The monochord does not allow the fundamental mode to resonate between three points, either. Unless in the very special case when the string is stopped in its very center and vibrates on both sides of the stopping device, as when the octave is played as a harmonic on the guitar. The fundamental of any vibrating string is that corresponding to the vibrating length, and the overtones are harmonics of this fundamental.

The relation from string length to fundamental frequency of course depends on the density, the section and the tension in the string. This is the famous Mersenne (or Taylor) formula.

The wave form in the string is complex. It can be viewed merely so (a complex waveform), but Fourier demonstrated that any periodic function (in our case the vibration) can also be described as the superposition of simple (sinusoïdal) functions (in our case vibrations). So "the idea that the string is the sum of all the possible modes of vibration" is only partially wrong: it is the sum of a set of possible vibrations, not necessarily all of them. And, in addition, you may or may not consider this fact -- it is not necessary to consider it, nor to perceive the overtones (which requires some experience and, often, specific filters such as Helmholtz resonators); but it is a fact, whether you consider it or not.

The frets do not define "multiples of the fundamental", they merely define different fundamentals (as do the different lengths in the piano strings). The frets by no means define nor correspond to overtones of the fundamental of the unstopped string length.

I like your answer because it summarizes the 300 or 400 year old theory called classic string theory of Mersenne.  But you are not correct..

Your error, my good friend and college, is that you are assuming that multiplication is defined by  real numbers.  This defines the harmonic operations without first defining the set on which the operation is defined.

Wikipedia page "music set theory" implies that music is not really set theory.  So every one wants to start with a function like pitch or position but not a set.  Then, with your real arithmetic, you think everything is just like the real number system.

You are right, the monochord does not allow the fundamental to vibrate between 3 points.  Actually, it extends the fundamental to a second string if the fulcrum is positioned at a node where the string does not move so there is no interference.  But the fact remains whether the string is driven or detained by fulcrum, if the string can vibrate at the node, then the overtone is destroyed.  You just cannot argue with the fact that interference does not allow 2 vibration modes along the length of the string.  That does not mean the string cannot have other overtones from other fundamentals that are independently determined and therefore incidental.

Imagine that the fundamental and the octave overtone are combined using Fourier.  Then we have a bimodal concatenary where the point at the middle of the string can move.  It is clear then the superimposed fundamental and octave modes combined will degenerate to the fundamental.  It like plucking the string at two points along the string: you will never get the octave harmonic. Also, the octave overtone never degenerates to the fundamental,

It cannot be true that the string can have more than mode at a time because that would create interference.  The fact that electric currents behave like a Fourier series does not mean that n overtones can be "superimposed" on a string.  Clearly the string cannot be deformed like an electric current.

I think what is happening is that Hooke's Law is applied to the deformation of the string concatenation, rather than the string deformation.  This solves the interference problem.  The concatenation has no time derivative, so it is called isochronus.  But the deformation of the fundamental concatenary is a harmonic motion which is not isochronus.

Imagine the angle of the string at the nut and bridge.  That angle is a constant boundary which is the equilibrium position of the string.  When the string is plucked there is a string triangle formed by stretching the concatenation at a point.  When the string returns to the boundary condition it overshoots the boundary.   Imagine that the string is a shape like a bell, and that shape itself has it own vibration.

You say the fret are not multiples.  Lets change the language.  The frets, I think you will admit, are sums on the fundamental.  That means the fundamental is zero, then add 1 to make the first overtone.  For example, if we add the tonic, third, and fifth we get a chord.  We never say that we multiply to make the chord, but we can if we want.  Bases add, exponents multiply.  Same result, either way.  But in topology the product is more useful as a generalization.

There is only 1 operation which is at once addition and multiplication.  This is possible because the octave identity (0, 1) includes the identity 0 for addition and 1 for multiplication.  The identity is the operation in music that results in zero change.

The frets multiples of the fundamental which all have the same factor 2-12.   You cannot assume that multiplication means the multiplication of real numbers.  I know that seems strange but that is how set theory works. 

