The answer is No. It is a standard fact that the complete graph K_5 on 5 vertices (i.e.a_0, ...,a_4 with all pairs of vertices joined by edges) is non-planar, i.e. it does not admit an embedding in the plane. This is also true for the complete bi-partite (3,3) graph.  It is a non-trivial theorem of Kuratowski that any non-planar graph contains one of these two as a subgraph. See the wikipedia article https://en.wikipedia.org/wiki/Kuratowski%27s_theorem and references therein.

u mean no embedded R2

Thank you.  Thank you very much.

I want to show that descriptive set theory applies to music so the integers are induced on the system fundamental.

At the present time neither mathematicians or musicians can figure out how to define pitch as a real-valued function.  Pitch is thought to be "integerized by tempering and the arithemetic is simply the arithmetic of the real number topology.

Of course, non-planarity is the answer I expected but the significance is everyone defines music in R2 in a trivial (arbitrary) way, so if music is 3-fold then it makes a paradigm for music that is different from what most people think.  Instead of the real number topology with the absolute value metric we have a subspace topology where every element is a singleton.

Euler said the musical key and the pitch value make a tone net as if they can be independently generated by [0, 1] and [1, 0].  That makes no sense because both are defined on the fundamental mode of vibration.  What we actually have is a Boolean atom where [0, 1], [1, 0]. and [1, 1] are equivalent.  The metric is established by the fundamental and the octave identity matrix This can only be so and the evidence that it is true is every where if you look at any graph.  The octave makes a curve-lifting function such that p = f and p = log 2 f have exactly the same number of points.  In turn, this makes the guitar tuning and guitar intonation inverse functors related by a left adjoint matrix.  This is all rather like "pin-hole camera theory" and the fundamental matrix.

For example: A proof the guitar model is 3-fold.

We number the strings 1 to 6 and the frets 0 to 20 so that the order of the index numbers increases with increasing pitch. We count 2 directions of tone movement: 1) by string, 2) by fret.  In these directions the pitch values and the string and fret values are strictly increasing or decreasing.  Now draw a line from the lowest note on the first string connecting all the guitar position with the same pitch, called an isotonic line. Typically this line has 5 points on 5 strings. Along the isotonic line the pitch is constant but the string and fret index numbers move in opposite directions.  Clearly the isotonic direction is not by fret or by string but by a composite function.  It follow the isotonic line is orthogonal to the frets and strings lines.

The strings are the open cover of the guitar because every point in the guitar model is in one of the strings, and if the strings are changed in any way the model changes.  But the frets can be reduced or increased, and the pitch values can change without changing the guitar tuning.  Therefore the guitar model is convex, compact, and everywhere dense. (That is, between any two notes is an interval, and between any two intervals there is a note.)

The guitar strings are closed because any set less than 12 steps or 1 octave can fit in any string and the strings are open to union with the system fundamental by the gluing theorem.  This makes a Baire existence proof the guitar is a metric space.  The interior of the guitar tuning space is the union of the strings and the closure is the intersection.  For some reason, the guitar model requires an existence proof, but it is so simple that somehow no one is motivated.

Now I think from Mobius we have only two possible  manifolds, either a sphere or a torus.  In the literature they call it a twisted 2-torus, which is a torus plus one more dimension, it seems.

If you follow the development in Maunder's Algebraic Topology you realize that the torus can be an approximation of the sphere projected in two dimensions.  He talks about how the S1 circle "classifies" the Z/Z12 system as an approximation.

Since we know that the guitar is a subset of music, if guitar music is defined in projective space (with an affine cover), then it must also be true that music theory is a Baire Space.  That is a novel idea, don't you agree?