Alain Connes Noncommutative Quotient Space is the inverse frequency of Dr. Ruth E. Kastner's Spacetime Interval wavelength

 

Alain Connes Noncommutative Quotient Space is the inverse frequency of Dr. Ruth E. Kastner's Spacetime Interval wavelength

OK so let's parse through the music noncommutative discrete ratio argument of Alain Connes on the 5^th dimensional noncommutative torus starting on page 388 of

https://www.its.caltech.edu/~matilde/coll-55.pdf

 

So by this phrase  we need to go back to what this refers to.

By C-algebra Connes means the noncommutative matrix math of Clifford algebra. So the above quote is on page 388 and we go back up to 386 to find this:

 

So then we find this:

So now it's been broadened to include the crossed product, just as Dr. Ruth E. Kastner is relying on the cross product of the squared time and squared spatial measurement.

So now we see that the music frequencies are inserted for p=2 to the 19^th x q=3 to the 12^th as the cross product. Based on logarithms then the quotient is the subtraction of the exponentials, just as Dr.

Ruth E. Kastner's spacetime interval relies on subtracting the squared exponentials. Only in the case of Connes the frequency is the inverted instead of using wavelength.

How do we know this? We have to go back up to his claim.

So the quotient is based on the noncommutative inverse frequency as 3 to the (1/19) x 2 to the (1/12) such that the quotient space is (3/2) to the (1/7).

 So the capital P is the cross product of the squares as the discrete rational frequencies and thus 2 to the 19^th x 3 to the 12^th.

So that has the value of subtracting the inverse frequencies for the quotient of the logarithm. Such that the quotient space is (3/2) to the (1/7) since the inverse of the exponential is

noncommutative and thus 2 to the (1/12^th) and 3 to the (1/19^th). Thereby subtracting the difference of 3/2 as (1/7).

This is then the noncommutative wavelength as the Quotient Space that is inserted back into the imaginary negative frequency axis constant equation Connes devised.

So that's way back on page 364.

So since it's greater than 1 then the subtraction to the negative 1 = always "strictly less than any positive number" and hence the Pythagorean discrete noncommutative "double quotient" ratios

Have a geometric dimension of Zero - as the noncommutative chords within the noncommutative torus (representing the quantum Bloch Sphere).

"The ear is only sensitive to the ratio, not to the additivity ... multiplication by 2 of the frequency and transposition, normally the simplest way is multiplication by 3 ... 2 to the power of 19 is almost 3 to the power of 12...time [spacetime] emerges from noncommutativity ... What about the relation with music? One finds quickly that music is best based on the scale (spectrum) which consists of all positive integer powers qn for the real number q=2 to the 1/12th~3 to the 1/19th. Due to the exponential growth of this spectrum, it cannot correspond to a familiar shape but to an object of dimension less than any strictly positive number. "

(Connes, 2012)

 OK that's my take on it...

thanks

p.s. and thus Connes spiral is also the Pythagorean Infinite Spiral of Fifths!