Polynomials

The Western music system uses 12 different notes to represent pitch; C, C#/D♭, D, D#/E♭, E, F, F#/G♭, G, G#/A♭, A, A#/B♭ and B. These notes then repeat each “octave” (an octave is how the sets of 12 notes are divided) with notes of each new octave having twice the pitch of the last. This exponential property's also consistent within octaves; each new note being 2^(1/12) times the frequency of the last. This allows us to model the frequency of notes as b x 2^(x/12), b being the frequency of the base note and x the number of semitones (notes) above b.

In the Western music system, b is defined as A4 (the note A in the fourth octave) at 440Hz (oscillating 440 times per second). Using this formula and value of b, we can convert numbers from our series into musical notes, however large numbers in our series will generate incredibly high pitch notes outside of our hearing range. To avoid this, we first take the modulus 12 of x (dividing x by 12 and only keeping the remainder) before using it. This means that we can't hear what octave the note is in but this is fine as equivalent notes across octaves sound similar already.

Polynomials consist of various powers of a variable each multiplied by a coefficient. For example, 5x³ + 3x² - 4x + 9 would be a cubic polynomial.

To calculate polynomials, simply substitute your value (or series term in the context of series) into the equation. For each term of the polynommial, raise your value to the relevant power and multiply by the coefficient. Then sum all of the terms to get your result.

Polynomials are often written in the form axⁿ + bxⁿ⁻¹ + cxⁿ⁻² + dxⁿ⁻³..., n representing the highest power of x and each other letter representing its coefficient. To input the polynomial you want to hear, simply write in each coefficient separated by spaces.