In your last sentence, what the frets define is the pitch value of the string detained between the fret and the bridge. So there are 3 fundamentals in the system: the system fundamental, string fundamental, and fret/pitch value fundamental.  When you hear guitar then only thing you know is the pitch value fundamental.  The sound detaches from the fret that defined it.

If you think about Mersenne in a skeptical way, then you will see that these things like the fundamental is 2L, not L, and there are two moving waves that reflect to make a standing way, make no sense.

I am grateful for you willingness to discuss this further.  Thank you.  The best thing you said is the octave is a special case.  I would add it is also the only case we need.

Dear friend, your error is to think that the fundamental of the whole string must still be considered when the string is vibrating in part(s). When the monochord string is divided in its middle by a fulcrum, or a guitar string by the light touch of a finger, then two halves (two strings) of equal length are vibrating together, but each length is half the total length. When the monochord (or the guitar) string is divided at 1/3 of its length, then it vibrates in three parts, with a node at 2/3 even although there is nothing limiting the movement or the string at that point.

You may claim that "there are 3 fundamentals in the system: the system fundamental, string fundamental, and fret/pitch value fundamental", but the only acoustic fundamental is the one produced by the vibrating string length(s). The actual pitch produced by 1/3 of the string is the same as that of the string vibrating in three thirds, because the vibrating length is the same in both cases. The only difference is that the string vibrating in three thirds actually is three strings vibrating at the same time: this is what produces the special timbre of this note often described as a "harmonic".

Mersenne's law merely says that the period of the vibration is the time needed for a deformation in the string to run twice its length (go and return): in is in this sense that the total length is 2L; the speed at which the deformation travels depends on tension and linear mass (or density times section). But to say that the fundamental is 2L has no sense: the fundamental is not a length, it is at best a frequency, the inverse of the period of the deformation travelling 2L at a constant speed determined by the string constants. Mersenne's law has nothing to say about standing waves -- which does not meand that standing waves don't exist.

The phenomenon by which a string divides in vibrating sections can only occur if the string is divided by an integer, otherwise the sections would not be isochronous. And this is (one reason) why the series of integers is called the harmonic series (or the set of integers the harmonic set).

Fourier's theorem does not concern electric currents anymore than it concerns strings. It is purely mathematical and it concerns mathematical functions exclusively. It would not really interest us, was it not that our ears can act as Fourier analyzers and can hear overtones in the complex waveforms reaching them.

Guitar frets do divide the string by factors of 2-12 and, as the pitch is inversely proportional to the length, the resulting frequencies do multiply by the same factors. But each pitch so produced corresponds to a complex waverform that could be analyzed in overtones, which (if the sound is periodic) correspond to integer multiples of the fundamental. The monochord bridge similarly can be positioned at points dividing the total length by factors of 2 -12, with the same results. I fail to see any problem with this.

The problem with dividing the string into smaller parts is there is no way to know when the process stops.

There must be an atom in music which is the smallest possible element that can be used to construct the whole.

I realize that you are representing the standard theory taught in every school but your 300 year old theory was never settled in the first place.

The problem is that no one believes the string can vibrate in more than one mode at a time but it seems the sound wave emitted by the string forces us to conclude the string has more than one mode. 

Let's try to find a way to agree on something.  I have tried a number of different arguments, but so far I have not motivated any change in your thinking.

First, I think you will agree that the monochord does not favor all n modes equal.

The fundamental is the lowest energy and the most stable mode.  The higher modes cannot degenerate down to the fundamental, which we would expect if all modes are allowed. The octave harmonic stays in that mode. The fundamental jumps to the octave only when a node is forced.  Nodes and waves cannot co-exist.  So if an octave and fundamental are added, the octave node is destroyed and the mode will degenerate to the fundamental.

There seems to be a monochord limit at about 8 sub-divisions and 7 is not stable while 2, 3, and 4 are always the strongest.

But how many degrees of freedom are there in the string? Because according to Bernoulli's Law the number of modes cannot be greater than the degrees of freedom.  Sound waves have many degrees of freedom in air so frequency can assume any values.  This means that waves that might seem to interfere otherwise can simply add.  But frequency on the string is restricted.

