We have the tuning is a set of programmable (because we can chose the tuning intervals, typically 2 to 7 steps between strings.

Everything about the tuning is defined by the intervals between adjacent strings, which the irreducible Zariski Vector, and equivalently by the summation Vector.

Thus for the tuning EADGBE we have the n-tuple (0, 5, 5, 5, 4, 5) or in summation (0 5 10 15 19 24).  These instructions partition the guitar under the axiom of separation.  We have the Spec S (for the string) maps to Spec G for the guitar so each string is inside the tuning.  The guitar tuning is a subset of the musical key formed by re-iterating the addition of point wise limits in standard construction.

There are at least a dozen tunings in common use, perhaps 100 are known.  Each tuning has its own Zariski vector, a list of computable instructions that intialize the guitar.

The guitar tuning is a subset of the musical key that does not contain the musical key but instead an image of the musical key projected by the Zariski vector. The tuning intervals are the prime ideals of the closed algebraic field of the guitar fret board (which is projective not just affine).

The tuning is a turing machine that can recognize when the tablature satisfies the guitar tuning theory.  If you give me tablature without the tuning, I can find the tuning because the tunings are well-ordered.  If you give me piano music the turing machine can make the data strip into tablature according to how the guitar is initialized.

The guitar strings are the open cover of the guitar because every point in the model is contained in at least 1 string.  The model is converx, compact, and complete.  The union of the strings ins the interior and the intersection of the strings is the closure.  The lattice signature is intersection and union.  Since I can find the guitar tuning, the ring signtaure (0 5 5 5 4 5) is an invertible homorphism.

The idea that the musical key is a universe form is dis proven by Skolem, isn't it?

Remember, the guitar is finite. Want to see the curve-lifting map?

Hi Terence

If you want think about the earliest computing aids, fingers immediately spring to mind, followed by the simple tally stick. The Bonobo bones were said to be used some 30,000 years ago to help women record their menstrual cycles in Aftica. The abacus was used by the Babylonians and later the Greeks some 2,000 - 4,000 years ago. The Antikythera mechanism, from just over 2,000 years ago was used by the ancient Greeks, and was thought to be used to plot the movements and positions of the planets, often considered as the earliest known mechanical analog computer. All before we get to AD. And there are a huge number to follow from then onwards. I once did a project on this for a computing science course. There were enought to fill a 20 minute video.

So there are a great many contenders for the title of earliest programmable device.

Hope that helps.

Regards

Bob

The loom and notched card that passes over a set of mechanical stops dates to the early 1700's. I believe.  Tablature, called scordatura or bad score, is a well-known mathematic problem without solution dating to late 1500's.

The point is that tablature is a game played with marks on paper that are manipulated using a language that satisfies the guitar tuning theory.  That is way different that counting.  In fact, it is precisely the two-countable nature of guitar that makes it a turing machine.  Every note on guitar is counted twice because is is a simple multiple of the system fundamental and the string fundamental.

The strings are the open cover of the guitar but the frets are not.  If we eliminate frets using a capo then the language of the tablature is not altered, but if the tuning of a single string is altered by the smallest increment then the tablature must be completely re-learned.

The algebraic language of the guitar is internal to the affine cover, so the observer thinks that the path of tonal movement in the tablature has no meaning because in R2 all paths are equivalent.  But in the guitar model sequences with the same pitch values do not produce the same result.   The inner abstract measure of the guitar tuning vanishes on intonation.

Music is older than mathematics but mathematics cannot explain music because the 300 year music model has not been updated to modern theory.  What we have in the literature today is the idea any mathematics apply to music except for set theory! 

If google does not have the axiom of separation then that is a limitation on google knowledge.

I suggest using either "Set Theory and the Continuum Hypothesis" by Paul Cohen or the more accessible "Set Theory and the Continuum Problem" by Smullyan and Fitting.

The axiom is the one that allows pitch to be partitioned in a n-ary function.  It applies to the 12-tone set and to the guitar strings.   It tells us that pitch space is disconnected because it is made of disjoint sets.  Only singletons are allowed and every distance is 1 because every tone regardless of rank is a distance of 1.  To go from Open D to Open G tuning you change 3 strings, but it is still 1 step.  It is called a log space.

I don't want to type the axiom here, but if you look at Smullyan page 170 axiom 2 we have axiom schemes in the form where we have (in words): For all pitch y there is a unique model z such that for all guitar positions the guitar position is in the model if and only if the guitar position is an element of pitch and the guitar tuning sequence is true.

The axiom of extension is obviously true and the axioms of the null set, union, and power set are hard to deny. 

It's ironic google can recognize the ZF axioms because ZF must apply to google even if google does not know it!

To believe music is not ZF is like not believing in Boolean atoms.

OK, good let's talk elementary sets.

Suppose we have a single guitar string with 20 frets.  So we have only integers up to 20. Finite. Each natural number is defined by induction as a multiple (or sum) of the system fundamental.  That is, the null set is 0 and the next element is 1.  An so on.

The string set is defined by a map between frets and pitch.  In projective space the string line is naturally a circle.  The circle classifies the pitch value set as Z.  (The idea the pitch values are "intergerized" may be why you think the sets are not constructable. But the octave is in fact precise to a point so that makes the 12-tone set a cauchy sequence.

