pitch a); the horizontal and the vertical in music texture; up and down in the scale; etc.

In the Middle Ages, music was grouped together with arithmetic, geometry, and astronomy in the quadrivium. Music was not considered an art in the modern sense but a science allied with mathematics and physics (acoustics). mathematics of a little higher level was used in the calculation of intervals, which required the use of logarithms, and the problems of temperament required the use of continuous fractions.

Except to experts, the application of some mathematical concepts to other aspects of music, for example in the analysis, the aesthetic aspects, the composition and the mathematical music theory is practically unknown. Let us see how some selected (very few of them, due to the limited length of this paper) mathematicians and musicians have applied mathematical concepts to music.

2 Mozart’s Dice Game.

Mozart, in 1777, at the age of 21, wrote a »Musical Dice Game« K.294 (Anh. C) to write waltzes with the aid of two dice, without being a musician and without knowing anything of composition (c.f. Mozart, 1956). He wrote 176 bars, adequately selected and put them in two tables of 88 elements each. The game starts throwing the two dice, so we have 11 possible numbers (from 2 to 12), and we make 8 throws obtaining distinct bars except the ones in the last column that are equal (these last ones with two possibilities: one for the repetition and the other to continue to the next table). The second table looks like the first one, but contains different bar numbers, and the bars of the last column are all equal. So, by means of a simple calculation using combinatorics, there are , that is, approximately different waltzes. If each waltz is played, with repetition of the first part, in 30 seconds, it will be needed , that is, approximately seconds, or days approximately, or years approximately in playing all, one after another without interruption. That is, a world premier of Mozart every 30 seconds through 361 billion years! (Remember that the Stone Age began only about years ago). Mozart was a fan of mathematics and his enormous talent showed once more. With this simple game, left the impossibility that an interpreter could play his integral work or that a CD company could record it!

Even more, it shows us what a small idea we have about large numbers such as . Today, with computers and combinatorics, we still can’t handle even very small portions of motives, since the amount of isomorphism classes is exorbitant. For example, Fripertinger’s formula gives the number of affine orbits of 72 element motives in , which is of order . The number of stars in a galaxy is estimated as . So, as Guerino Mazzola says in his »Mathematical Music Theory--Status Quo 2000« (c.f. Mazzola, 2000, 2004, in this volume), »the musical universe is a serious competitor against the physical universe, in its quantity as well as in its quality of spiritual antagonist« . Hence the usage of statistical methods to control the variety of cases is predicted since not even the next generation computers will have the calculation power to check all possible cases, concludes Mazzola.