Terence,

A most interesting question! FWIW, I have done some research into the nature of wave harmonics.

("Dynamic spacetime imposes matter continuity," available on RG)

Abstract

We identify a least wave harmonic in 1-dimension continuous spacetime, a ground state source by which every succeeding global state is locally connected. Continuous spacetime allows generalization, in Hilbert space, to a single wave state that manifests physically as a spacetime soliton. We propose a validating experiment.

(pp. 12-13)

" ... Perhaps there is a true ground state that can be indirectly detected in an interaction that, upon completion, is compelled to equilibrium—what we mean is, that instead of gaining or losing kinetic energy on interaction, the energy is invested in an equilibrium state—a perfect damping function that is constrained on a single point 1/2 in the interval [0,∞). And this is true, if the Riemann Hypothesis is true.

A particle is only free, when randomly fluctuating around an equilibrium point.

The exact result 1/2 is the equilibrium point, called the critical point in the literature.

This mathematically precise number bound in the interval [0,1], is free to fluctuate before arriving at equilibrium."

Best wishes for your research. Tom

The decay of the sound involves more than just the wave equation: there has to be some kind of damping. Damping can be due to different effects. For example, the motion of the string through the air results in energy being lost to sound in the air. The rate of energy loss would be proportional to the square of amplitude*frequency. As a result waves of different frequencies would decay at different rates, and the envelope would not likely be an exponential; in fact it would drop faster initially and then slow down as the higher frequency waves damped out faster and became insignificant.

Things can be more complex in more complex instruments; pianos typically have more than one string for a note with the same (or nearly the same) lengths. This results in some near-resonance effects that can change the envelope.

Note that the Wikipedia page

https://en.wikipedia.org/wiki/Piano_acoustics

says in the section on Multiple strings:

"All but the lowest notes of a piano have multiple strings tuned to the same frequency. This allows the piano to have a loud attack with a fast decay but a long sustain in the Attack Decay Sustain Release (ADSR) system."

By the way, notes plucked, bowed, or hammered on a string result in a range of different frequencies that are all multiples of the original frequency (the 3rd mode mentioned in your attachment *is* a harmonic). However, 2- and 3-dimensional media such as drum surfaces and bells do not have harmonic vibrations (harmonic=all frequencies multiples of the fundamental frequency), and so do not sound as "nice" or melodious as string instruments. Woodwind and brass instruments are also essentially one-dimensional (although folded) and so have (nearly) harmonic frequencies when a note is played.

These are great answers! The first brings up Hilbert Space and the second uses the topology of real numbers in the frequency domain.

So we agree the most important principle here is that all harmonic values are simple multiples of the fundamental.

I think we also agree the Octave is exactly 2 times the fundamental.

Then it follows the space is Hilbert.

There is nothing more to say, really, but most people don't get it. The string space is topologized by the fundamental and the octave. The rest just follows mathematically.

The important difference between the Stewart and Ray answer is how the operation of multiplication is defined on the fundamental F. Stewart's answer assumes the harmonic operation of multiplication is the same thing as the multiplication of real numbers. The topology of real numbers makes the assumption the frequency is a real valued, continuous function the same as the real number line. But the frequency is a discrete value, not contiuous. But the octave space is continuous because the octave interval is precise to a point.

In fact the harmonic multiplication table that is defined completely by the Octave rule is not real valued. This makes a log 2 space. Every point is a distance 1 from the fundamental. Not intuitive.

The topology of the space is not altered when the octave is further divided to 12 tones because S12 = 1 is a Cuachy series that is also precise to a point.

So it follows directly from the octave rule that the tonal space created by the string and its octave is a special kind of Hilbert space in which every distance is 1 and every angle is 90 degrees. This is a Hilbert Space because a = a2 is true when a can only be 1 or 0. A disconnected space of singletons called the discrete topology.

This space can be directly observed as disconnected spaces in drawings of musical objects. Deawings of musical objects are useful but cannot be drawn accurately in 2 dimensions (Haas Theorem; think logic cuircuits). Look at any graph of pitch values (say a graph like Euler's tone net which plots the pitch value x versus musical key y). These graphs are easily shown to be non-planar graphs.

