The string is a real closed field minus finitely many points.

If pitch is a continuous function defined on the string interval [0, 1] then there must be a fixed point in the interval.

Each mode of vibration is represented by an n-tuple that is a subset of Rn so that the waves are intervals between nodal points in projective space.

The null set is just the endpoints, no string, which are the points (-1, 0) and (1, 0) represented by S0 .

The string is connected like a circle missing a single point which can be used to glue the circle into the origin of RP3 - 0.

There is a curve-lifting equation that proves the string at rest and in the perturbed state match point for point, which proves the force acting upon the string is orthonormal to the string axis.  Without a force in the direction the string axis there can be no traveling wave! They kinetic and potential energy go to zero at the endpoints.

The string assumes the same shape regardless of how it is struck and clearly the fundamental has the lowest frequency and therefore the lowest engergy.  Why would several energy levels co-exist?  Why wouldn't higher modes degenerate to lower modes if they can co-exist.  Clearly the modes are singletons that are all 1 step away from the fundamental.  Isn't it obvious that the string is given a subspace topology?

The string has an atomic structure that is the union of wave and not waves, and so on.

Now my question is, if the sound wave that radiates from the string is a purely algebraic object that is a  function of one continuous variable pitch(frequency) and the wave is a polyhedron with n + 1 vertices, n edges, and 1 face in R2 , then why isn't it clear already that the string is a semialgebariac ring, and not just a frequency transducer.

We have a graph (the string with nodes and wave) and a tuning function f and intonation function g.  What else is needed here?  Why isn't clear that each n-tuple is a different system and the n-tuples cannot just add?  Pitch cannot be divided into 0 and 1 without an algorithm. Clearly the string is partitioned into a finite number of points and intervals in the real closed field.  Isn't that enough, by itself, to make a new theoretic model for the string that makes sense?

It is just astonishing that people believe in things like traveling waves reflect to make standing waves (clearly the boundary condition for this (1, 0, 1) cannot exist); or that  the string can just be divided into smaller and smaller fundamentals that all co-exist in the string as independent modes.

Nodes and waves cannot co-exist. Period. If evidence shows they can, the evidence needs to be re-examined.  The monochord proves that string can have only one mode at a time according to a fixed point on the string and also that higher modes exist only when the string driven in a higher energy state.  But each fulcrum point is a different system. 

Notice the monochord string has the property that it recognizes when [0, 1] is equal to a simple multiple the string fundamental. That is a boundary condition for harmonic motion.  Recognition is a property of a finite state machine.  The string recognizes it's finite modes. Finite state machines have only one state at a time.

I cannot understand why such fallacy as classic string theory is allowed to persist in science.  Such a profound illusion that no one gets this!