The guitar is a machine operated by a robotic arm. The tuning is the state by which every other state on guitar is known. Therefore, it must be possible to learn the mathematic structure of guitar music action directly using the tuning and nothing else.

In Standard tuning, we are given that the guitar strings are tuned E A D G B E from low to high.

The tuning breaks down into two forms, f = 0 5 10 15 19 24 and g = 5 5 5 4 5. Form f is obtained by numbering the pitch values beginning with zero for the lowest note. Form g is derived by taking the difference in pitch between adjacent strings.

Form g has one dimension less than the pitch values and is therefore a surface induced by f as pointwise limit in R6.

Addition is defined on these vectors as well has multiplication by two, but really there is only one binary operation which can be view as either a product or sum. Then we have f + g = 1. That is the composition of f and g defines a unit interval on guitar.

Forms f and g are opposites. Form f represents frequency values (points on the real number line). The points on the guitar machine can be assigned numbers or letters as tokens for the real numbers. Then the guitar has the decidability of real numbers (that is, tablature is analytic).

Form g represents frequency intervals (in which no pitch values exist). Every real number on the guitar interval is then either a pitch value or not pitch. That is, the guitar is a nondegenerate two form.

Whatever is true for f must also be true of g since they are opposite.

The tuning is differentiable since form g is obviously the derivative of f (since g is the gradient of f), and by the law of opposites it must also be that f is the integral of g.

Writing tablature depends on 5 5 5 4 5 (which does not depend on pitch since intervals do not depend on their endpoints) while 0 5 10 15 19 24 is merely a naming function. If 5 5 5 4 5 takes music to tablature and 0 5 10 15 19 24 takes tablature back to music, then the composition of f and g is an identity and f = 1/g.

It seems to me that the fact that f and g are invertible by itself established the topologic nature of guitar in a unique way.

Maybe you can explain how I have this wrong? Is it all nonsense?