Piano strings with reduced inharmonicity Jean-Pierre Dalmont, Sylvain Maugeais To cite this version: Jean-Pierre Dalmont, Sylvain Maugeais. Piano strings with reduced inharmonicity. 2020. hal- 02166229v2 HAL Id: hal-02166229 https://hal.archives-ouvertes.fr/hal-02166229v2 Preprint submitted on 6 Feb 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Piano strings with reduced inharmonicity J.P. Dalmont1), S. Maugeais2) Le Mans Universit´e,Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France 1) Laboratoire d'acoustique (LAUM, UMR CNRS 6613) 2) Laboratoire Manceau de Math´ematiques(LMM, EA 3263) defect could be rectified by using an inhomogeneous 42 1 Summary winding on the whole string, in order to minimise in- 43 44 2 Even on modern straight pianos, the inharmonicity of harmonicity. The string is thus considered to be inho- 45 3 the lower strings is rather large especially for the first mogeneous that is with a non uniform linear density 46 4 octave. Consequently, the timber of these strings can ([8]). From a theoretical point of view, the problem 47 5 sometimes sound awful and chords on the first octave translates into finding an \optimal" non uniform lin- 48 6 be highly dissonant. The idea of the present study is ear density for a stretched string with uniform stiff- 49 7 to show how this defect can be rectified using an in- ness. Here, the optimality condition amounts to being 50 8 homogeneous winding on the whole string in order to as harmonic as possible. This problem is solved using 51 9 minimize inharmonicity. The problem is solved using an optimisation procedure, initialised with the char- 52 10 an optimisation procedure considering a non uniform acteristics of a real string. The diameter of the opti- 53 11 linear density. Results show that the inharmonicity mised string is allowed to vary between the diameter 54 12 of the first partials could be highly reduced by a non of the core (supporting the winding) and about twice 55 13 uniform winding limited to a quarter of the string. the diameter of the reference string. An area with uniform winding will be kept to reduce the amount 56 of work during the manufacturing of the true string. 57 Moreover, it is proposed to limit the non uniform 58 14 1 Introduction winding to one side of the string. 59 15 So-called harmonic strings are largely used in mu- 16 sic because a uniform string without stiffness, and 2 Euler Bernoulli model 60 17 stretched between two fixed points, naturally have 18 harmonic eigenfrequencies. When considering the The model chosen for the string is linear and only 61 19 string's stiffness the eigenfrequencies are no longer involves tension, bending stiffness and mass per unit 62 20 harmonic. The consequence on instruments like harp- length. According to Chabassier, this model is suffi- 63 21 sichord and early pianos is limited. However, the de- cient for low frequencies (cf. [1], remark I.1.2, there 64 22 velopment of piano making during the 19th century is no need to add a shear term), small amplitudes. 65 23 saw a tendency of increasing string tension by a factor Moreover, even if this not completely true (see [3]), it 66 24 4, and the mass in the same proportion (cf. [2]). A is considered that the increase of stiffness due to the 67 25 consequence is that the inharmonicity cannot be con- wrapping can be neglected. So, the stiffness is that of 68 26 sidered negligible anymore, which reflects on the tun- the core (cf. [1] I.1.5) and is therefore constant along 69 27 ing of the instrument and the timbre. For the lower the string. Finally, the model is taken without any 70 28 strings a solution has been found in order to increase losses as only the eigenfrequencies are of interest. 