Harmonic math can be first understood by means of analogy: art is to geometry (as a sub-discipline of mathematics) as music is to harmonics. The field of geometry endows us with powerful conceptual tools to model situation, design structures, and solve problems in reality. It is able to do so because as a language, re-enforced by logical and mathematical rigour, it corresponds to a range of formal structures that exist in reality.
If this is possible with geometry, as a visual phenomena, then why ought it not be possible with harmonics, as an acoustic phenomena? A comparative examination of the most basic principles of geometry and harmonics sheds light on the potential for the use of music as a conceptual tool to perform mathematical thought, much in the same way the geometry has been used for centuries to understand mathematics.
Let us examing the simplest of geometric and harmonic axioms. In geometry, this refers to the existence of the point and the line (composed of two points). In harmonics, this refers to the existence of the frequency, the sonic point, and the interval, the sonic line. If we consider the unit line where geometric line AB is composed of geometric point A and geometric point B, then the length or distance between A and B is equal to 1. Similarly, the length or distance between B and A is also 1. No matter the direction of the line, AB or BA, the length of the line equals 1.
In contrast, if we consider the unit interval where the harmonic interval AB is composed of frequency A and frequency B, then the harmonic distance of interval AB is equal to 1; however, the harmonic distance of interval BA is equal to -1. Where geometry calls attention to the absolute value of the distance measure, harmonics calls attention to the relative relationship. The unit interval of harmonics has a harmonic distance of both 1 and -1, which is to say that the harmonic model contains a property of vectorization which is embedded in harmonic cognition on a more fundamental level than is the case with geometry.