Abstract
In modern straight pianos and half-grand pianos the inharmonicity due to the bending stiffness of the strings makes the sound of lower notes of poor quality. The question of whether this can be corrected has been addressed in a previous paper and it was shown on the basis of numerical simulations that this might be possible. In the present paper we show using analytical models that fixing a single mass near the end of the string is a simple and efficient solution. Indeed, the experiments show that theoretical predictions are relevant. So, in practice, on the basis of a simple formula involving the inharmonicity coefficient, it is easy to deduce the position of a given mass in order to reduce significantly the inharmonicity of the bass strings.
Highlights
The use of harmonic resonators with harmonic eigenfrequencies – here called harmonic resonators – is the basis of most musical instruments because the harmonicity is needed so that an unambiguous pitch is assigned to a periodic or quasi-periodic signal
This phenomenon is marked on upright piano with strings around 1 m length
The study conducted here allows us to understand the results from the digital optimization proposed by Maugeais together with the present author in a previous paper [11]
Summary
The use of harmonic resonators with harmonic eigenfrequencies – here called harmonic resonators – is the basis of most musical instruments because the harmonicity is needed so that an unambiguous pitch is assigned to a periodic or quasi-periodic signal (i.e. with decreasing amplitude). Instrument makers use naturally harmonic resonators such as homogeneous strings or cylindrical pipes. Overall the inharmonicity increases with the pitch of the note but in the first 2 octaves this inharmonicity increases when the pitch of the note decreases [5, 6] This phenomenon is marked on upright piano with strings around 1 m length. The tonal quality of the notes of the first octave is poor and it is difficult to assign a pitch to the first notes [7] This phenomenon is undoubtedly together with the mechanics, one of the reasons why professional pianists are turning away from upright pianos
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