Equal temperament

Even though Marin Mersenne produced a correct and systematic description in 1635, equal temperament was not adopted until nearly 100 years later in Germany, and it was not commercially established in England until 1841.

Equal temperament paved the way for music to develop to where it is today because of its simplicity of use. It allowed jazz players to roam unrestricted among tonal centres. The music tended toward chord sequences with a widening definition of chromatic chordal resources. Given these advantages, it is understandable why musicians sacrificed pure harmony for convenience.

Calculating equally tempered intervals:

We know that an octave is a doubling of the pitch or frequency. To calculate the values of 12 steps within the octave, rising between the numbers 1 and 2, mathematicians came up with the formula:

where n = one of the 12 notes.

As the 12th root of 2 = 1.059463094, the second interval will be 1.059463094 squared, the third interval will be 1.059463094 to the power of 3, and so on. If you want to start with the standard A=440, then A# will be 440 x 1.059463094 = 466.16 cycles per second.