a soundboard fixed and then inharmonicities δn with a soundboard unfixed. The latter are larger in a bass section than the former, and in mediant and treble sections there is no difference between the two. The vibration of a piano string is not interpreted as independent vibration, but as coupled vibration. Theoretical treatment of coupled vibration of a string with the soundboard leads to the following solution.δn-δon=1200 log2(1+εn-ε1)_??_1731(εn-ε1) [cent]and εn_??_(n2π2M/ρl-kl/T)-1where M ; the equivalent mass of a soundboardk ; the spring constant of a soundboardρ; the linear density of the springT ; the tension of the string l; the speaking length of the string, The above experimental facts are well explained by these theoretical calculations.