Linguistic Analysis of Western Music Theory: Harmonic Focus

This mapping provides the user of the system with a mathematical lens to comprehend certain aspects of music theory; specifically within western music theory in regards to harmony and rhythm. This theory is a combination of a plethora of different theories in mathematics and music; and are as follows: Analysis of harmony and rhythm American military music (1), Schoenberg 12-tone theory (2), Linear Algebra (3) and Differential Geometry (4) approach to music theory and tuning, the Pythagorean Lambdoma (5). While the musical and mathematical concepts aren't novel, the application of both fields to Western Harmony and Rhythm is in a novel practice in this configuration. 

The visual representation of Western Music Theory are as follows:

a. Standard Notation

b. Algebraic Expressions

c. Geometric Proofs

There does exist the wave analysis and the biomechanical aspects of music productions. However, these visual representations are not fundamental to understanding the three aspects of the theory listed above. 

The following concepts will be explained with the three types of visual representation:

A triad in A Major, (A, C#, E), in root position and a set of rhtythms where there are 4 eighth notes with 2 quarter notes enclosing the eighth notes. 

a. Standard notation

Figure 1. 

b. Algebraic Expressions

A sub-set of positive integers, from 0-12, share a cardinality with the set of musical note names. If the letter A (note name A) is the starting point for music theory, then the number 1 will be its correspondence. This occurance is as follows: 

                A=1, Bb/A#=2, B=3, C=4, 

                C#/Db=5, D=6, D#/Eb=7, E=8, 

                F=9, F#/Gb=10, G=11, G#/Ab=12,

                as nothing occurs=0.

The expression derived from the one-to-one correspondence for the melodic expression mentioned in section a. is as follows: 

                (A, C#, E)=(1, 5, 8)

The algebraic expression of the A Major triad can be mapped to all 12 notes as the bass note, or tonic, as the generation of the numerical mapping:

                Major Triad = (n, n+4, n+7)

This generative expression of major tonality also maps to the minor tonality:

                Minor Triad = (n, n+3, n+7)

With this generative, algebraic expression, all possible 3-note combinations can be represented via a starting note, represented by the variable n. However, this generation does lead to issues with the foundation of the theory which has a numerical limit of 12; 

                A subset of the set of positive integers, (0-12). 

Since the sub-set of integers (0-12) have a limit to the number 12, certain algebraic expressions have to be in place for any set of combinations. This phenomenon highlights two deficiencies in the theory; the first is the limitation of mapping up to 3 notes at one time, and the second is that no number can be greater than 12 in the algebraic expressions. While the limitation of 3 notes per analysis is a limit of the system, the number 12 can be mitigated through subsequent algebraic expressions:

                A subset of the set of integers; 0-12 have a one-to-one correspondence to the note names with A being the first note in western harmony. 

                A=1, Bb/A#=2, B=3, C=4, 

    C#/Db=5, D=6, D#/Eb=7, E=8, 

    F=9, F#/Gb=10, G=11, G#/Ab=12,

        as nothing occurs=0.

For the major triad, the algebraic expression is as follows:

        Major Triad = (n, n+4, n+7)

With the A major triad, the algebraic expression works:

                A Major: A=1; (1, 1+4, 1+7), (1, 5, 8)

However, for the G# major triad, the algebraic expression has a sum greater than the limit of 12:

                G# Major: A=12; (12, 12+4, 12+7), (12, 16, 19)

In order to resolve this issue, another algebraic expression should be used in combination with the original cardinality:

                A=1, Bb/A#=2, B=3, C=4, 

    C#/Db=5, D=6, D#/Eb=7, E=8, 

    F=9, F#/Gb=10, G=11, G#/Ab=12,

        as nothing occurs=0.

        For all algebraic expressions, the sum of n cannot be greater than 12. If the sum of n is greater than 12, then the sum should subtract 12;

                G# Major: A=12; (12, 12+4, 12+7), (12, 16, 19); (12, 16-12, 19-12); (12, 4, 7); (G#, C, D#)

The additional algebraic expressions allow for the use of each note name, or n, to begin a harmonic phenomenon without exceeding the numerical limit of 12 in this system. 

c. The geometric expression cab utilize the resultant algebraic expression of a major triad, (n, n+4, n+7) in terms of a Cartesian Plane with the corresponding coordinates (x,y,z) for (root, third, and fifth) for this algebraic expression; 

                A major triad; (1, 5, 8)

To fully incorporate the aspects of western harmony into the system, all 6 combinations can be expressed in the (x,y,z) format:  

                A major triad; (1, 5, 8) 

                A major triad; (1, 8, 5)

                A major triad; (5, 8, 1)

                A major triad; (5, 1, 8)

                A major triad; (8, 1, 5)

                A major triad; (8, 5, 1).

Each arrangement of the triad corresponds to the possible arrangements of triads in western harmony. 

For single note lines, each vector will contain the note value. For instance, if n = 1, then (x,y,z) will be represented as (n,n,n) or (1, 1, 1). 

For the full theory, visit my researchgate linked below:

Cartesian Mapping of Chords and Scales

For information on how to apply this theory to music composition, see the book linked below:

Cartesian Mapping Book


1. Sousa, J. P. (n.d.). Works of John Philip Sousa.

2. Schoenberg, A. (2011). Theory of harmony. University of California Press.

3. Amir-Moéz, A. R. (Ali R.). (1990). Algebra lineal. S.l. : s.n.

4. Berger, Robert. (1967). Differentialrechnung in der analytischen Geometrie [von] R. Berger [u.a.]. Berlin, Heidelberg, New York, Springer.

5. Kayser, Hans. (1950). 61. at <>. In Lehrbuch der Harmonik (Vol. OL14793653M, pp. 61–62). essay, Occident Verlag, Zürich.