Banjo Break Angle Tension Modulation as Parametric Oscillation
HHDP: 20 – 01
Banjo Break Angle Tension Modulation as Parametric Oscillation
David Politzer ∗ (Dated: January 9, 2020)The motion of the floating bridge of the banjo, in conjunction with the break angleof the strings over that bridge, produces string tension modulation that is first orderin the amplitude of the string motion. This note refines a previous suggestion regard-ing the impact on the frequencies of the strings’ and bridge’s motion. For a givenmode frequency pair of string and bridge, the resulting tension modulation producesa new, additional motion characterized by the sum and difference of the original ones.Strictly speaking, this corresponds to canonical “frequency modulation” only in thelimit of modulation slow compared to the string frequency. The more general resultis precisely an example of what is known as “parametric oscillation,” first analyzedby Rayleigh. The qualitative impact of tension modulation on banjo timbre remainsas suggested previously. It is only the precise math and physics that warrants thiscorrection. ∗ a r X i v : . [ phy s i c s . pop - ph ] J a n Banjo Break Angle Tension Modulation as Parametric Oscillation
BACKGROUND
As they pass over the bridge, the strings of the banjo form a “break” angle — typicallysomewhere between 3 o and 16 o — which is determined by the height of the bridge and thegeometry of the tailpiece, as illustrated in FIG. 1. bridgeneck string rim tailpieceheadto nut break angle FIG. 1It is long-known to players and builders that increasing that angle enhances the “ring”and “clang” of the instrument. A direct demonstration of that phenomenon was presentedin ref. [1], which contains sound samples of a particular instrument where the break anglecan be either zero or 13 o . Note that zero degrees requires a specially designed bridge toallow the strings to function normally without buzzing, as shown on page 1. Links to therecordings themselves are also given in this note’s appendix.For a given length and linear density, the fundamental pitch frequency of a string isproportional to the square-root of the tension. It was an obvious leap to imagine thattension modulation produced frequency modulation. The sound of audio range frequencymodulation has a rich and wonderful history since its 1973 discovery by John Chowning,the licensing to Yamaha to become Stanford University’s second highest earning patent,the production of the Yamaha DX7, and all that followed.[2] Chowning described his firstexamples as sounding metallic.While the string stretch associated with small vibrations and fixed ends is second order inthe amplitude, a non-zero break angle produces stretching that is first order in the amplitudewhen the bridge moves. A natural assumption was that this phenomenon produced the soundof Chowning’s frequency modulation.[3]But frequency modulation is usually understood to have a precise mathematical meaning.An initially sinusoidal oscillation in time is modulated in frequency at some other sinusoidalfrequency and amplitude. When the modulation is weak in amplitude and much lower infrequency, its principal effect is to add side bands at the sum and difference frequencies.(This is the basis of FM radio transmission.)An explicit analysis of the basic equations of motion given below shows that the physicalmechanism of tension modulation literally produces frequency modulation only in the limitthat the modulation frequency is much less than the initial string frequency. In the caseof the banjo, there are certainly frequency components of the tension modulations that arecomparable and higher than some of the initial string frequencies. And then a purely sinu-soidal tension modulation does not translate into a purely sinusoidal frequency modulation.What does in fact happen is an example of what has come to be called parametric oscil-lation, which was first analyzed by Rayleigh.[4] For a given modal frequency pair of stringand small amplitude bridge motion, the result is new, additional string and bridge motionscharacterized by the sum and difference frequencies of the original ones. TENSION MODULATION IN THE STRING EQUATION OF MOTION
The zeroth order string motion can be expressed as a sum over the ideal normal modes.It could be just one of them or plucked triangular traveling waves or any other possible freemotion of the string. The tension modulation (again, assumed to be simultaneous along thelength of the string) is a function of time. It need not have the periodicity of the string oreven be periodic. It is the net result of the forces of all modes of all the strings on the bridgeand head. In any case, it has a Fourier transform. Even if there is only a single string’smotion determining the bridge motion, in Fourier space we generally have to consider termswith string and bridge having different frequencies.In the following, a dimensionless parameter b is the relative amplitude of the tensionmodulation and is assumed to be small. As shown in ref. [3], b depends linearly on thestring amplitude, A . I work to lowest non-trivial order in b because keeping anything higherthan b would be inconsistent — such things having been ignored in the ideal string equationin the first place.So (as they say) without loss of generality, we