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<   No. 4356   2020-10-05   >

Comic #4356

1 Salesman: Hey friends! Wanna buy a hyperguitar, with relativistic whammy bar?
1 Iki Piki: How does that work?
2 Salesman: String vibration frequency depends on tension and mass. Regular guitars change strings’ tension to bend notes. This one changes their mass.
3 Salesman: It spins each string at a few billion r.p.m. so the outer layer is moving close to the speed of light.
4 Iki Piki: Isn’t that... dangerous?
4 Salesman: Heavy metal bands love them!

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A whammy bar is the small lever you can see on many electric guitars, which the player uses to alter the pitch of notes as the strings vibrate, producing either a vibrato or pitch bend effect. The whammy bar is part of a larger vibrato system that includes the mechanical components necessary to change the tension of the strings.

The frequencies at which taut strings vibrate are fixed and reliable, due to the effect of resonance. The mechanical system has an intrinsic frequency at which it will vibrate when excited, and vibrations of that frequency are reinforced while those at other frequencies are damped. The result is that plucking a string—or bowing it in the case of violins and similar bowed instruments—produces a predictable musical note. The whole concept of string instruments (including guitars, violins, and pianos among others) rests on this physical principle - if if were not true then such instruments would not exist.

The frequency at which a string will vibrate can be calculated according to Mersenne's laws, first formulated by the French mathematician and music theorist Marin Mersenne in 1637. His laws state that the fundamental frequency of a stretched string varies as:

  1. the reciprocal of the length L of the string
  2. the square root of the tensile T force in the string
  3. the reciprocal of the square root of the mass per unit length μ of the string

Combining all these, and using SI units, the fundamental vibration frequency f in hertz is given by the following equation:

f = 1/(2L) sqrt(T/μ)

Guitar strings are tuned by varying the tension on the string using the tuning knobs. You can also use different weights of string, but changing that (on a conventional guitar) requires changing the string, so is a slower and more labour intensive process.

When playing, the use of a whammy bar may vary the length of the string by a very small amount, but primarily it works by changing the tension on the string. Pulling the string tauter makes the tension and hence the pitch higher, while relaxing the tension makes the pitch of the note lower.

Changing the length of the string is doable mechanically by moving the bridge and the nut[1] further apart or closer together.

But naturally[2] I wondered how one would go about implementing a whammy bar by changing the mass of the string while you're in the process of playing the instrument. Special relativity to the rescue!

As objects accelerate to close to the speed of light, their relativistic mass increases by the Lorentz factor γ. Moving a string in a straight line at close to the speed of light would risk destroying the instrument[3], so I came up with the idea of spinning the strings about their long axis, which keeps them in the same place and maintains the integrity of the guitar[4].

[1] The bridge is the part of a string instrument that supports the strings at the lower end and transmits vibrations to a soundboard (for an acoustic instrument - an electric instrument picks up the vibrations electromagnetically and them amplifies them electronically). The nut is the part that supports the strings at the top of the neck, near the tuning pegs. These are the two parts of a guitar (or other string instrument) between which the strings are stretched and vibrate, defining the length of the stretched string.

[2] For some definition of "naturally".

[3] To put it mildly.

[4] More or less.

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