Algorithms/Spiral Method

From MusicIn2D

The first algorithm I developed for Geia was based on my own Spiral Composition Method, which expands a musical sequence using rotation, or cyclic permutation. The basis of a composition in this method is an ordered set consisting of 8 notes, one of each of the modal tones of a particular 7-note scale and the octave of the tonic. The set forms the basis of the melody and also directs movement through dimensions. In its earliest form, the algorithm had fixed dimensions, although recent evolution of the method allow for more choice concerning the order, type and number of permutations.

The original dimensions were:

Dimension 1: Monophonic melody
Dimension 2. Harmonic movement
Dimension 3: Tonic modulation via standard or pitch axis transposition

Originally, a composition could consist of 1, 2 or 3 dimensions, with increasing complexity and duration with each added dimension. The algorithm currently allows for up to 4 dimensions, although Geia can theoretically support more. The problem of course is the exponential extension of the resulting composition, with a 3-D composition at 120bpm lasting a little over 4 minutes, while 4 and 5 dimensional pieces would run 34 minutes and 4 and a half hours respectively.

As an example, if our sequence was in the key of C Major, a 3 dimensional spiral composition would consist of a melody composed of the notes C, D, E, F, G, A, B and the octave of C, in that order, being the first dimension. The melody would be repeated enharmonically moving through the modes of the tonic scale starting in C Major and moving to D Dorian, E Phrygian, etc, in the order of the sequence, that being the second dimension. The 3rd dimension via standard transposition would entail modulation of the result of the 1st and second dimensions through the key centers defined in the sequence, in this case transposing from C Major, to D Major, to E Major, etc. Using pitch axis modulation, the 2nd dimension would be transposed from C Major, to C Dorian, to C Phrygian, etc.

Other limitation factors of the algorithm are that the first note of the row must be the tonic and that each note appear only once within the row, with the exception of the tonic octave. Those restrictions provide a focus on melodic movement without ambiguity. Confining the permutation operations to rotation preserves the intervallic structure of the initial sequence, which is the central postulate of musical set theory.

Examples:

My first 3D spiral composition was Deadly Germs in 2006. This song has a very simple arrangement, so the progression through the dimensions can be heard fairly easily. This composition was actually pre-Geia and was arranged note by painstaking note.

Another less apparent example is the title track from Spiralology which takes a minute to get into the modal sequence progression and only features the "theme" or 1st and 2nd dimension on the 1 (intro), 8 (third section) and again at the end as a closer. The structure of the piece, however, is based firmly on the 3rd dimension via harmony and key modulations which follow the modal row.