https://arxiv.org/pdf/2105.12233.pdf
Self-affine is a kind of self-similarity, where the scaling is different in each of the axis. Self-similar objects on the other hand grow at the same magnification factor in all three axis in same time frame. Thus, a self-affine object changes as we zoom in, unlike a self-similar object.
Converting the exponential to the logarithmic addition replaces the symmetric self-similarity with a noncommutative self-affine "operator." http://fac-staff.seattleu.edu/boersema/web/WCOAS/Slides/Landry.pdf
the process never terminates:
no power of (3/2) is ever a whole number of octaves
(or indeed any integer multiple of the first frequency)
...
So we fudge the fifth to make the equal-tempered scale:
find a ratio r roughly equal to 3/2 so that
some small power of r is a power of 2.
This amounts to finding good integer approximations
for the solutions of (3/2)^x = 2^y,
which we rewrite as 3^k=2^l, or l/k=log3/log2=1.584962501...
The theory of continued fractions tells us how to do this:
Form the continued fraction expansion of this real number,
stop at certain points, and reevaluate the fraction l/k
which will approximate log3/log2.
...
So here it is: log3/log2= cont.frac[1,1,1,2,2,3,1,5,2,23,...],
which gives the following optimal approximations:
1/1, 2/1, 3/2, 8/5, 19/12,
https://www.valdostamuseum.com/hamsmith/musPhys.html 
So the continued rational fraction - just as in music theory - is noncommutative that then converges or "recovers" the self-similar symmetric fractal!!
https://arxiv.org/pdf/2105.12233.pdf
We construct Dirac operators and spectral triples for certain, not necessarily self-similar, fractal sets
built on curves. Connes’ distance formula of noncommutative geometry provides a natural metric on the
fractal. To motivate the construction, we address Kigami’s measurable Riemannian geometry, which is a
metric realization of the Sierpinski gasket as a self-affine space with continuously differentiable geodesics.
As a fractal analog of Connes’ theorem for a compact Riemmanian manifold, it is proved that the natural
metric coincides with Kigami’s geodesic metric.
https://math.ucr.edu/~lapidus/confidential/LapSar.pdf
In fact, despite all these advantages, non-fractal antennas can reach or exceed the performance of their
fractal counterparts. This is in accordance with antenna theory. In 1999, a characterization to make antennas’ frequency invariant was established.
the harmonic Sierpinski gasket can be
used as a geometric configuration in antenna design. Clearly, the antenna design will use a pre-fractal
version of the harmonic Sierpinski gasket, that is the IFS is built with a finite number of iterations. As
already mentioned in Section 2, a self-affine fractal is given by contractions that scale the set by different
factors, horizontally and vertically. Accordingly, this self-affine geometric configuration can provide
further flexibility in design.
Fractals are symmetric but not self-similar typically - vid
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