# Composing with the Thue-Morse Sequence

The Thue-Morse sequence is a binary sequence and was first used by Eugene Prouhet in 1851 by applying this sequence to the study of integers.  However, it was Axel Thue in 1906 that used this sequence to create the field of study called, “combinatorics on words”.  In 1921 Marston Morse applied the sequence to differential geometry.  Morse used this sequence to prove that geodesics (mapping of straight lines to curved space) are recurrent but non-periodic on certain surfaces of negative curvature (i.e. hyperboloid).  Here, the sequence will be used to construct a rhythm pattern for composing music.  The Thue-Morse sequence can be generated in several ways.  A very simple way of generating the sequence is to let 0 -> 01 and let 1 -> 10.  The first eight digits of the sequence can be built up in the following manner:

0 -> 01 -> 0110 -> 01101001

this is just taking the initial state 0 -> 01 and replacing the 0 with 01 and  and replacing the 1 with 10 to get 0110, and so forth.  Other ways to generate the sequence is with using bitwise negation, this is an operation that yields the bitwise complement of the operand.  Every bit has an opposite.  For example the bit 0 has the complement (opposite) equal to 1.  Applying this technique to the above operator 0 -> 01, complement of 01 is 10, this is then added to the sequence to create 0110, moving forward, the complement of 0110 is 1001, and this is added to the sequence to create 01101001, which yields the same result as above.  Remember that 0 and 1 could represent anything.  For example, 0 could represent a word such as “hello” and 1 could represent another word such as “goodbye”.

The Thue-Morse sequence has some unique properties.  The sequence is self-similar as represented in the following diagram: as shown above in the diagram, the 0’s and 1’s are replaced with colored square to get an idea of how this sequence can be visualized.  The property of self-similarity is apparent, by striking out every other number the original sequence is produced.  The self-similarity property of this sequence leads to its fractal nature.  Another interesting property of the Thue-Morse sequence is that it contains no cube sequences.  For example, the sequences 000 or 111 will never be found.  However, the square sequences such as 00 or 11 are present.  Now we will focus on creating a rhythmic structure for composing music.

Lets choose a trichord as a first example, we will use a C major chord; CEG.  Then we will apply the Thue-Morse sequence as the rhythm pattern for performing arpeggios of the this chord.  If we let 0 = 1/16 notes and 1 = 1/4 notes, and every time a square sequence is encounters, such as 00, or 1/16 1/16, we will add these two together to get an 1/8 note.  The following diagram represent the rhythm of  just the single not C: this is interesting to look as an educational tool and can by itself produce a pleasant sound.  Moving onto the trichord, here is the representation of this same rhythm patter applied to the major chord CEG: CEG_squares

This is starting to look more like a piece of music.  Now lets try something slightly more complicated.  We will use the diminished chord, Cdim7, and apply this rhythm sequence then at the end of the piece the diminished chord will resolved to the dominant form.  Here is the rhythm pattern applied to Cdim7: Cdim7_resolved_Cdom7

and there you have it.  There are many ways to use the Thue-Morse sequence as applied to music composition, one could imagine that instead of using the sequence for rhythm, it would be possible to substitute notes for 0 and 1, or maybe short squares could dictate what note is used.

Other properties of the Thue-Morse sequence are very interesting.  For example, this sequence can be used to construct the Koch snowflake.  The following diagram expelling how this is accomplished: You could write a composition where the Thue-Morse sequence is used in combination with the Koch snowflake pattern to produce a piece of music.