OK, now please try and follow this part.  When you divide the string into small units, the fundamental gets smaller and smaller, too.  This means that the series of overtones on the monochord are not harmonic.  Each overtone is a fraction of the string fundamental and not a multiple.  Now, there is one overtone -- the octave -- that is consistent with the system fundamental because the octave is 2 times the fundamental.

But when the string is divided in to 3 or more waves, these overtones are not harmonic because the units 1/3, 1/4 etc are not computable on the fundamental as multiples!

I think part of the problem is that multiplication is defined in mathematic by a multiplication table.  You know, you have 1, 2, 3 above the columns and beside the rows and then you fill in each square in the table to define what multiplication means.

When the string is divided in to two parts by the octave, the multiplication table is set.  It is just like a ruler with 1 unit divided into 12 parts.  After that all the other over tones not not in the system.

You take the octave ruler and make a multiplication R X R, then you map the multiplication table back to R.  The guitar is R X R, what you hear is in R.

I think you will agree the shape of the string is always the same regardless of how it is struck.  But when the string is plucked near the bridge, the a symmetric triangle drives more energy in the string that plucking the middle.  The over tones are on top of the string mode and so do not create interference pattens.

I am absolutely certain the monochord cannot make the 12th overtone, as you claim.   It does not appear that anything over 8 is possible.

You should watch this video because it shows the string can only be in one mode at a time and we can see that the overtones are the result of the vibration of the standing wave.  My wife calls it noise on top of the mode.

https://search.yahoo.com/yhs/search?p=String+vibration+videos&ei=UTF-8&hspart=mozilla&hsimp=yhs-001

https://search.yahoo.com/yhs/search?p=String+vibration+videos&ei=UTF-8&hspart=mozilla&hsimp=yhs-001

The second video show the same thing but explains more.

I have a string that contains all music as 1, but you have an infinite series that adds up to an infinity, not a 1.

Terence, can you read me for five minutes, without jumping to conclusions, or without reading things that I didn't write? Let me repeat:

A vibrating string first shows a deformation at the point of plucking, bowing or striking, and this initial deformation produces complex forms, the "waveform". In normal conditions, the string vibrates as a whole and there is nowhere a node (a point of no displacement) during its vibration. The vibration is a function of time, usually a periodical one.

What I claim is that this complex periodical function can be described, or analyzed, as a superposition of simple (sinusoïdal -- this is the definition of what I call "simple") forms. It CAN be so described, but it needs not. If you don't want to, don't. If you do not want to hear overtones in the sound produced, you probably won't.

If you want to hear the overtones, however, you will. That is because your ears have the strange faculty of acting as Fourier analyzers: they do hear (or at list they can hear) the complex vibration of the string as a superposition of simple ones. I addition, it can be shown (and it has been) that these overtones are not merely subjective -- that is, they do not arise in the ear exclusively. They have an existence even outside our ears, an existence which can be evidenced by a variety of devices, filters or resonators -- they are "objective".

You ask how many degrees of freedom there are in the string. Well, I suppose that depends on its actual physical properties. The theory is about ideal strings and therefore says there is an infinite number of degrees. But I don't mind because in any case we are concerned only with what we can hear. Can a string produce its 12th overtone? I presume it can, but I don't mind because that overtone would have a frequency of 2048F (where F is the fundamental frequency of the string) and nobody could hear it. (For a fundamental of 20Hz, close to the low limit of our hearing, the 12th overtone would be above 40000Hz, or even 80000, depending on how you count the overtones).

You say that overtone 3 and above cannot be harmonics as 1/3, 1/4, etc, are no multiples. But overtone 3 has a frequency of 3 times the fundamental, not 1/3! You should consider that the string length determines the period of vibration (the time taken by one vibration, i.e. the time needed for the deformation to travel in both directions in the string, V/2L, where V is the speed). The period of the 3d overtone is 1/3 that of the fundamental, so that the frequency is 3F (the frequency is the inverse of the period).

Is that more or less clear? It is not a theory 300 or 400 years old, it is not really a theory, it merely rests on the very definition of these terms, period, frequency, length, vibration, overtone, etc. You can build another theory, but then you should use other terms, because these already are defined.

