Equal Temperament (ET) is a system that gives equal distance & balance to every interval inside of its respective octave. ET is globally attractive because it's relateable to all 12 keys signatures at once.
ET tuning scheme is tempered,
meaning that the intervals are altered in order to keep
equal distance from octave to octave.
In numeric definition this means that 1+1=2 (c1),
2+2=4 (c2 another octave),
4+4=8 (c3 another octave),
8+8=16 (c4) and so on..
This is considered as the Law Of Octaves.
1+1=2 is conceptually the same as:
Using the note name of C vibrating at 128Hz
the Law Of Octaves continues as:
After 12 repetitions of this cycle (12 octaves) a drift in tuning occurs between the original tone & the final tone. The numeric representations of these types of discrepancies are known as Commas & Diesis.
The extra distance after 12 Octaves is equal to the Pythagorean Comma (PC). In order to rectify this extra space, ET tempers each octave by 1/12 of the PC, the extras are compensated for.
In other words,
ET folds the extra space equally
& then disperses it back into itself.
The tuning scheme of Equal Temperament allows for the 'mechanics & reasoning' that govern the natural motions of harmony to become self-evident. It's a harmony purely for the sake of harmony itself. This makes the ET system 'out of tune' (or artificial) in comparison to the vibrational functions of nature that Just & Pythagorean systems both outline. ET is unnatural.
Here is a chart that shows the differences among the 3 systems: