COMBINATORICS

Pascal's Triangle

The numbers in Pascal's Triangle are the number of distinct ways to reach them by following the arrows from the top 1. The first 5 rows of it (which were used in the piece of music) are shown in the picture.

The piece of music has three parts:

1) A treble part where the center numbers 1, 2 and 6 are the tonic D of the D alternating octatonic scale. The other vertical lines of numbers represent one scale step; a descending scale step to the left and an ascending scale step to the right. All ways of reaching all the numbers in the triangle are mapped to consecutive sixteenths, starting from the top and going down south west first to every new row, inwards. This part is repeated.

2) The main treble part. It has the structure A1 B A2 B. It starts with the left side of the symmetric triangle, including the center (A1), the numbers are the number of scale steps to ascend (south east arrows) or descend (south west arrows). After an intermission (B) where this part doubles the first part, the right side is initiated (A2) and it is the mirror image of A1.

3) A walking bass part (quarter notes), same as the first treble part but consecutive equal tones are tied. This part is repeated.

Playing time: 4' 47".

Permutations

A permutation is a way to order of a set of items. For example, if the items are the digits 1, 2 and 3, then all the permutations are (1 2 3), (1 3 2), (2 1 3), (2 3 1), (3 1 2) and (3 2 1).

This piece is based on the 120 permutations of the C pentatonic scale. It has three parts:

1) The digits 1, 2, 3, 4 and 5 are mapped to the tones C, D, F, G and A respectively. All numbers containing 1, 2, 3, 4 and 5 in the interval 12345 - 54321 are taken in ascending order.

2) On top of the last four notes of every permutation in the first part, the fourths are played. so all the 24 permutations of the current four notes are presented.

3) On top of the last three notes of every permutationsin the second part, the thirds are played, so all the 6 permutations of the current three notes are presented.

Playing time: 2' 30".

INDEX