Introduction
One of the anomalies which lies at the heart of tunings systems used in western music can be demonstrated like this:
Where you end up is a major third higher than where you started. You can see this on a piano keyboard:
However, if the perfect fifths are all just, the resulting major third will not be just -- it will be sharp (if you'd like to see the arithmetic behind this, click here).
Musicians have coped with this discrepancy in many ways. Some systems abandon the idea of having the intervals of a certain type (e.g. all perfect fifths) be the same size (if you have a Macintosh computer and would like to explore some of these tunings, click here). In several tuning systems, however, many intervals of a certain type are the same size.
Equal-interval tunings
In Pythagorean tuning (used in medieval music), 11 of the 12 possible fifths are just (with the result that 11 of the 12 major thirds are sharp). In equal-tempered tuning, all 12 fifths are tuned a little flat (and the major thirds are somewhat less sharp than in Pythagorean tuning, though still sharp). In meantone tunings, the fifths are flattened considerably, so as to bring the major third closer to just.
It is possible to arrange all equal-interval tunings in a continuum, like this:
Here, each vertical line corresponds to one possible tuning. The position where that line crosses the "Major 3rd", "Minor 3rd" and "Perfect Fifth" lines indicates the size of those intervals in that particular tuning; the "just intervals" line indicates where an interval is "pure" (a whole-number ratio). Each vertical grid division corresponds to a one cent deviation from just (1 cent = 1/100 of a semitone).
In the Pythagorean tuning, the perfect fifth is pure, the major 3rd is very sharp, and the minor 3rd is very flat. In equal temperament, the perfect fifth is a little flat, and the thirds are slightly better than in Pythagorean tuning.
As we move to the right from equal temperament, we enter the realm of meantone temperaments; in these, the tuning of the fifth is sacrificed in order to bring the major and minor third closer to pure intervals. In 1/4-comma meantone, the major third is pure.
Once we get a little to the right of 1/3-comma meantone, the thirds and fifths all diverge further and further from pure intervals.
When the fifth gets 1 cent smaller, the major third gets 4 cents smaller (which is why its line is so much steeper). A triad (major or minor) is made up of one major and one minor third, adding up to a perfect fifth. When you make the fifth a little smaller, the major third gets a lot smaller -- and the minor third gets bigger to take up the slack.
What about the minor third?