On inharmonicity in bass guitar strings with application to tapered and lumped constructions

Abstract

In this study, the inharmonicity of bass guitar strings with and without areas of lowered and raised mass near the saddle is studied. Using a very high sample rate of over 900 kHz enabled finite difference time domain simulation to be applied for strings that simultaneously have nonzero stiffness and linear density which varies along the length of the string. Results are compared to experiments on specially constructed strings. Perturbation theory is demonstrated to be sufficiently accurate (and much more computationally efficient) for practical design purposes in reducing inharmonicity. The subject of inharmonicity is well known in pianos but has not been studied extensively in bass guitar strings. Here, the inharmonicity is found to be low in the lowest (open string) pitch on the five string bass guitar (B 0) given typical standard construction. Conversely, the inharmonicity is high (around 100 cents at the 10th partial) when that string is sounded when stopped at the 12th fret and very high (around 100 cents at the 6th partial) when that string is stopped at the 21st fret. Bass guitar strings were constructed with three different constructions (standard, tapered and lumped) in order to demonstrate how incorporating a lump of raised mass near the saddle can achieve close to zero inharmonicity for the lower frequency partials. This also has potential in terms of improving the use of higher fret numbers for musical harmony (reducing beating) and also in controlling pitch glide that has, with some exceptions, often been attributed solely to nonlinear behaviour.

Piano bass strings with reduced inharmonicity: theory and experiments

“…Sanderson [9] used strings where a winding was applied towards the end of the string and then reversed over the rest of the string to leave an increased mass near the piano string end. Kemp [10] used a separate layer of winding of smaller cross-section placed under the main winding(s) of bass guitar strings. The risk with over winding is that it may increase locally the stiffness which may counterbalance the desired effect.…”

“…These strings are marketed as Sanderson Accu-String. In a recent paper, Kemp [10] applied about the same idea to the bass guitar. Modeling the string using finite difference time domain modelling as well as an analytical perturbation method he succeeded in reducing significantly the anharmonicity of the first partials.…”

The acoustical behavior of a bass guitar bridge with no saddles

“…The most straightforward way of doing this is to simulate the effect of the core knot (where two thicknesses of core wire are present instead of one) by adding a fictitious section of cylindrical winding of diameter d f within the calculation such that it contributes an equal mass per unit length to that of main core wire in the vicinity of the knot. Equation (2) from [ 8 ] gives the ratio of mass per unit length in the constructed string to the mass per unit length of the core, and cancelling terms to simplify gives: where d 1 is the diameter of the core at its widest point and d 2 is the width across the core plus the first section of winding and d M is the width across the full constructed string with a core plus M − 1 layers of winding etc., and is the ratio of cross sectional area to radius squared assuming a hexagonal core wire (whereas γ core = π would be used if simulating a cylindrical core wire). If we set the mass per unit length of the fictitious winding equal to the mass per unit length of the core (setting M = 2 and τ = 2 to temporarily consider the knot only), we get the relationship: The diameter of the “fictitious winding” (which is the diameter of spiral winding around a straight core that gives the doubling of linear density expected of a knotted core) is then: …”

“…The mode frequencies were then calculated using the perturbation theory method by application of the following equations [ 8 ]: which gives the frequency of the p th mode, in terms of the (unperturbed) mode frequencies of a stiff string: with where E is the Young’s modulus of the (steel) core, S is the cross-sectional area of the core, is the radius of gyration of the hexagonal cross-section core, L is the sounding length of the string, τμ core is the mass per unit length of the main sounding length of the string, and is the ideal fundamental (if both stiffness and perturbations had been ignored). Finally, the perturbation in Eq 4 requires the factor [ 8 ]: where J = 3 is the number of sections in the case of vibrations parallel to the body. The vector x j consists of the x coordinates of the changes in density with x 0 = 0 and …”

“…The reduction in inharmonicity is expected to be smaller for the thicker strings as the core size does not usually increase linearly with the overall width of the string (in order to prevent excess stiffness and prevent reduced engineering strain in the core). An example of a B 0 string (usually the thickest string on five string electric bass guitar) is seen in recent work [ 8 ] as having τ = 16.0 in the main section of string, and adding a knot (modelled using Eq 3 ) within the sounding length of such as string would give τ = 19.5 there, hence giving the factor ( τ ( j ) − τ ( J ))/ τ ( J ) = 0.22 to 2 significant figures. This means that strength of the factor acting to reduce the inharmonicity for such a B 0 string is almost of halved in comparison to that for the G 2 string discussed above.…”