The question is, why would anyone change our music intonation system and add a number of tones more than doubling the repository of tones per octave? The 12-tone system we are nowadays using has lasted for several centuries now and survived various music epochs and is currently about to survive the music style pluralism we are facing today.
Besides from academic or not-so-academic curiosity (just wanting to know how different intonations sound) there are real technical and historical reasons why one would to rethink the decision that has been taken so long ago about the number of tones to use.
Another question, why this odd number 31, and not somthing like 24. In fact, there is an intonation system using 24 tones (Quatertone Music), and it has the appealing advantage that you can divide it in 2, 3, 4, 6, 8, or 12 parts. In contrast, you cannot divide 31 by any number.
Let us first think about how our current equally tempered 12-tone system developed.
A long time ago in ancient Greece the philosopher Pythagoras discovered the psychological nature of intervals with tones played on chords with simple length ratios. It is that time a relation between well-sounding choords and simple frequency ratios came into consciousness. Basically Pythagoras said any interval to be constructable by stacking perfect fifths (exact frequency ratios 3/2). Read more about the Pythagorean Tuning here
The history of many centuries after Pythagoras (he died somewhat after 510 b.C.) more or less circled around the music theory founded by Pythagoras. Only in the 16th century and maybe some time before that, a new tuning scheme arose. Its name was Meantone Temperament. As a fact it is a whole family of tunings, of which probably the Quarter-Comma Meantone is the most famous one. Here the basic idea is to detune the perfect fifth in favor of perfect major thirds. This procedure was probably first mathematically accurately described by Gioseffo Zarlino in the late 16th century. Read more about Quarter-Comma Meantone here .
The problem with the Meantone Temperament was its inability to provide for modulations between different signatures. Say you have a Meantone series starting from Eb (here also given the ratios if equally tempered, explanation below):
While for Eb-Major this is a quite accurate tuning, with the major third being perfect and the fifth bE-Bb being only 5.4 cents off the perfect fifth (a cent is the 100th-part of a semitone, on a logarithmic scale), the major third of B-Major is 1.2800 insted of 1.25, which is 42 cents apart from the perfect major third. This is an intolerable deviation, so, as a fact, there is no usable major third above the ground tone in B major, and henceforth a modulation form Eb Major to B Major impossible.
Also, if you take the fifth between Eb and G#, it has a ratio of 1.53127, which is 36 cents off the perfect fifth. Again unusable, and since that time this interval was called Wolf Fifth.
It is indeed the case, that a few triads sound very well in the Meantone Temperament, while some others sound very ugly and are unusable. And there is nothing like equality between different scales given the same tuning.
This inequality was removed by J.S. Bach and contemporaries in the late 17th and beginning 18th century. They evened out the differences between the major and minor scales given a tuning, which was then called Equal Temperament . Actually there were some people using it long before that in the late 16th century, so it has lived besides the Meantone Temperament. But only in J.S. Bach's time it has gained major influence and superseded its competitor. In Equal Temperament all intervals are perfect stackings of a half tones each given as a frequency ratio of SQRT12(2) and by that more or less "impure", but impure to the same degree. Go here if you want to know more about the Equal Temperament
So what about the 31-tone scale? It is given by simply dividing the octave into 31 perfectly equal parts. As a matter of fact, it has rolled back history to the time before J.S. Bach and taken a different approach to overcome the inadequacies of the Meantone Scale without losing its advantages. Indeed we will see, that it contains the Meantone Temperament as well as the free transposability of the Equal Temperament. Let's see how this can happen. Go here to see how a 31-tone tuning develops