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From Exploring Theory with Practica Musica:

Temperaments are various methods of tuning the notes of the octave in order to make the pitches come out "even." You would think that the ideal way to tune your piano would be to make each of the consonant intervals sound exactly in tune, but that doesn’t work out very well on a keyboard with only twelve keys to the octave. For example, you can tune your A so that it will make a very nice sixth above your C, and the G as a good fifth above the C, and the D to make a good fifth below A, but then you’ll find that the G will be out of tune with the D. In other words, it doesn’t come out even; you can’t tune all the consonances to be simultaneously exactly right unless you have two different keys for some of the notes, such as one D tuned to agree with G and another D made to work with A.

This problem was solved in many different ways over the years. The first solution was to tune the fifths exactly right and not to worry about the thirds and sixths, just as Pythagoras, the semi-legendary discoverer of musical mathematics, had done. This Pythagorean tuning was the basis of organ tuning and music theory until around the late 14th or early 15th century, which is why thirds and sixths were regarded before that time as dissonant intervals.

But when musicians began to want to use the consonant third a compromise had to be found, and they responded by tuning that troublesome D to the mean (average) of the two that were needed, which produced a system in which most of the thirds were very good and most of the fifths were only slightly off; this we now call meantone temperament (to “temper” an interval means to adjust its tuning a little bit away from perfect).

Meantone was used for several centuries and is still enjoyed by connoisseurs of early music, but it had one problem: since four of the less-common thirds were very discordant (and one fifth, the wolf fifth, was really awful), meantone didn’t allow the free use of all the intervals in all the keys. So musicians began to alter the compromise a little with the hope of making the wolf and the bad thirds less objectionable. The Vallotti temperament is a variant of meantone in which the wolf is gone, the bad thirds are not quite so large, and the good thirds are, in trade, not quite so good as they were in normal meantone. Other people invented schemes in which some chords have both perfect thirds and perfect fifths and others have varying shades of inexactness; the Kirnberger temperament is an example. In these tunings that grew out of meantone there begins to be some variety in the quality of the acceptable chords, which may have contributed to the notion that different keys have different characters — for example, C major was thought to be pure and cheerful, while F# major was harsh and dark.

The temperament that won out in the end, at least so far, was equal temperament, in which all the thirds are just barely OK but all the fifths are very good. Equal temperament has much to recommend it; besides being the same in all keys its large major thirds are well-suited to melody, which is an important factor in our music. For some reason musicians often prefer large thirds for melody even though the smaller thirds make smoother harmony. Fortunately, most of us are able to accept the larger thirds in harmony, too, especially since the modern piano does not emphasize the higher overtones the way the harpsichord did. Nonetheless, many musicians experiment with tuning, and there may yet be something of a revival of alternate systems, even ones with more than 12 notes to the octave, which are much easier to implement with electronic musical instruments than they were with mechanical ones.

Exploring Theory contains charts of each provided temperament, if anyone is interested. For the present purpose it is enough to know that the Kirnberger 1/2 comma is the most irregular temperament provided, with a D-A interval that is noticeably discordant, and that the "extended" temperaments each have 35 notes per octave, without enharmonic equivalents (D# cannot be used as Eb, etc.). Extended Meantone is probably the best temperament for harmonic purposes owing to its low thirds, while the extended Pythagorean might be thought best for melodic purposes.

If you are using a Macintosh computer you can listen to your composition in various historical temperaments, including extended meantone, which has more than 12 pitches to the octave. Unfortunately the music synthesizer on Windows does not support microtonal adjustments.

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