I don't think I am miss reading you because I am familiar with you literature.  I apologize if I have misrepresent your point of view by trying to restate your position. We have a better focus on issues now.  My terms are defined, but you reject the mathematic terminology and never accept definitions.

I think what we need now is to make a list of statements where you say one thing and I say another.  The idea is that instead of using a line of reasoning that is hard to follow, we juxtapose fact M and fact A side-by -side

Fact M: The vibration is a function of time. 

Fact A:  The vibration is a standing wave that is isochronus.  Frequency of vibration does not vary with time, only the amplitude decays. Frequency and amplitude are independent.  The vibration is a constant because it is a state of system.

Fact M:  This complex periodical function can be described, or analyzed, as a superposition of sinusoidal forms.  

Fact A:  It is well-known objects can emit frequencies not defined as harmonic.  Sounds waves can superimpose mathematically in air or electromagnetic fields but the string has only one degree of freedom so the string can have only one state of system at a time, and therefore the string can have only one frequency. 

Fact M:  The sinusoidal function is the mathematic definition of  simple. 

Fact A:  This is simply not true!  The definition of simple is 0, 1. The spectral resolution theorem says that the fundamental cannot be defined by a point in the frequency domain. When the fundamental mode is deformed it is subject to its own harmonic motion but these are vibration are not simple multiples of the fundamental and therefore are not defined in the Corps Sonore even if they exist in the string itself in some form (without creating nodes!)

Fact M: The string length determines the period of vibration (frequency). 

Fact A:  Guitar strings are the same length but different frequency.  The length of every string is 1.  The string defines 0 and 1.  By the way, the fundamental is L, not 2L.

Fact M: F is the fundamental frequency of the string.

Fact A:  The fundamental is defined by a system F initialized to a frequency f so that taken together system F and state f make a state of system.  Frequency cannot change without changing the state of system.

Fact M: The period of the 3d overtone is 1/3 that of the fundamental, so that the frequency is 3F. 

Fact A:  Yes, we agree on this! But the problem is if the octave is the second overtone, then the series is 1, 2, 4...  and not the same as just 1, 2, 3.... 

Say the fundamental frequency is 100.  Then the frequency goes 100, 200, 400. Then 300 does not exist in the system!  This is an absolute contradiction, to mix 2 and 3 unit systems (might work for meter but not tonal series).  We end up with fractions of chromatic steps.

The third overtone is the focal point of our disagreement about the corps sonore. I think you want overtones to come naturally from the string overtones series (as if from god) but I want the string to be the smallest possible set in music that is an integral domain of Liebnitz: or as they say in mathematics, the string is an arrow with a 1, or just a 1.

I have a hard time imagining how you understand tablature without understanding the tuning used to write tablature.  Have you ever tried to actually write tablature as an exersize?  Writing tablature is where i come from.

Harmonic numbers are pleasing (I think we agree on that) and what is pleasing can be said to be the fact that the sequences 1, 2, 3, 4 ....  and 1, 2, 4, 8, .... turn out to be the same.

Now since the series 1, 2, 3, .. works out just fine in music, you may assume that music numbers are just ordinary numbers.  When music has a 1-1 relation between pitch and device position the System M works fine!  But when there relation between the pitch we hear and the position on a musical instrument are not 1-1 then the number system in music makes a big difference.

If we use your numbers then there is no way to understand how to write tablature.

The definition of simple (as in simple multiples of the fundamental) is counting numbers, not sin waves.  In music every angle is 90 degrees so every trig function is 0 or 1.  That is simple.

Tablature does not depend on frequency in any way.  So if you take the frequency away,  then you have nothing left to explain guitar.   This makes the guitar music absolutely foreign to you.  If I try to describe the guitar it sounds like nonsense I made up because you have not heard it before.

I used my example of Lucy in the Sky with Diamonds at AAAS meeting in San Diego.  I told them if the tablature came from outer space without the guitar tuning to play it, I could find the tuning by trying to play the music is every possible tuning until I got to the one called Open D.  Then I would say "Oh, the meaning of the tablature statement is Lucy in the Sky with Diamonds,  Simple observation, but it show the guitar tuning recognizes the tablature is written with the property of the guitar.

