Mozart Musikalisches Würfelspiel

It seems like it's hard not to talk about Mozart this year. Here, however, we're focusing on an aspect of Wolfgang's work that's perhaps a little more playful, but still very interesting.
In 1793, two years after his death, a Musikalisches Würfelspiel (musical dice game) was published to great success. It allowed one to compose a minuet without any musical knowledge, using two dice and a series of predefined measures. It's an example of algorithmic composition ante litteram, although it's worth remembering that the second half of the 18th century saw a certain flourishing of musical games of this type.

Mozart's authorship of this work has long been debated, but it now seems established.
The rules to follow are very simple. A minuet, at the time, had a nearly standardized form, in 3/4 time, composed of a first part (the minuet), followed by a central section, more melodic and relaxed, called a "trio" because it was originally performed by three instruments and concluded with a reprise of the original minuet.
Both the minuet and the trio were 16 bars long, and in creating their internal structure it was possible to use a precise scheme. The minuet from Kleine Nachtmusik, for example, is composed of a 4-bar phrase (A) followed by a variation of equal duration (A'); then we have a second phrase (B), also 4 bars long, which concludes with a reprise of A'. The trio opens with an eight-bar melody (C), followed by a different four-bar motif (D), which concludes with a shortened reprise of C.
It was structures of this type that allowed Mozart's compositional automation. The game is based on two tables, one for the minuet, for which two dice are used, and the other for the trio, for which a single die is sufficient, and a booklet containing a list of 176 bars composed by Mozart. Each table has the bar number on the x-axis and the result of the dice roll on the y-axis. The bar number to be used is found at the intersection of the x-axis and y-axis.

The user simply rolls the dice once for each measure and transcribes the corresponding measure by copying it from the list and placing it after the previous one.
Thus, by rolling the dice 16 times, the 16 measures that make up the minuet are obtained. The table for the minuet, in fact, has 11 rows (the dice range from 2 to 12) by 16 columns, and in this case, Mozart wrote 11 possible measures for each of the 16 measures of the minuet (11 * 16 = 176).
To "compose" the trio, however, a single die is used, and the table has only 6 rows. For each measure, therefore, Mozart wrote 6 possible measures.

Mozart's skill, in this case, was to write bars that could be logically linked without creating significant disruptions to each previous one. Since this is tonal music, this is less complex than it might seem at first glance (the key is always the same, D for the minuet, G for the trio), but it remains quite challenging when considering that a composition of this type does not allow for major sequential irregularities.

Finally, it would be interesting to know whether Mozart, in arranging the bars in the tables, took into account the different probabilities of the outcome determined by the roll of the dice. This problem does not arise in the case of the trio, where a single die is used and each number has a probability of 1/6, but rather in the minuet, where two dice are used with different probabilities for the 11 possible outcomes. With two dice, in fact, the 7 has the highest probability because it can be obtained with six different combinations. The 6 and 8 are followed by 5, the 5 and 9 by 4, the 4 and 10 by 3, the 3 and 11 by 2, while the 2 and 12 are obtained with just one.
Ultimately, however, with this system, 11¹⁶ can be generated, that is, 45,949,729,863,572,161 minuets, and 6¹⁶, that is, 2,821,109,907,456 trios. Since the minuet as a musical form is composed of a minuet and a trio, we have a total of 129,629,238,163,050,258,624,287,932,416 different possible compositions. Obviously, many of them will be almost identical, as the difference will be only one or two measures; Nonetheless, out of such a large total, there are still many interesting combinations to listen to.
The problem, however, is that, even leaving out the final reprise of the minuet, each piece lasts about a minute, and therefore it would take 246,630,970,629,852,090,228,858 years to listen to them all.

By clicking here you can listen to one of these minuets, here in MIDIfile, while here you can find a site that implements the game.

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