The Mathematical Guitar

According to the New Oxford American Dictionary, the [acoustic] guitar is “a stringed musical instrument with a fretted fingerboard, typically incurved sides, and six or twelve strings, played by plucking or strumming with the fingers or a plectrum”. The same reference describes the electric guitar as being “a guitar with a built-in pickup or pickups that convert sound vibrations into electrical signals for amplification”.

There are many types of acoustic guitars (like the folk, the twelve-string and the jazz guitars) as well as of electric guitars (where, besides the shape of the body, the type and location of the pickups are key features). The strings can be made of nylon or of steel, and many distinct tunings exist. It is also large the number of parts that make up the instrument, especially in the electric version.

Physically speaking, the acoustic guitar is a system of coupled vibrators: sound is produced by the strings and radiated by the guitar body. In electric guitars, the vibrations of the body are not of primary importance: string vibrations are captured by pickups and “radiated” by external amplifiers. Pickups can be electromagnetic or piezoelectric. Piezoelectric pickups are also common in acoustic guitars, eliminating the need of microphones, although microphones capture better the “acoustic” nature of the sound produced.

In summary, a guitar is a six-strings musical instrument, acoustic or electric, with the default tuning 82.41, 110.00, 146.83, 196.00, 246.94 and 329.63 Hz, from the top to the bottom string, according to the point of view of the audience. The mentioned fundamental frequencies correspond to MIDI notes E2, A2, D3, G3, B3 and E4, respectively. It is also supposed that frets are separated in the fretboard according to the equally-tempered scale.

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