Future redshifting and effect on the 'pitch' of CMB radiation After discovering this question exploring the sound of a blackbody, I started wondering about the sound of the Cosmic Microwave Background radiation from the Big Bang, specifically what the current pitch might be at its peak frequency and how that might evolve over time due to redshifting. Here's what I've been able to deduce so far:

$$
pitch_{CMB} = 160.4\,GHz \approx D_{33} = 157.7\,GHz
\\
pitch_{CMB}\,lowered\,by\,30\,octaves = {160.4\, GHz \over 2^{30}} = 149.38\, Hz \approx D_3 = 146.83\,Hz
$$

Building on this result:

How much time will need to elapse before the CMB redshift will cause this pitch to drop by an octave (or even by a semitone)?
How much variability would this pitch have depending on the dipole anisotropy of the CMB from our perspective as well as due to more generalized anisotropies in the CMB?
What pitch would the peak value of this blackbody have had at earlier points in the history of the Universe?


I've tried to apply some of the information in this question and elsewhere, but would greatly appreciate hearing from someone with more expertise in the subject area if this unorthodox comparison has piqued your interest like it has mine.
The wavelength of the CMB increases linearly with the scale factor $a$, which is defined as 1 today, so the "sound" of the CMB has dropped an octave when the Universe has doubled in size (in all three directions), i.e. when $a = 2$, which will happen when it is roughly 25 billion years old (see e.g. Fig 1 of Davis &amp; Lineweaver 2004).

The magnitude of the dipole of the CMB is $3.4\,\mathrm{mK}$ (Kogut et al. 1993), i.e. $\sim10^{-3}$ of the peak value. The primordial fluctuations are even lower, roughly a factor $10^{-5}$ of the peak value. The dipole is partly due to our motion round the Milky Way, and so will fluctuate somewhat with the period of a Galactic year, which is 225 million years.

The figure shows how the peak frequency evolved in the past. Its current value is 160.4 Ghz. To calculate how a human would perceive the sound at a given time, I suppose you'll have to multiply the spectrum with the frequency-dependent transmission function of the human ear, but that's physiology which I know nothing about.



The figure was produced by calculating the frequency at a given time as
$$
\nu(t) = \frac{160.4\,\mathrm{GHz}}{a},
$$
where $a$ runs from $1/1100$ to $4$,
and calculating the age by integrating the Friedmann equation, assuming a Planck 2015 cosmology.