Combinatorial algorithms

By combinatorial algorithms we simply mean different ways in which objects can be chosen or arranged. For instance, the symbols * + – can be arranged in different orders as follows:

* + –
* – +
+ * –
+ – *
– * +
– + *

Playing with all possible arrangements or permutations of musical structures gives a systematic way of varying musical material, be it melodic, harmonic, rhythmic etc.

The earliest known example of a compositional procedure which might be called combinatorial is described in Guido d’Arezzo‘s treatise Micrologus de disciplina artis musicae (c. 1025-1028). The method, introduced as a way of setting a Latin text to music, is similar to the following example.

We start by considering the vowels of a text, and how to assign a note to each of them. There are more notes in the 12-tone scale than vowels in Latin, so at least some vowels can be assigned to more than one note. For instance:

a: do, la, sol#
e: re, ti, la#
i: mi, do#
o: fa, re#
u: sol, fa#

Now, consider a text to be musicalised. Here we take the Hymn to St John (in fact, the original source of the familiar solmisation syllables):

UT queant laxis,
REsonare fibris,
MIra gestorum,
FAmuli tuorum,
SOLve polluti,
LAbii reatum,
Sancte Ioannes.

Extract the vowels from each line:

uueaai
eoaeii
iaeou
auiuou
oeoui
aiieau
aeioae

Then use the vowel-note identification table above to associate each letter with a note. Since the note assignations are not uniquely determined for each vowel, choose one randomly from the available set. So, for the first verse, a possible choice would be:

sol  fa#  re  sol#  la  mi

An entire poem can be set this way, for voices or even for instruments alone. The patterns can also be transposed according to a chosen modulatory scheme. Moreover, in exactly the same way as with St. John’s Hymn, a different labelling of the notes can be proposed according to the chosen poem.

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