So you have to ask yourself "Where do the fret numbers in the tablature come from in the Corp Sonore, if not from pitch?"  What makes A4 if not concert pitch?

Terence, I am extremely sorry, but I understand nothing of what you write. I am a historian of theory.

As such, I have read a lot about Rameau's theory of the corps sonore and, as a result, I utterly fail to see what you mean by this term in your context.

I also studied the history of tablature writing, and I utterly fail to understand what you mean when you write "Tablature does not depend on frequency in any way. So if you take the frequency away, then you have nothing left to explain guitar."

I spent quite some time studying the history of the idea that consonance should be "pleasing" and I am most puzzled by your statement that "Harmonic numbers are pleasing (I think we agree on that)". If you mean this to be true in music, then no, certainly we don't agree on that.

You add "what is pleasing can be said to be the fact that the sequences 1, 2, 3, 4 .... and 1, 2, 4, 8, .... turn out to be the same." Well, we can't agree on that either. I fail to see in what sense these two sequences would "turn out to be the same".

And when you ask "Where do the fret numbers in the tablature come from in the Corp Sonore, if not from pitch?", I undestand strictly nothing. I don't know what the "fret numbers" are, I don't see how they could reside "in the tablature", I don't understand in what sense they could come from anywhere "in the Corps sonore", and I fail to see how they could relate to pitch.

I am very sorry, Terence, I can but conclude that all these terms have utterly different meanings in your vocabulary and in mine, and I fail to see how we could go on discussing if we don't speak the same language.

Why dont you just do a test of this with FFT software and a mockup of the strings and supports?

Thank you for your comment.

We already know measuring the frequency of the strings will show the two series are not the same. I think everyone agrees the string modes and 12-tone system do not have the same frequency spectrum. The first three overtones on the string are in agreement, but higher modes (if they exist) are clearly discordant. Some say only the octave is true.

Rameau’s assertion is the 12-tone system is derived from the string overtones.

I say there is actually no mathematic relation that can be defined between the two series. So there is no derivation of 12 tones form natural overtones.

The octave is a property of the fundamental. We can tell by ear when two fundamentals are in the same octave class. We don't need to invoke the octave mode.

Using machines to measure the frequency of strings in the laboratory will not solve the problem of understanding where the mathematics of music theory come from. Music is not a general property of frequency, and music is the same at every frequency level. To understand how Open G and Open D guitar tunings are different, measuring frequency is useless.

It is clearly a mistake to think the shape of the string is a sine wave. First, a sine wave is defined in a plane so it is difficult to understand how a collection of sine waves can make a coherent string orbit. Like the string overtones, a Fourier series has no limit. What the waves add up to cannot be determined. That is incoherence.

Second, the highest principle here is energy conservation. The shape of the string is determined by the lowest possible energy level. A sine wave implies that curvature of the string can vary at points on the string. Some parts of the string are then compressed and other parts are stretched. Then the tensile force on the string is not everywhere the same. Then the energy of the string is not at equilibrium.

Mersenne's law is not a formula that can calculate the frequency of the string using the length, tension, and density. It is merely a statement of proportionality.

Frequency goes up with tension, down with length.

I think it is clear that mathematicians physicists not not understand these two series. More accurate measurements of frequency will not make them think straight until they abandon this 300 year old mathematic fallacy. The music of the spheres is still alive in modern science!

I think that visualizing the string orbit would be more interesting than measuring the frequency. The string action does not depend on how the string is embedded in Euclidean space.

Everything about the string can be learned by inductive or deductive reasoning because on the experiments have been done long ago.

We know what music theory is, but where does it come from?

The interesting problem on the string is how it can have both a discrete frequency spectrum and a secondary spectrum that is not discrete. I suspect this is related to curvature and torsion.

What you are saying boils down to this: "natural" consonances, as they result from the harmonic series (by a blending of partials) are not the same as tempered consonances (resulting from a division of the octave in 12 equal parts).

To say that "there is actually no mathematic relation that can be defined between the two series" is mere nonsense.