We have two defined sets: the pitch value set indexed by integers i and the the fret position value set indexed by integers j.  The sets are disjoint and can form a union in which the fret numbers and the pitch values run together like railroad tracks that vanish on the string fundamental which is the null set value.  The fret and pitch numbers have a constant relation because on the first string of the guitar the pitch-position relation is constant.  When the musical key is introduced as a base-point then the operation of musical key transposition is induced on the string as a second operation.  It is immediate clear the musical keys are projective and not affine sets because the tonal center induces a base-point topology on  the string.

Now we add a second string.  The 2 string sets, each a semi-algebraic set, can form a union because they can be glued together at the system fundamental which is the point of origin in projective space. The union of the strings is closed to intersection and therefore forms a Baire metric space.  The key element here is the octave closure which makes a Lebesgue measure of 12.  That means any set 12 or small (that is, any set  that fits in an octave unit interval) is contained in every set.  The musical key on two strings induces a new set of operations.  So the musical key on one string is a k = 1 construction while the musical key on 2 strings is a k = 2 construction.  The keys in the tuning space make an ascending chain so the k+1 key on 2 strings is a subset of the k = 1 musical key.  The effect is surprising because we think there is only 1 musical key for all music.  This is impossible in set theory.  There is no set of all sets.

The strings are the cover of the guitar in the axiom of separation that says the pitch value set can map to the  guitar because in is partitioned by the strings into subsets of the pitch value set.  That is, the fret numbers that are the typed alphabet of the tablature algebraic script are always less than or equal to the pitch value numbers.  That makes the guitar music a subset of the music that is determined by the the interval between the two strings, and nothing else.

So now the 2-string set of string depends only on the interval between the two so the the quantification by pitch is eliminated.  That is, the intervals on guitar like all intervals do not depend on the interval end-points.  In fact, if guitar music is defined by a set of intervals that are whole numbers then it is immediately clear they are Borel Sets in which F(x, y) = 0.

The guitar fret board is a closed algebraic field with a nullstellsatz, because the guitar tuning is a 6-tuple associated with a field of zeros that are the string fundamental.

The fret are not the cover of the guitar because they can be eliminated without changing the non-trivial algebra of the tablature.  It is hard to see that tablature is a language unless you say, try to change one tuning to another.  Then you see how algebra alone fails but semi-algebra does not.  Predicate logic is exactly what it is: consistent withe ZF.  That means if you know guitar, then you can tell when the tablature is correctly proofed.  The sequences in music are proofs!

You  may think that all guitar tunings are the same because surely if all the notes are correct in the pitch value then the sequence of notes on guitar does not matter, but in fact the sequences do matter. The important thing here is that music sequences are finite and that means you can learn guitar from music because guitar is a subset of music.  You only need the tuning to initialized the turing machine.

For example, in the guitar tuning with the tuple (0, 1, 1, 1, 1, 1) almost every sequence in music is false (that is, unplayable).  That is why no one uses that tuning.  The fact that only 10 to 100 tunings out of many thousands are used shows there is a Borel probability distribution on the well-define set of guitar tunings.  The more true statements, the better the guitar tuning rank.  When the Borel distribution f(x, y) = 0 on [0, 1 x [0, 1] is observed by auditory surveillance in 2-dimensions on [0, 1] it always seems the relation pitch=position is trivial.

I think you make not realize how pitch is defined as a continuous function and how the octave defines a curve-lifting function.  This is key to showing that the intonation and tuning functions are inverse functors.  It is the left-adjoint matrix that allow the turing machine to recognize when the tablature is correctly written.

Incidentally, intonation means to adjust the pitch of the system fundamental, and tuning means to adjust the interval between 2 or more strings.  Therefore, intonation is specific to pitch and independent of tuning, and vice versa.  The intervals make a set of diaphanous equations that define a commuting algebra of Hilbert projections.

By the way, the answer to my question is that it is impossible to prove musical sets are not constructible if  ZF is consistent.

Here's an abstract I am working on. I would appreciate any comments.

Music is a collection of finite sequences based on a nontrivial disjoint pairwise pitch-position equivalence relation f(x, y) = 0 as a standard model. The initial element F0 is abstracted from pitch, not in the frequency spectrum. The mathematic category of music is a theory, a structure, and a workplace. Writing music sequences is a mathematic workplace game using predicate logic to manipulate marks on paper. Probability distributions observed in music are anomalous to affine theory since the probability of each 12-tone set element is not 1/12. Borel analytics assume musical sets are Baire spaces. The elemental operation is the induction of integers on the initial element. Real, rational, and integer submodels are accurate Borel approximations for music quantified by pitch, but these trivial measures of pitch do not triangulate guitar sequences correctly. Tablature is a rich algebraic language where sequences of ordinals are interpreted in an algebraic closed field using a model of cardinal prime ideals. Pitch is the abstract inner measure of music algebra sequences. The units 0, 1 are identical to 0 because tuning measures vanish on all singletons at intonation. The abstract measure of pitch has nothing to do with simple measures of frequency on the real line. The tuning ring signature (such as [0 5 5 5 4 5) is invertible and can be recovered up to intonation isomorphism using the guitar as typed alphabet finite state machine. The guitar model recognizes when the algebraic language of tablature satisfies the abstract inner measure of the guitar tuning theory. Music theory is a Zermelo-Fraenkel universe.