Working on the boundary problem for the vibrating string, so I want to apply simplex theory.
I figure the Euler number for the string v - e + f is calculated as 2.
The reason is the there are n + 1 nodes for n waves and the concatenation itself is the face. So we have n - (n + 1) + 1 = 2
This seems to apply to guitar, too. We have 6 strings which are the vertices, 5 intervals which are the edges, and one face which is the pitch value set.
Can anyone dispute the calculation?
Dear G. Kumar
Thank you very much for correcting my algebra!
I have the same result for the guitar with 6 strings (vertices defined by frequency points), 5 intervals between strings (edges), and 1 face which is the set of pitches on guitar. So the string is a polyhedron, and the guitar is a polyhedron whose vertices are also polyhedrons. That is, the guitar is a simplex whose vertices are also simplices.
This is a novel result that is not reported in the literature and defies the commonly held notion that the structure of music theory is 2-fold, which Euler established in 1739 with his tonnetz, meaning that he thought the musical key and the pitch value made a tone net that defined a Euclidean plane. However, if the Euler number is 2, then the structure cannot be realized in R2.
Since it is not reported, I am glad to see that you can follow the Euler number calculation. It is so obvious, but a theory benefits when others are convinced its true.
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