Banjo Break Angle Tension Modulation as Parametric Oscillation

Abstract

The motion of the floating bridge of the banjo, in conjunction with the break angle of the strings over that bridge, produces string tension modulation that is first order in the amplitude of the string motion. This note refines a previous suggestion regarding the impact on the frequencies of the strings’ and bridge’s motion. For a given mode frequency pair of string and bridge, the resulting tension modulation produces a new, additional motion characterized by the sum and difference of the original ones.Strictly speaking, this corresponds to canonical “frequency modulation” only in the limit of modulation slow compared to the string frequency. The more general result is precisely an example of what is known as “parametric oscillation,” first analyzed by Rayleigh. The qualitative impact of tension modulation on banjo timbre remains as suggested previously. It is only the precise math and physics that warrants this correction.