Temperament

Starting from a base note³, Pythagoras discovered that the interval with ratio $1:2$ is most harmonious, called the perfect octave. Next, the interval with ratio $2:3$ is also harmonious, called the perfect fifth. The principle is: simpler ratios yield more harmonious intervals. If we let $𝑟=3/2$ then:

Pythagorean tuning works well but isn't ideal for traditional composition and performance. Specifically, C♯ and D♭ have different pitches, causing difficulties in transposition, modulation, and chord derivation. To solve this, musicians developed twelve-tone equal temperament. Its principle is dividing an octave into 12 equal intervals, each with ratio $2^{1/12}$. Letting $𝑟=2^{1/12}$, we get:

The trade-off is imperfect intervals, e.g. the perfect fifth becomes $1.4983𝐶$ instead of $1.5𝐶$.

1. Wikipedia: Pythagorean tuning
2. Wikipedia: Pythagorean tuning
3. In DAWs like FL Studio, this note is usually A, and $𝐴=440\text{ Hz}$