Approximate the perfect fifth Starting at 1-TET, give equal temperaments that have better and better approximation of the perfect fifth(just ratio 3/2). (OEIS sequence A060528)

The formal description of the sequence, copied from the OEIS:


  A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of two tones of musical harmony: the perfect 4th, 4/3 and its complement the perfect 5th, 3/2.


Note that by symmetry, the perfect fourth doesn't matter.

Let's say we know that 3 is in the sequence. The frequencies in 3-TET are:

2^0, 2^⅓, 2^⅔


Where 2^⅔ is the closest logarithmic approximation of 3/2.

Is 4 in the sequence? The frequencies in 4-TET are:

2^0, 2^¼, 2^½, 2^¾


Where 2^½ is the closest approximation of 3/2. This is not better than 2^⅔, so 4 is not in the sequence.

By similar method, we confirm that 5 is in the sequence, and so on.

When given an integer n as input, the output must be the first N numbers of the sequence in order. For example, when n = 7, the output should be:

1 2 3 5 7 12 29


Sequence description by xnor

The irrational constant \$ \log_2(3) \approx 1.5849625007211563\dots\$ can be approximated by a sequence of rational fractions

$$ \frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{11}{7}, \frac{19}{12}, \frac{46}{29}, \dots$$

A fraction is included in the sequence if it's the new closest one by absolute distance \$ \left| \frac{p}{q} - \log_2(3)\ \right|\$, that is, closer than any  other fraction with a smaller or equal denominator.

Your goal is to output the first \$n\$ denominators in order. These are sequence A060528 (table). The numerators (not required) are given by A254351 (table) 

Rules:


Do not import the sequence A060528 directly.
The format doesn't matter as long as the numbers are distinguishable. In the example above, the output can also be:

[1,2,3,5,7,12,29]
As this is a code-golf, the shortest code in bytes wins.
05AB1E, 19 18 bytes

µ¯ßNLN/3.²&lt;αßDˆ›D–


Try it online!

µ                      # repeat until counter == input
 ¯                     #  push the global array
  ß                    #  get the minimum (let's call it M)
   N                   #  1-based iteration count
    L                  #  range 1..N
     N/                #  divide each by N
       3.²             #  log2(3)
          &lt;            #  -1
           α           #  absolute difference with each element of the range
            ß          #  get the minimum
             Dˆ        #  add a copy to the global array
               ›       #  is M strictly greater than this new minimum?
                D–     #  if true, print N
                       #  implicit: if true, add 1 to the counter