From Logarithms to Entropy: More music acoustic perception lies of science

In other words, perceiving a perfect fifth as consonant does not mean that our ears like the numerical value , it rather reflects the circumstance that the third partial of the lower tone coincides with the second of the upper (see Fig. 1).

Yeah that's nice only the perfect Fifth is also 2/3 not just 3/2 and if is it 2/3 then the geometry of the perception is not the same pitch is it? In other words the third partial of the lower tone is G but if the Perfect Fifth is 2/3 then the F clashes with the G that is the third partial of the C, the root tonic. Just as the third partial of the upper tone of 3/2 is D which clashes with the C. And yet C to F as 2/3 is also a Perfect Fifth. So why is this not included in the examination of the Perfect Fifth perceived pitch interval?

And so Wikipedia in their equal-tempered tuning entry cites this 2017 lecture notes of a University of Chicago website on Digital Sound:

At first approximation, the pitch of a sound is the logarithm of its frequency. The human ear detects frequencies from about 20 Hertz (cycles per second) to about 20,000 Hertz. But the change from 20 to 21 Hertz is perceptually much greater than the change from 19,999 to 20,000. Multiplying, rather than incrementing, a frequency is perceived as adding to the pitch.

What is being referred here with the phrase "perceptually much greater" is actually the harmonic beats of the clashing overtones but this can also create a new undertone, the "difference tone." So that the beats sound ...as uncertainty

How long do you need to determine whether two notes are in tune?  But you probably found that, when the frequencies differed by 3 Hz, you needed (very roughly) about a third of a second. When they differed by 1 Hz, you needed more time. So, roughly speaking, if the frequencies differ by Δf, then you need a time of 1/Δf to notice. In other words:

Δf.Δt > ~ 1     or, in non-mathematical language:

(time taken to measure f) times (error in f) is on the order of one, or larger. This result was known to Joseph Fourier In this context, I call it: The musician's uncertainty principle. Because musicians know this, qualitatively at least. If the chord is short, or if you are playing a percussive instrument, the tuning is less critical. In a long sustained chord, you have to get the tuning accurate. And of course oboists in orchestras play notes for tens of seconds while the other instruments tune carefully before a concert.
So far this calculation is just an order-of magnitude. One can imagine doing a little better than  Δf.Δt > 1  by careful measurement. (Have a look at the diagrams on What are interference beats?). The uncertainty principle is usually written with an extra factor of 2π: it takes about one radian rather than a whole cycle. So:

(time taken to measure f) times (error in f) is greater than about 1/2π.

The difference tone:

 Actually, this frequency is far too fast for the brain to follow individual oscillations, and so the brain does not interpret these beats as a periodic change in the loudness of the sound. In fact, under the right conditions, the brain interprets the beat frequency as its own frequency.

1. When it comes to sound, beats frequencies in this range sound rough and dissonant – they are simply unpleasant to listen to. The pitches are too far apart to consider as one pitch, but too close together to distinguish. The brain cannot follow the individual beats but it is not fast enough to interpret as a new pitch. Simply put, when beats are present in this frequency range, the brain registers dissonance.
2. When the beat frequency is greater than about 60 Hz, the brain can distinguish the pitches of the two notes and the beat frequency is interpreted as a difference frequency, if at all.

But this "uncertainty" of time and frequency is actually due to noncommutative phase with a dimension of zero - or less than any real positive number.

 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 ${\displaystyle q^n}$ for the real number ${\displaystyle q=2^{\frac{1}{12}}\sim 3^{\frac{1}{19}}}$. 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. As explained in the talk, there is a beautiful space which has the correct spectrum: the quantum sphere

 http://noncommutativegeometry.blogspot.com/2012/10/the-music-of-spheres.html

So a sphere is normally 4pi or  Δν Δt ≥ 1 /(4π) but for sound as time-frequency is it 2pi 
or Δν Δt ≥ 1/( 2π )

This means that in the "sphere" of the Perfect Fifth phase we can still have 2 different frequencies of 3/2 and 2/3 as different phase cycles yet both the Perfect Fifth. The ear will hear no difference and yet the visual spectrum is different - as is the root tonic or fundamental pitch in relation to the octave. In other words if we double 2/3 to 4/3 now the root tonic of 4/3 is the G - in relation to the overtone series with 1 as the C and 2 as the C and 4 as the C. But the 3 of 2/3 is an F! So suddenly our doubling of the octave changes the pitch perceived. This is why the Perfect Fourth of 4/3 is not part of the harmonic series. It is noncommutative phase that has a geometric dimension of zero.

Δp_x.Δx > ~ h/2π.
(uncertainty in momentum) times (uncertainty in position) is greater than h/2π.

 So the Planck's Constant is the average mean energy of light as frequency per second with the seconds as a wavelength radian as inverse to the momentum as directly proportion to frequency. But it is assumed that the Joules per second can be canceled out with the wavelength as second (cycles/second as frequency/time). But since it is noncommutative phase then it can not be canceled out as a unit of radians (dimensionless!).

In other words.... this 50% is a divide and average statistic but the 2pi is noncommutative so can not just be doubled to 4pi.

 In another treatment, the "uncertainty" of a variable is taken to be the smallest width of a range which contains 50% of the values, which, also in the case of normally distributed variables, leads to a lower bound of
Δν Δt ≥ 1/( 2π ) for the product of the uncertainties

ΔxΔp ≥ h ⁄ (4π)