While the following art piece operates as a mathematical proofing based on the Cartesian Mapping of Chords and Scales, the resulting proof is very artistic. There are symmetries whose beauty is hard to ignore. Below is the an interactive 3D rendering of the proof.
The explanation of the theory can be found on the on my research profile. The theory is as follows:
Each note in the Western Harmonic Practice can be reduced to one octave. In this octave, there are 12 notes (A-1, A#/Bb-2, ..., G#/Ab-12). With this numeration, one can apply an algebraic sequence to this set of 12 notes. In total, there are actually 13 elements in this set of music notes with silence being represented as an empty set ∅.
The algebraic sequence will be that of the major scale:
Major Scale: (n, n+2, n+4, n+5, n+7, n+9, n+11)
(n-12 iff n>12):(12, 2, 4, 5, 7, 9, 11)
The above algebraic sequence can be read as:
For the major scale of any key, there is a starting note named, n. This scale has 7 notes based on the original starting note. Starting from the point n, add 2 to n for the next note in the major scale sequence. For the third note, add 4 to n. This continues until the last note where one would add 11 to n.
NOTE: the resulting algebraic sequencing cannot produce a number greater than 12. If and only if n is greater than 12, subtract 12 from n.
Here's an example of the A Major scale:
(1, 3, 5, 6, 8, 10, 12) -> (A, B, C#, D, E, F#, G#)
This algebraic sequence can be extended out to all 12 notes to gain the full algebraic sequence for the major scales.
Here's an example of the G#/Ab Major Scale:
(12, 12+2, 12+4, 12+5, 12+7, 12+9, 12+11)
(n-12 iff n>12)
Therefore,
(12, 2, 4, 5, 7, 9, 11)
The same process can be performed for major triads:
Major Triad: (n, n+4, n+7)
(n-12 iff n>12)
A Major triad: (1, 5, 8)->(A, C#, E)
While not expressed algebraically, there are 6 total permutations of three elements. This is true for Western Music Theory as well.
The inversions are as follows:
(1, 5, 8), (1, 8, 5), (5, 8, 1), (5, 1, 8), (8, 1, 5), (8, 5, 1)
At this point, all of the triads found in the A Major tonality can be expressed algebraically; and thusly, mapped onto a 3D plane (x, y, z) with the different permutations of each of the 7 triads associated with A Major (see the interactive proof above).
While the visualization can be overwhelming, the use of the associated values like distances between chords, or the area of the circle for each of the inversions can be used to write music.
For instance, if one were to choose a random number to map onto the cartesian mapping system, one could choose the square root of two. This number has a non-terminating decimal; which means there is an infinite amount of qualities to attach to the mapping.
Linked below is an example of using the Cartesian Mapping system, the square root of two, and one's ear to write a piece of music.