These charts show how the directional derivative on guitar is analyzed at a point in standard tuning. The example is pitch value E4 to C#4. Arrows show possible movements. This shows the internal structure of the manifold which is otherwise difficult to imagine.

Manifold, directional derivative? Where is it?

All that we have some discrete values that are connected by a summation recurrence relation.

Where are the embeddings and the charts of your assumed manifold?

I don't found any precise theorem that supports your claims.

The attached article is a clear, readable article.

How do explain the rich, expressive language structure of guitar?

Correct, the value are discrete (integers) but the category defined by the guitar position values and the arrows which connect the values (the directional derivative at a point) have a comprehension principle based on least action.

A pertinent question is why is 0 5 5 5 4 5 tuning so highly conserved overtime, while 0 5 5 5 5 5 (an old lute tuning) is almost never used. The answers is that major and minor chords, especially 7th chord, are too difficult to play.

Out of thousands of possible tunings there are probably less than 10, certainly less that 30, that are conserved. Why? Because they are optimal solutions. Lots of good chords to play.

Clearly the guitar tunings have a probability measure that is not explained in modern science.

The tuning is a manifold formed on the first string which is constant (that is, the same for all tunings). This makes a Baire space. The tuning is the union and intersection of the strings (also Borel). A smooth manifold is created by the connecting map on the first string. Tablature is the free language of the guitar tuning monad. The problem is there is no simple way to understand least action on guitar without many experiments on the 6 dimensional object.

I appreciate the reference to David Hornbeck's work, which I have seen before. But he uses only a single tuning and so he misses the importance of re-writing rules for changing the guitar tuning, key, and pitch level. Like Hormbeck, many mathematicians use rational number to approximate pitch integers. But rational numbers do not make a coherent theory of language, while integers do.

The vibrating string is obviously a smooth manifold but not understood by physicist or mathematicians, either.

The fact that I cannot explain the manifold to your satisfaction reflects on my ability as a mathematician but does not mean the numeric script of tablature is not purely mathematic.

It should be obvious the tuning is a simplicial complex. A set of interval is by definition Borel.

Where is the manifold? At a point in R6 with an R5 hypersurface as a face.

If the guitar is not a smooth manifold, you can prove your assertion!

* "The vibrating string is obviously a smooth manifold but not understood by physicist or mathematicians, either."

If so, show us the equations of such manifolds.

All that we have is a few discrete values follow some recurrence pattern.

It is not clear how such values are connected with the homotopy spheres Sn.

First, according to the Gauss-Bonnet Theorem, the standing wave on the string has an ordinary differential equation. We know the equation of motion exists.

Second, he basic principle (stated clearly by Taylor in 1715) is isochronicity: When a point on the string arrives at the center of motion, every point on the string arrives at the center at the same point in time. Also, when a point arrives at the limit of displacement, every point arrives at the same time. This means the image of the string (in the cross-section of the string manifold) is the same at every point on the string any given point in time. So the sub-manifolds of the string are discs which can be glued together like a stack of coins. This gives a smooth manifold under Liouville integration. The submanifolds are countable but infinite.

Each point on the string acts like a pendulum which moves strictly perpendicular to the string axis. That is, according to Gauss's theorema egregium, the string bends but does not elongate. Then the manifold must have a smooth set of charts.

The tangent at a point p on the string is given by 1-period Euler-Lagrange solutions with critical point sets Ci = s0 where dF/dt|p = 0. [ that is, nodal points do not move]. The Lagrangian L[t, q, v]: ℝ/nℤ x TM →ℝ is a deformation retraction of the string power set on itself M:ℝm x ℝm→ℝm (a symplectic symmetric bilinear transformation).

The nodal points on the string are critical Morse points and between the nodes are an infinite number of smooth curves.

Points on the string cannot be critical and non-critical at the same time. This proves superposition is not possible because no two modes have the same critical point.

Since the frequency spectrum of the string is discrete and does not change over time it is clear that the equation of motion does not depend on time either.

Do you really doubt the string has a Hamiltonian operator with tangent-cotangent bundles defined at a point? But there is no Hamiltonian for the string given in the literature. In fact, equation's like d' Alembert's were derived before the Hamiltonian was described.

I am familiar with all concepts that are mentioned in your answer.

Unfortunately, I didn't see any connection between all of them and the mentioned recurrence values as well.

The solution of the p.d.e that represents a string vibration is well known.

According to your claim, we have a different nonclear issue.

We need to see one logical, precise theorem.

Inflation of different advanced concepts will never mean correct results.

Exact clear answers may resolve the issue.

What are the recurrence values? If you are talking about the distinguished vectors on guitar, then each string vibrates in the fundamental mode so each interval between the strings is a connecting map between two Hamiltonians.

So tablature is simply the action functionals on the string fundamental.

What is your equation for the string? How do you understand tablature? Does Gauss-Bonnet not apply?

No physicist or mathematician could answer my question about the curvature of the string. Obviously they don't know the answer! If I am wrong, then you tell me this: Is the curvature of the string constant? Does theorema eqregium not apply?

