# Equal tempered scale

The frequency of a note is not always strictly specified. Notes are tuned to each other and so, depending on which note we start with, what frequency we choose for it, and how we tune we may end up with various frequencies for the remaining notes.

Given an arbitrary starting note with an arbitrarily chosen frequency, the equal tempered scale, also known as "equal tempered tuning", is a way of computing the frequency of the remaining notes, such that the ratio of the frequencies of any two adjacent notes is the same and thus the octave is split in even intervals.

Suppose that we start with the note A = 440 Hz. The same note an octave higher will be A = 440 * 2 = 880 Hz. To create twelve equal semitones in this interval we do the following: A# / Bb = 440 * (2^(1/12)), B = 440 * (2^(2/12)), C = 440 * (2^(3/12)), C# / Db = 440 * (2^(4/12)), and so on until G# / Ab = 440 * (2^(11/12)) and A = 440 * (2^(12/12)) = 440 * 2 = 880 Hz. Note that with this computation the ratio of two adjacent notes is always equal to 2^(1/12) and thus all semitones are the same. If measured in cents, each semitone will be exactly 100 cents.

Thus the equal tempered scale splits the octave exponentially in equal intervals and the perceived distance (the ratio) between adjacent notes is the same.

The equal tempered scale differs from just tempered scales. Equal tempered scale semitones are all the same, but the note frequencies are not necessarily integer harmonics of the frequency that we start with. Just tempered scales are scales in which the notes are harmonics of the base note frequency, but the scale semitones are not the same. If we start with A = 400 Hz and we follow the equal tempered computation above we will have E = 440 * (2^(7/12)) = 659.26 Hz, whereas just tempered scale will usually set E = 660 Hz as such E is an odd order harmonic of A = 440 Hz (660 = 440 * 3 / 2). Both types of scales and tuning have advantages and disadvantages. Since equal tempered scales ensure equal semitones they allow the easy transposition of music from one scale to another. Just tempered scales sound better due to their use of harmonics, but do not allow for easy transposition as the semitones between notes differ.

An example computation of the 12 semitone equal tempered scale is provided in the topic on Frequency of notes.

Scale, Scale (index)

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