The mathematic relation is as follows: the octave is defined by a ratio 2:1 (because the fundamental of the higher sound blends with partial 2 of the lower one). The tempered semitone is then defined as the 12th root of this ratio, and the other intervals by the number of tempered semitones in them.

You should not confuse music theory with equal temperament...

Good to hear from you, Meeus! The idea of relation is so basic. Let's see if we can agree on the math here. Because the problem is the algebra of the modes does not make sense. We know what it means to put two notes together but what does it means to put two modes together?

Let's define the relation of the natural and tempered series as a formal relation between two sets. We have the natural overtone series as the set N = F0, F1, F2, F3, … and the tempered series is the set T = A1, A#1, B1, C1, C#1, …, A2, …n.

A relation between N and T is defined in algebra as some subset of the Cartesian product N X T.

• First, consider relations on the Cartesian product N X N. The Cartesian product of the overtone series is the set of all pairs of modes, which is like an excel spread sheet with N = F0, F1, F2, F3, … for columns and N = F0, F1, F2, F3, … for rows. In the table for N X N cells contain fractions, like 3/2 for (F3, F2).
• Then the table defines a series of smaller and smaller fractions without limit. There is then no smallest unit of measure. There is no continuity. Differentiation and Integration cannot be defined.
• This means the pairs in N X N are not contained in N. So no relation is defined.
• The table serves no music purpose. It is nonsense because it is not closed upon itself perfectly like music.

But every cell in T X T is a simple multiple of 1. T X T is closed upon itself because it is contained in a 12 x 12 spread sheet where every pair of elements is a whole number. No fractions! And the table is coherent because every note, interval, chord, scale, mode, key, tuning, score, and tablature can be constructed from the 12 X 12 table.

That is to say, the 12 x 12 product set is the sentence-writing function of musical machines. So T X T makes sense. We can learn to understand its grammar (with effort). T X T has a smallest set, the note and interval; and no sets are smaller than notes or intervals so music is continuous.

• Differentiation and integration are simple: The tangent from C to G is 7 up and 5 down; they integrate to an octave. Cartesian product set 12 X 12 is a spread sheet can be used to make the totality of music theory but the N X N spread sheet is actually nonsense that has nothing to do with music. We don't compose modes.
• The 12-tone set has a defined complement (intervals are the complement of notes). Intervals are analytic, but I do not understand how the complement of N is defined, let alone analytic. Is the interval between two modes defined if it is not contained in N?

I think the two sets N and T can be set in a trivial one-to-one relationship (as Cantor showed the number of integers and rational number is the same)) but the relation of integers and their fractions has no useful musical meaning. Are you claiming modes make sense harmonically in music theory?

• We have multiplication by 2, but what does it mean to multiply by 2 and 3? So then music is just ordinary multiplication? Nothing special?
• Once you give up the higher modes, it may seem the octave mode still defines the interval 2, but in fact we know when two fundamentals are octaves. Who needs the octave mode if we have the octave note by ear already? Of course, it is useful in guitar construction but tablature is only fundamentals (harmonics require annotations).
• We could make all music theory rules without the octave mode. So the octave mode cannot be critical to the octave interval.

We definitely agree on the mathematical rule of the 12th root of 2 as the anterior most law of nature. Can we also agree that there can only be that one relation where 12 = 1? No other rules allowed -- everything must follow from the 12 X 12?

I think you can expect the same harmonic series from a monochord as a guitar or any other bowed string. It is the bowing point or plucking point that determines the shape and strenghts of the harmonics and nothing else.

Terence B Allen, you should not compare pears with apples.

You write: " We have the natural overtone series as the set N = F0, F1, F2, F3, … and the tempered series is the set T = A1, A#1, B1, C1, C#1, …, A2, …n. "

You should realize that while the first series is a series of frequencies (which would better be expressed as N = F, 2F, 3F, 4F, ..., with F being the fundamental frequency), the second is a series of logarithms of frequency ratios.

If units of the tempered series are 21/12, then the logarithmic series that you call the "tempered series" becomes T = 1, 2, ~3, 4, etc.