71 29 the mass of the string without increasing too much 30 its bending stiffness: the wound strings. Neverthe- 2.1 Mathematical model 72 31 less, the inharmonicity remains rather large especially 32 for the first octave of medium grand piano, and even Let us consider the Euler-Bernoulli model without 73 33 worse on an upright piano: according to Young the losses for a stiff string of length L. 74 34 inharmonicity of the bass strings on a medium piano 35 is twice that of a grand piano, and that of a straight 36 piano twice that of a medium ([12], [7]). Our study 37 focuses on straight piano because we consider that 38 designing strings with reduced inharmonicity would 39 improve quite a lot the musical quality of these in- Figure 1: Sketch of the string and notations 40 struments. 41 The idea of the present study is to show how this The displacement equation is then given in the 1 Dalmont et al., p. 2 Fourier domain by (cf. [4]) In general, it is not possible to work directly with 106 the function µ, and a discretisation of the space is 107 @2y @4y 2 needed so that an approximation in finite dimension 108 −µω y = T 2 − EI 4 (1) @x @x can be used. 109 75 where µ is the mass per unit length (function of x), T 76 is the tension, E is the string core's Young modulus, 3.1 Numerical implementation 110 2 77 I = Ar =4 with A the core section, r the core radius When mass is non uniform, solutions of equation (1) 78 and ! is the pulsation. The string being simply supported at both ends, have to be approximated by a numerical method. The the boundary conditions are given by problem with non-constant µ is thus solved with a classical FEM in space (similar to the more complex 2 2 @ y @ y setting of [1], xII.1) using Hermite's polynomials (cf. yjx=0 = yjx=L = 2 = 2 = 0: @x x=0 @x x=L [5], 1.7) and a uniform discretisation of [0;L] for fixed L N and 0 < i < N + 1 with h = N+1 , xi = hi. 79 2.2 Solution with constant µ The projection of the operators \multiplication by µ", 2 4 T @ and EI @ then define three 2N × 2N matrices When µ(x) = µ is constant, it is possible to find an @x2 @x4 0 (µ), and so that (1) can be approximated by explicit solution of (1). The eigenfrequencies of the M T E 2 oscillator are given by (cf. [11] x3.4) ! M(µ)U = (T + E)U (2) s t 0 0 1 T π2EI with U = (y(x1); y (x1); ··· ; y(xN ); y (xN )), 111 p 2 0 fn = nf0 1 + Bn with f0 = and B = 2 : where the prime denotes the spatial derivative. 112 2L µ0 TL More precisely, the matrices T and E can be com- 113 p 2 puted using Hermite's polynomials and leads to the 114 80 The inharmonicity factor comes from 1 + Bn . formulas 115 81 This equation shows the influence of string's stiffness 0 1 .. 0. 1 . .. 82 on inharmonicity. A shorter string with small tension B 6 h C B C B − 5 10 C B −12 6h C B 2 C T B − h − h C EI B −6h 2h2 C 83 and high stiffness has a higher B, and therefore in- B 10 30 C B C 116 T = 2 B 6 h 12 C, E = 4 B C h − 5 − 10 5 0 h B −12 −6h 24 0 C B 2 2 C B C 84 harmonic eigenfrequencies. For a typical piano string, B h − h 0 4h C B 6h 2h2 0 8h2 C B 10 30 15 C @ A @ . A .. 85 the inharmonicity is minimum for the second octave .. −4 86 and in the range of 10 . It increases with the fre- and as well the matrix M(µ) is computed for func- 117 87 quency in upper octaves but also for the first octave tions µ constants on each interval ]xi; xi+1[ by 118 0. 1 88 ([9]). For a grand piano for the first note A0, B is .. B 9 13h C −4 B µi−1=2 70 −µi−1=2 420 C 89 less than 10 which leads to an inharmonicity of 20 B 2 C B µ 13h −µ h C = B i−1=2 420 i−1=2 140 C 119 M B µ 9 µ 13h (µ + µ ) 13 (−µ + µ ) 11h C 90 cents for the 16th harmonic but for a straight piano B i−1=2 70 i−1=2 420 i−1=2 i+1=2 35 i−1=2 i+1=2 210 C B 13h h2 11h h2 C −3 B −µi−1=2 420 −µi−1=2 140 (−µi−1=2 + µi+1=2) 210 (µi−1=2 + µi+1=2) 105 C 91 B can reach 10 which leads to an inharmonicity of @ . A .. 92 200 cents for the 16th harmonic (see [3]). With such where µi+1=2 denotes the value of µ on ]xi; xi+1[. 120 93 values of B, the sound of the lower string is awful and 94 chords on the first octave cause a lot of beatings in- Equation (2) is a generalised eigenvalue problem 121 95 ducing a high roughness.