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Just Temperament

Just Temperament is the pure natural musical tuning where each note is related to the fundamental frequency of the instrument by simple whole number ratios.

The simplest ratio is the Octave which is a ratio of 2:1 i.e. the octave above is exactly twice the frequency of the fundamental and an octave below is exactly half the frequency of the fundamental.

The interval of a Perfect Fifth is the ratio 3:2 (e.g. G to D or C to G).
The interval of a Perfect Fourth is the ratio 4:3 (e.g. G to C or C to F)

Together with the Octave (2:1) and Unison (1:1), these form the so called perfect intervals.

In a diatonic scale, the intervals are:

Unison 1:1
Second 9:8
Third 5:4 (major third)
Fourth 4:3
Fifth 3:2
Sixth 5:3
Seventh 15:8
Octave 2:1

The ear and brain can detect these frequency ratios very accurately and can tell when an interval is correct or slightly out. Why the brain is able to do this is not known, but Evolutionary Musicology follows the theory that it is the result of an adaptation for evolutionary advantage (from the ability to signal territorial rights or attract mates or socially interact). Related to this is exaptation. That is, it was originally an adaptation for evolutionary advantage, but is now redundant for that purpose and is providing a secondary role as an expressive medium.

This still does not explain why certain intervals are important and easily discernible to the human brain, so for now, we have to accept that we are built that way and our music is shaped by it.

A phenomenon that results from this is that starting from a particular fundamental, and building interval by interval in perfect fifths, will eventually get to a note that is the same but a lot higher in pitch. The same is true of other intervals too such as the octave. The problem is that each interval is correct and perfectly in ratio, but the end result if two different routes are taken, is not exactly the same frequency. The reason is that that the fifth interval is odd, and the octave interval is even. It doesn't matter what you do, they will not come out to the same frequency. This drove Pythagoras mad because it was close but not exact. This difference is known today as the Pythagorean Comma.

What it meant was, that with a multi octave instrument such as a piano, organ and other keyboard instruments, it was not possible to tune every fifth as a perfect fifth and have every octave sounding as a perfect octave. It just wouldn't work. The result of this in practice is that one fifth interval is left about a quarter tone flat if octaves are preserved and this 'Wolf Interval' was generally avoided in composition and performance.

All other temperaments are attempts to disguise or mask this discrepancy to enable multi octave and ensemble playing of instruments.