In "A Geometry of Music" by Tymoczko the author describes a lattice that "lives in what mathematicians call 'the interior of a twisted triangular two-torus,' otherwise known as a triangular doughnut."
Clearly the torus is isomorphic to a closed, bound rectangle in a plane. I am assuming that the twisted torus is a torus that has a composition function that forms a line that winds around the torus.
It seems to me then that the twisting torus is really a projection of a sphere in two dimensions. The winding number implies there is a curve lifting function.
My assertion is the torus T is an approximation of the sphere because the inclusion map i: Sn→ (R3 – 0) induces i: Sn/T→ RP3 and the map f:(Rn+1 – 0)→ Sn induces f:RPn→ Sn/T.
This seems to be what the author implies anyway but does not say: "Pitch-class space is formed out of pitch space when we choose to ignore, or abstract away from, octave information. As a result, many properties of linear space are transferred to circular pitch-class space."
Doesn't this last quote indicate that the pitch line is automatically a circle in RP3? Then the manifold is a sphere and not a torus, right?
My goal here is to show that music spaces are projective but have an affine cover with at single face in R1.
The torus is absurd: it cannot reduce to a point.
Dear Terence, my limited knowledge doesn't permit me to speak about those "music spaces". I am just surprised by a statement you made: """Clearly the torus is isomorphic to a closed, bound rectangle in a plane""". Not at all. Consider a rectangle ABCD. To get a torus, one has to identify AB with DC, and then identify the two circles such that A is identified with B. So you get ABCD / rel, where rel is an equivalence relation, and the result of this factorisation of a topological space is not at all an isomorphic (homeomorphic) topological space. For example, let M be the middle of AB translated with epsion to the right side and N be the middle of DC translated with epsilon to the left side. The rectangle \setminus MN is no more a connected topological space, while the torus \setminus MN is still connected. Of course, there are thousands of other reasons why they are not isomorphic. Similarly, rectangle, torus, sphere and projective space are pairwise not isomorphic (not homeomorphic).
To Mihai Prunescu
We start here with what Euler described as the tonnetz, meaning the tone net. The octave is a point and the compliment is the filter. The subfilter is the net.
Euler learned that, on Bach's tempered keyboard, the music key and pitch value seem to be values in the R2 coordinate plane such that the pair (musical key, pitch value) = (musical key + x, pitch value + y). This makes an affine space.
Since the pitch values and musical keys are closed by the octave, Euler says the resulting manifold in a torus. This makes the Cartesian plane into a real number topology with the absolute value metric.
Now the topologic manifold (by Mobius) is either a torus or a sphere. So what is the fundamental group?
I get lost following your argument that the plane is not closed and bound. The Heine-Borel theorem tell us it must be. The only question is how is the line closed as a circle?
Most importantly I think you are incorrect because in the case of the spherical manifold we have an isomorphism that includes a point, triangle, cube, square, vector, matrix, and all the other thing that follow when we have an atomic (Boolean) structure.
Clearly the manifold for music is dense: Between every two elements is another element. There is nothing keeping you from understanding music topology except that you don't consider it a valid mathematic field of investigation.
The sphere can map to a plane or a torus using a point-wise projection, but to recover the sphere from the torus requires an algorithm. (Tarski Theorem)/
To make a torus twist is to adjoin another copy of R to the torus that assumes x = y. I think that is what you said. This is what Euler called pitch value space. Just add copies of pitch as R to make Rn. It impossible.
Euler must be wrong because how can a harmonic system with 1 fundamental have 2 generators, or n generators? Prime ideals yes, but 2 generators - no.
Perhaps you're answer explains why mathematicians aren't interested in Euler's fallacy. They don't seem to think the tone net is a metric space.
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