Motivic Development

The aim of this study is to present the decimal values of square roots as potentially infinite melodies. This can aid the user of this system ideas to create new works for the 21st Century. While there are pieces written with square root models, this exercise will combine the works of Schenker(1), Schoenberg(2), Kirnberger(3), Slonimsky(4), and Hellmer(5). This writing exercise can also operate as a method for understanding square root values in general. The method used for converting square root values to music is called Cartesian Mapping of Chords and Scales(6).The method allows for integers in a sequence, much like the square roots, to have a one-to-one correspondence to the 12-tones. 


The first set of square roots to be studied in this manner will be between the squares of 6 and 7; which gives 12 distinct melodies starting on 6|D.


Here are the a few of the decimal values for the select set of square roots (Square root of 37-48):

The possibilities for melodies, harmonies, and tonalities are infinite. Although this is true with the diatonic scale(), the square root melodies provide many harmonic issues at the onset of creating motifs. Since the notes are derived from a numerical sequence rather than the harmonic series, the entire tonality of the selected string of integers via place values are left to interpretation of the user of the system. One way to confront this issue is to create a sequence of triads based on the set of notes found in the selected square root value. 

For this example, the square root of 48 will be used. 

This sequencing can be used to create motifs and periods. Another method is a Cantus Firmus; with either the upper voice or lower voice as the set melody. 

Both of the methods serve as a starting point for understanding how to compose with Square Root values as a sequence of musical notes. For an example of a piece written with a square root value as its inspiration, there is a piece written about the square root of 2 (see below).