can consider a single ideal string mode, withfrequency ω o and wave number k , as the zeroth order string motion, and a single frequencyof tension modulation ω . In particular, let the tension T ( t ) be T o (1 + b cos ωt ). Let ψ ( z, t )be the string transverse displacement. For small amplitudes, it satisfies ρ ¨ ψ = T ( t ) ψ (cid:48)(cid:48) .And let ψ = ψ o + bφ , with both ψ o and φ vanishing at both ends. Take ψ o = A cos ω o t sin kz .Even before trying to find φ , the zeroth order ψ o exerts an O ( b ) vertical force on the bridgeof bT o cos ωt ∂∂z ψ o = bT o Ak cos ωt cos ω o t = bT o Ak (cos ω + t + ω − t ), where ω ± ≡ ω o ± ω .To find the equation for φ ( z, t ) to lowest order in b : Let v o ≡ ( T o /ρ ) / = ω o /k . Keepingonly the terms linear in b , you get¨ φ = v o φ (cid:48)(cid:48) + v o cos ωt ψ (cid:48)(cid:48) o .Let φ ( z, t ) = f ( t ) sin kz . Then¨ f ( t ) = − v o k f ( t ) − v o k A cos ωt cos ω o t = − ω o f ( t ) − ω o A (cos ω + t + cos ω − t ) .So f ( t ) is simply a forced SHO. You add the particular solutions for the two forcing fre-quencies. In particular, with f ( t ) = f + ( t ) + f − ( t ) , f ± ( t ) = ω o A ω o − ω ± ) cos ω ± t = ω o A ω ( ∓ ω o − ω ) cos ω ± t .The general solution for f ( t ) also includes oscillation at ω o , independent of ω . That partshould be absorbed in ψ o ( t ). The pole at ω = 0 is just a permanent retuning of thestring. The pole in f − ( t ) at ω = 2 ω o is the standard parametric oscillator resonance. Inparticular, tension modulation at ω ≈ ω o drives the string mode with cos ω − t ≈ cos ω o t ,i.e., approximately at the original mode frequency.The resulting vertical force on the bridge due to φ ( z, t ) is bT o kf ( t ). Adding this tothe O ( b ) force on the bridge due to ψ o and T ( t ), the total O ( b ) force on the bridge is T o bA { (1 − ωωo (2+ ωωo ) )cos( ω o + ω ) t + (1 + ωωo (2 − ωωo ) )cos( ω o − ω ) t } . DECAY TIMES & RESONANCE WIDTHS A is the amplitude of a plucked string mode. So it decays in time more or less (or exactly)like an exponential. The modulation amplitude b also decays similarly, although potentiallywith a somewhat different decay time. (Strictly speaking, the decays can be the sum ofmore than one exponential and also may exhibit beats.) The tension-modulation-inducedamplitude decays like bA , i.e., faster than either separately. Hence, the tension modulationeffect appears in the attack or early part of the pluck sound.These decay times are also reflected in the behavior near the resonance in f ± ( t ). Theactual behavior will have a finite maximum and a related resonance width. This couldbe exhibited explicitly by including damping in the equations of motion. But it is also anecessary consequence of the fact that the resonance amplitude can only build up to a finitevalue in a finite amount of time. THE PROBLEM WITH FREQUENCY MODULATION
The starting equation, ρ ¨ ψ = T ( t ) ψ (cid:48)(cid:48) = T o (1 + b cos ωt ) ψ (cid:48)(cid:48) , suggests that the whole effectmight simply be viewed as a modulation of v o , i.e., v o → v ( t ) = v o (1 + b cos ωt ) or amodulation of the flow of time, e.g., t → τ ( t ) = t − b ω sin ωt . However, any such ansatz forthe wave ψ does not satisfy the basic equation of motion. The discrepancy is O ( ωω o b ), i.e.,it goes away if the modulation is slow compared to the original frequency.In terms of traveling waves, tension modulation changes the speed of travel but not thewave shapes in space only if ω (cid:28) ω o . More generally, tension modulation changes the waveshapes as well. CONCLUSION
The implication of all this is that the general motion of the banjo floating bridge witha non-zero break angle can introduce components into the early part of the sound that arenot the harmonics of the plucked string. Note that even for frequencies already present inthe zeroth order string motion, additional sounds at those same frequencies will alter thetimbre. I would welcome any suggestion of an alternative explanation for the differences insound that are presented in the Appendix, below. A proper analysis would require a modelof the total bridge motion and a double sum over string and bridge motion modes. The lowfrequency string modes generally produce the largest bridge motion ( b ), so it’s expected thatsignificant contributions come from ω (cid:28) ω o and this gives the sound of canonical frequencymodulation. APPENDIX
Here are links to the sounds of the 0 o and 13 o break angle configurations. The banjo is ayear 2000 Deering Goodtime, played with a solid disk back spaced 1 / (cid:48)(cid:48) ~ politzer/zero-break/A.mp3 and then switch A to B,C, and D. If you can’t hear which is which, this exercise is pretty bootless. [1] D. Politzer, Banjo Ring from Stretching String: A Zero Break Angle Demo ~ politzer.[2] J. Chowning, The Synthesis of Complex Audio Spectra by Means of Frequency Modulation , J.Audio Eng. Soc. (7), 526 (1973). And some Yamaha PR: history of FM synthesizers athttps://usa.yamaha.com/products/contents/music production/synth 40th/history/chapter02/index.html[3] D. Politzer, Banjo timbre from string stretching and frequency modulation ~ politzer — or directly as String Stretching,Frequency Modulation, and Banjo Clang .[4] Lord Rayleigh,
On Maintained Vibrations , Phil.Mag.S.5,15