This is a very interesting question.

Euler and Bernoulli were able to show that Taylor's assumption that the fundamental shape of a vibrating string sinusoidal is false; the motion of a vibrating string is much more complex than a simple sinusoid.   See page 32 in

https://xenharmonic.wikispaces.com/file/view/Christensen+-+Eighteenth-Century+Science+and+the+Corps+Sonore.pdf

Daniel Bernoulli entered the picture in 1753 by giving yet another solution of the Vibrating-String Problem: see page 6 in

http://www.mr-ideahamster.com/classes/assets/a_evfcn.pdf

The concatenation function is given in Section 4.1 in

http://arxiv.org/pdf/math/0612234.pdf

These are interesting references but so complex! Music sequences are only integers so I can't accept surreal numbers before we rule out the real numbers.

I am confused by the literature.  For example, in "From Music to Mathematics" by Gareth Roberts, Johh Hopkins, 2016 says "The overtones series is simply the sequence of multiples of the fundamental frequency.  For example, if the fundamental frequency is f = 100 Hz, then the overtone sequence is given by 100, 200, 300, 400, 500 ...

But I thought the sequence would be 100, 200, 400, 800 ... That is how the piano would be tuned from A4 = 440. The multiple is 2, not 1.

In The "Jazz of Physics" by Stephon Alexander Basic Books 2016 on page 148 the author gives a formula for the wavelenght of the fundamental "if we know how fast the wave is moving and its period of oscillation.  By multiplying by an integer, we get the higher harmonic (or higher partials) that also play an important role in the ... timbre."

But in the accompanying figure the author shows that the wave is a standing wave so the wave has no speed!  The frequency of the string is variable but the string length is fixed.  I don't see how the frequency can be a function of wavelength for a standing wave.  The frequency is programmed by adjusting tension. The string is computable if the frequency is known. The wavelength is constant.

Furthermore the figure show that the overtone series continues as if the sequence is 1, 2, 3 ... waves on the string when in fact the 3rd overtone (twice the octave) is not even on the string.

If the metric space is defined by the octave then every space of the octave has the octave metric and the higher overtones are not defined.  The system is defined completely on the fundamental and the octave, and the inergers come from the octave as the definition of 1.

Is it possible there is a map between the first and second overtones (F0 and O1) that is enough by itself to make the 12-tone system?  I think the modern theory of music assumes the frequency spectrum is a sine wave which is tantamount to the assumption that pitch equal frequency.  But what happen to p is log 2 f?

But if p = f and p = log 2 f are both true that can only be true when log 2 1 = 0 and log 2 2 = 1.  The we have 0 and 1 and that is all we need. We have a curve-lifting function.  At every point in the frequency domain there is a point on the stair case 1 octave above. That makes pitch a continuous function of frequency. Doesn't that make sense?  (Fig 3 and 4 in my attachment)

On the issue of whether vibration modes are sine waves or concatenations; I think we sometimes confuse the static deflection curve of the string and the vibration mode shapes. The deflection curve of a simply-supported string is a concatenation. Yet, the eigenvalue problem produces mode shapes which are sine waves. Now, the deflection of the string at any point (which is a point on the concatenation curve) is simply a linear combination (or modal expansion) of those sine functions that represent the significant mode shapes.       

To Yehia A Khulief

I appreciate your answer but I am still confused.

If the string is the linear combination of all the sin waves, then how is the string closed and bound?  Whatever makes the concatenation, it must add up to 1, right.  The concatenation is defined on the interval pi, and not like a sin wave on 2 pi.

The problem I have is how can the wavelength be twice the string length? That implies that the fundamental mode is decomposable into another wave that has twice the observed wavelength.  But this defies the definition of the fundamental as the lowest possible mode of vibration.  None can be lower than the greatest least boundary.

I also cannot see how the string is defined by the sum of 1/n overtones.  I think the string is defined by F0 and Octave1.  There must be a bound and closed space by the Heine-Borel Theorem so I don't think there is a infinite series require to make the numbers 1 and  2.  If the wavelength is 2L then there is a multiple of F0 that is not defined (since only simple multiples are allowed). If W = 2L then we have 1 = 2.  Makes more sense if the string is 1 = 1 + 1 (because every a = a2).  We have 1 string = 1 octave + 1 octave (Tarski's Theorem:  Projective sets are closed to projection. A projective set plus a projection is another projective set.)

But to return to your answer, is it correct to assume the the sine wave is emitted is orthogonal to the string? Then the string length and the radiation are independent vectors?  What is it that makes the Eigen value of the string always 1 (since the pitch graph always  has a slope of 1)?

I am asserting the string is the union of the pitch value set and the fret position value set which make a disjoint pairwise equivalence relation on the string.  The string is defined independent of pitch.  Pitch is a witness function.  The string theory is what remains when pitch is eliminated as the string quantifier. The string is the fundamental and not the sum of all the modes.

The theory of the vibrating string is archaic and does not fit with Zermelo-Fraenkel axioms.  This recent spathe of books on music and math are ridiculous and I don't understand why mathematician do not object to such idea as pitch is subjective and beyond mathematic definitions.  Pitch is a real valued function! Not a sine wave. Frequency is Zahalen.  I think you are assuming the string is a sin wave because you are assuming frequency and pitch are the same thing.

If there is a map between pi and 2 pi, then that makes a homotopy, doesn't it?

Thanks again for the answer.