In Pythagorean tuning, the pitches are generated by compounding perfect (just intonated) fifths, one way to do this is to go symmetrically outward from a central pitch, octave reducing.
Note that Gf, Fs generated this way are very close in pitch (enharmonic). The reason why is happens is a numerical coincidence
(3/2)**12 is approximately equal to
2**7, i.e. going out enough fifths (approximately) gets you some number of octaves above your starting pitch. This makes 12 notes a "natural" stopping point in considering the spine of fifths. Also note that the 5-note pentatonic and 7 note diatonic (B-flat major in this case) scales, appear in the sequence of scales generated in this way.
In many cases one says Gf and Fs are "close enough" to be considered equivalent, one of them is dropped, and you have a 12 note chromatic scale. Note that one can start at a different starting point (e.g. start on B) and get a 12 note scale where some of the notes are replaced with their enharmonic equivalents.
If you generate 12 tones by extending along (or outwards) the spine of fifths, you can generate 12 notes that might serve as a workable scale (I'm not sure to what extent pure Pythagorean tuning has been used in practice -- this theory is more applicable to 7 note diatonic scales), and the note names associated with those pitches will be in the sensible order.
Although rarely considered, one can continue extending the "line-of-fifths" to gneerate, e.g. G-sharp, or A-flat-flat and so on; it is formally it is infinite.
There is no reason why one couldn't start at a given pitch, and then grow outwards using an alternative "generator", e.g. the golden ratio as alluded to in the question to generate
and so on (I just made up symbols for these notes); aside from some numerical coincidences that one could look for, there is no reason to think that the pitches generated in this way will have any particular relationships to the pitches generated via Pythagorean tuning;
this even includes the idea of getting an approximate octave-equivalence that would allow one to close the "circle-of-phi" and thus have a workable number of scale degrees.
The following Python script demonstrates how to generate a table of relative pitches given a single generating interval.
while f > 2:
while f < 0.5:
p_steps=xrange(12) # generate a 12 note scale
p_basis=3.0/2.0 # pythagorean is generated by compounding just intonated fifthe
p_frequencies=[ octave_reduce( p_basis**p ) for p in p_steps]
# output the Pythagorean scale in of pitch, which is just their scale order
print [x for x in sorted( zip(p_frequencies, p_names))]
# one could make a golden ratio based scale
# in essentially the same way:
g_frequencies=[ octave_reduce( g_basis**p) for p in p_steps]