Thirteen Aspects of Mathematics

My previous post "64 Aspects of Mathematics" assumed 8 elementary theories and predicted higher order theories through a non-commutative multiplication of the octet by itself.

This post is a speculation on the resemblance of the octet to the major scale of the 12 tone equally tempered music. The octet classification has some imperfections that may be addressed better in a stack of 13 (12+1).

The major scale has a modular symmetry when the octave of the root is appended (2C gives a feeling of completion);

(((C,D),(E,F)),((G,A),(B,2C)))

This symmetry can be heard easily by trained ears. The Stern-Brocot Tree easily explains this modularity. Modular Group, SL(2,C), is mentioned in many papers dealing with the music theory.

The equal tempered music is a marriage of two symmetries; the modular group and the logarithm. The cycle of fifths almost meets the octaves in 7 octaves;

((3/2)^12)/2^7 = 1.0136

This is the miracle that makes the equal temperament possible. I think this is not an accident.

The 13 tone stack is;

C,#C,D,#D,E,F,#F,G,#G,A,#A,B,2C

The mathematics octet can similarly be expanded by filling in the missing theories;

Set
-Logic
Analysis
-DifferentialG ... ?
Geometry
Algebra
-Combinatorics ... ?
Probability ... QM
-StochasticP ... QFT
Statistics ... SM
-Inference ... Td
Computation ... SR
Information ... GR

Logic for #C and so on. The most interesting part of this comparison is the relationship of the first to the last;

(Set,Information) ... (C,2C)

Obviously there is a very simple relationship between C and 2C. Is there such a thing for the Set and the Information Theories? There are some studies on this subject. A notable one is by Gregory J. Chaitin, "Gödel's Theorem and Information";

http://www.cs.auckland.ac.nz/~chaitin/georgia.html

I have come across a few studies (proofs?) of Riemann's Hypothesis via the concept of entropy. But as far as I know these are not taken to be the last word yet.

When the experiment part is compared to a classification of physics, IT corresponds to General Relativity. Perhaps this is why the Black Holes have a simple entropy equation. The Modular Group and General Relativity relationship is also all over the net.

The Thermodynamics (Td) and Special Relativity (SR) neighborhood is also interesting;

http://arxiv.org/abs/0712.3793
"On the Relationship between Thermodynamics and Special Relativity"