The pitch value graph seems to be x-y planar, but Euler's graph is topologically the same as a torus in a closed space. The torus is the real number topology but pure nonsense because the key and the pitch values must be multiples of the same fundamental and cannot not be two independent real values.

Euler's torus cannot reduce to a point which is the fundamental. There are only two possible manifolds here, totus and sphere. The sphere is obviously correct and Hilbertian.

So in Euler's tone net graph, if you count the number of steps around the trinagle that is formed, the hyptonuese has the same number of steps as the x- and y-axis. The steps are the multples of the fundamental that make the distance formula for the space. In this space the triangle formed has an equal distance on all 3 sides; so all angles are equal and 90 degrees, too. (For instance, in a graph of an octave the triangle formed is an octave on all sides: a2 + a2 = a2) This triangle must be on the surface of a sphere, because any planar triangle the angles add to 180 degrees but on a sphere there can be an equilateral right triangle where angles add to 270.

The result is a wierd arithmetic of octave rings in which there is one operation that is at once both addition and multiplication and every equation has the form 1 + 1 = 1.

My prespective on the string is based on the guitar string. A am influenced by the string condisered with frets and musical keys. The musical key on the string is projective, not affine. That is, the difference keys on the string cannot be brought into coincidnece by a change in intonation.

So I see the string topology as completely defined by a map between pitch value numbers 0, 1, 2, ... N and fret value numbers 0, 1, 2 ... n. Then there is the musical key defined by any pitch value on the string which is the tone center defining the key. There are no real numbers here, and the numbers in use are far more than counting numbers because they are closed and open sets. The clopen nature is highly aesthetic in the sense of pleasing number systems!

The space created by the string, octave, and frets is clearly a disconnected space of singletons. The pitch values are by definition multiples of the fundamental and the frets map on to the pitch values so the fret numbers and the key (or scale degree of each note within the key) are also multiples of F, as are the intervals between notes and frets.

The space is continuous because between every point there is an interval 1 and between every interval there is exactly 1 point.

What makes ther guitar tuning so remarkable is the string is closed and open. The tuning space is Borel and Barre, both the intersection and the union of the strings! The string is closed to multipication/addition operations by the ocatave. Any string larger than 12 frets can contain every key. But the string is also open to union with another string whose fundamental is a simple multiple of the original fundamental.

(It is still not possible for any system to have two independent fundamentals, which violates our first law.)

The guitar tuning is a collection of intervals that defined the tuning completely, say 0 5 5 5 4 5 is the ever popular EADGBE tuning called standard. The 0 is a place holder for the fundamental which is the lowest note on the lowest string.

The 0 5 5 5 4 5 intervals make a summation vector 0 5 10 15 19 24 which gives the pitch value numbers of the EADGBE expression. So on each string the fret number is the same as the pitch value of the open string plus the fret number. This makes a vector equation "sumation vector + fret value vector = pitch value vector" which is an example of 1 + 1 = 1 equation.

Here's anothe example. To change standard tuning EADGBE to Drop D tuning DADGBE use the vector 2 0 0 0 0 0 that add two frets to each note on the lowest string. To change the musical key from E to D subtract 2 frets from each string using the vector -2 -2 -2 -2 -2 -2. To change standard to Drop D and lower musical key we have the vector equation 2 0 0 0 0 0 + -2 -2 -2 -2 -2 -2 -2 = 0 -2 -2 -2 -2 -2. These 3 vectors are orthogonal and they form a triangle with equal sides. This is not a space of real numbers.

I give these examples because they show the mathematics are not real number theory. They are also not Z/Z12 nor are they based in a system of rational numbers. The topologic space of the string is atonishing because it is very obvious and clear but simply outside the accepted paradgm. But what is clear is the mathematic literature on music shows there is no moderm mathematical theory of music. The theory of the string and its multiple modes of vibration, sometimes called the corp ronds, is an arachaic joke.

I think it is clear that the string thoery is not properly understood in modern literature. Real analysis is so compelling but violates the spectal resolution theorem: You cannot assume that the frequency domain is a continous real values function. The problem I have is that to see why this is important you have to learn tablature arithmetic. Tablature seems like nonsense but it is a consistent system of numbers.

"The sphere is obviously correct and Hilbertian."

This also has deep implications for a self-organized universe. I like the idea of casting the shape of spacetime in terms of the sound envelope.