The fact is clear that physicists do not understand string mechanics in a coherent way. So even if I am not coherently explaining this, that does not prove you understand the string closure properly.

Why don't you explain guitar theory to me? Can you prove the string manifold is not smooth? Can a manifold not be smooth?

What is the difference between Open G (0 5 7 5 4 3) and Open D (0 7 5 4 3 5) tunings? Why are they so highly conserved?

Guitar is mathematic object that is not understood by modern science! This is how false paradigms persist. If I am wrong, you can easily prove it. But you can't just say I am wrong because you don't understand.

From Open G to Open D, one answer is tangent (0 -2 0 1 2 0) takes Open G to Open D and cotangent (0 2 0 -1 -2 0) moves Open D back to Open G. Of course, these vectors are purely algebraic operators. You still need to re-write the tabs according to the destination metric second-order logic. Re-writing is what makes the theory of guitar so difficult to learn. The algebra is trivial. The logic is non-trivial and takes years to learn.

By the way, if you listen to popular music, you have almost certainly heard these tunings but of course cannot recognize them without special learning. This makes the tunings seem to be nonsense because you do not understand the paradigm. Correct me if I am wrong.

I may not understand the math, but I definitely understand the tunings. See my You-tube lectures on Beatles guitar.

OK, so there's your recurrence values. The tablature sequences are re-written until the re-writing rules produce no further changes.

The curvature of the string re-writes itself with each cycle until no further change in angular momentum occurs.

It is called minimization.

The tablature is just weakly-determined.

We are talking in different languages. Show me equations and formulas.

A precise mathematical paradigm may help to judge your ideas and not a general description.

OK.

First, the only equation allowed is a + b = c. This generalizes transposition of the musical key as a Category 3 object.

Second, the metric of the space is established by the octave. The octave rule is improper -- does not apply anywhere else. The rule is sometimes stated p = log2 f (equivalent to 2p = f). Actually the rule says dividing the string in 2 multiples the pitch by 2. So we have 1/2 = 2, equivalent to 1/a = a. So the octave is idempotent. Universal. Every note has an octave and every interval has an octave inversion.

The octave partitions the space. The 12 tones are a subfilter.

Third, there are no sets smaller than notes and intervals but the smallest set that contains the entire theory is 12. The octave metric closes the space because the measure-preserving a + b = c transformation is modulo 12.

On a single guitar string there is a union two dysjoint sets with the subspace topology: the fret index numbers S and the pitch index numbers A. The string defines the cartesian product A X S which goes to A when the notes are sounded at a specific pitch.

The musical key is the simplicial complex of edges connected to a point on the string as the center of motion. The musical key defines a topology that is different at every point because of the boundary defined by the lowest and highest notes.

There is a vector product equation "pitch value vector = fret value vector + summation vector." Another: "Tuning-change vector + Key-change vector = Tuning-Key-Change Vector". Under the octave addition and multiplication are the same, so you may want to see these vector equations as products.

The start state s0 of the string is defined by tension, length, and density. In a local region these parameters create a Euclidean space so that these parameters have the same vector basis. That means if the tension goes up by a and the distance goes down by a the frequency does not change.

So the equation for the string manifold is M = (A, S, s0, A x S to A). You will recognize this as the formula for a finite state machine.

When you said the distinguished standard tuning vectors 0 5 5 5 4 5 and 0 5 10 15 19 24 are just related by summation you are missing the topologic space that is formed by the intervals.

The tuning is an open subset of the octave intervals. That by itself proves the space of the tuning is Borel.

The tunings like 0 5 5 5 4 5, 0 5 7 5 4 3, and 0 7 5 4 3 5 are highly sophisticated regular languages. They have extraordinary intellectual value. They are not random. They recur.

This is always the problem with tablature, a theory of integers that seems like nonsense! Who can think in 6 dimensions?

I think you know the real numbers are rich enough to define a highly expressive calculus of variations that I am trying to explain.

The last answer is more evident than the previous ones.

But one needs to see and follow some articles that show the preliminaries of the whole theory.

There are no articles! You have to think about it for yourself. Its all deductive from simple principles. It is easy but difficult to see how it is easy.

For instance, contravariance should be obvious: Whatever is true for notes is also true for intervals. That is what they teach in music theory.

The amazing thing if the way the guitar is linear over the entire fretboard (but not the entire string).

My original point was mathematicians and physicist do not understand the theory of the vibrating string.

I believe I have proved my case. Even if I am wrong and the octave is not idempotent.

You cannot explain why certain tunings recur.

Kolmogorov's theorem proves there is a theory of tablature, but it doesn't mean it will be easy to explain.

All of this is utter nonsense! Some sentences are really funny:

- "according to the Gauss-Bonnet Theorem, the standing wave on the string has an ordinary differential equation" (so, when separating variables, you should write G(x)B(t) instead of X(x)T(t), to apply Gauss-Bonnet, ha, ha!)

- "The Lagrangian L[t, q, v]: ℝ/nℤ x TM →ℝ is a deformation retraction" (as in Bourbaki, functions are just sets :) )

- "a symplectic symmetric bilinear transformation" (so it is identically zero?)