If these two series ressemble each other, it merely is to the extent that a series of whole numbers may ressemble a series of logarithms. However, if you want to express the series of whole numbers in terms of the logarithmic series with basis 21/2, then it becomes, to the approximation dictated by the logarithmic base, log(N1/12) = 0, 12, 19, 24, ... – or, in numeric values of the logarithms, 1, 2, 2.997 (=~3), 4, etc.

This is not high level mathematics, it merely is low level arithmetics. It has nothing to do with the vibration of strings, the arithmetics would remain exactly the same for notes (or frequencies) produced by any other means (e.g. wind instruments, or singing, or whatever).

In response to Anders Buen

But the monochord has one more point of force acting on the string.

This is inertial in the sense that the string vibrates in the fundamental mode every time -- unless acted upon by an external force that kicks the fundamental into a higher energy mode.

Similarly, the octave mode continues until acted upon by a force. But if the octave and fundamental can coexist, then the octave would degenerate to the fundamental. After all, the string seeks the lowest energy level and energy increases with frequency. So the fundamental is the lowest energy level possible,

Modes on the string cannot add because they have different critical points. If the octave and the fundamental coexist, then the point at the middle of the string is both moving and fixed. Clearly that is impossible.

So the curvature of the string, and hence its frequency, is constant. It does not matter how the string is plucked.

But the torsion of the string (motion that is perpendicular to the direction of motion on the tangent) depends on how the string is deformed. So the non-discrete frequency spectrum of the string depends on where it is pluck or bowed. The bowed string is in overdrive so it does not achieve an equilibrium.

Think of how the bass string reads on a guitar tuner. Sort of a buzzy "Wooow" as the wobbly string settles into an equilibrium. That is the torsion dampening onto the uniform action.

Similarly, consider this experiment: A string is bowed or repeatedly plucked as a finger touching the string lightly at a point moves across the string. This induces a number of overtones at positions which are clearly not simple multiples. I don't see these overtones are explained anywhere in the literature.

Also remember that musical instrument and guitar pick up coils are engineered to resonate and create frequencies that are driven by the fundamental, but no properties of the string. Without a sound board, maybe the higher frequencies do not even exist.

Thanks again for your comment. It was helpful to think about.

N = F0, F1, F2, F3, … is the sequence of modes that are subspaces of Euclidean space defined by the string action. Frequency is the algebraic property of F. But N is the same at every frequency.

Tempered was your word. The bit about the 12th root of 2 is a theorem that can be proven geometrically from the rule that 12 =1. I would call the set the pitch value alphabet set.

What we are interest in is not simple math, but the richly-expressive added mathematic structure of the musical key. The musical key is mathematical but where does the math come from?The product set R12 x R12 is astonishingly rich. String modes in fact have no elements in common because they do not share critical points.

You seem to suggest that a set of integers is induced by the function p = log2 f. I accept this transformation. But in fact the only integers induced are log2 0 = 1 and log2 1 = 2. Those are the only simple logs. log2 3 is not simple. We don't allow it. Then we have two points defined - (0, 1) and (1, 2). This defines a line. On this line we establish a correspondence between pitch set A and the set of states on a musical instrument S.

A correspondence means there is a function from pitch to the music instrument and f:A to S and a function g:S to A where f and g are inverse function.

Now consider how the pitch value alphabet set and the set of string modes might correspond. We have then a function f from the pitch value alphabet to the mode and another function g from the string mode set to the pitch alphabet. Then the functions f and g have an identity where fg = 1.

I can't see how such a correspondence can be established.

The correspondence you suggest has the order of the string modes that induce the 12 tones in the order 0 12 19 24 .... ? So in the first four elements we pick up one that is not in the octave class of the fundamental.

There is also a problem with symmetry: The modes are bilaterally symmetric -- they extend across the string.

The 12 tones are not symmetric because the second octave takes only a 1/4 of the string. There is no highest note but the notes cannot be extended across the string.

In my figure I am trying to show that there is no correspondence between mode 3 and tone 7 that is 1-to-1.

Too much text and philosphy. A monochord is simply an instrument with one string bowed or plucked with bands to hold down the finger on so different notes can be played. The action of that string is not different from any other musical string And it will not give a different harmonic series than other strings given the same relative bowing or plucking point.