Square Root Melodies

For the use of Square Roots in music composition, the process is very similar to that of 12 tone rows and the 12 tone manner of writing. As said in Schoenberg's Fundamentals of Music Composition, the first step in creating larger forms is "the construction of simple themes." At this point of writing, it is more important write within the pitch class as melodically as possible. Using this example, I will work backwards from a complete series of pieces on piano, to the writing of simple themes and motifs. 

The first example will use a certain pitch class that is derived from the distance between two chords in a theoretical 3D space: A Major (A, C#, E) and (F#, G#, 0). In the theoretical 3D space, the coordinates of these two chords are (1, 5, 8) and (10, 12, 0). The pythagorean distance between the two points is equal to the square root of 210. Utilizing the first 12 integers as note values (A=1, ..., G#=12), the following note sequence appears: 14.4913767461 -> <A, C, C, F, A, B, Eb, D, Eb, C, D, A>. 

Now, the current scale of the square root of 210 will be used in part. However, I will combine the elements of the two chords with the sequence in an arbitrary manner. So, instead of having the above sequence, the sequence used in the following two pieces are as follows: <A, Bb, B, C, E, F, G#>. This ordered sequence of notes are derived from the above square root sequence, the two chords expressed earlier, and the concept of writing in two keys (mostly A and Bb Major) throughout the collection of piano pieces. 

Below, is the sequence presented and two works: a Waltz in A and POG March.