  We present a spectroscopic characterisation and a new orbital solution for
the binary system beta Cyg Aa/Ac (MCA 55), the primary component (beta Cyg A)
of the well-known wide double star Albireo. By matching evolutionary tracks to
the physical parameters of all three Albireo stars (beta Cyg Aa, Ac and B) as
obtained from a spectroscopic analysis of TIGRE and IUE spectra, we confirm
that they are likely coeval. Our final orbit solution is based on
radial-velocity measurements taken over a baseline exceeding $120$ years,
combined with relative astrometry from speckle interferometric observations and
the absolute astrometry from the Hipparcos and Gaia missions. Our final orbit
solution has a period of $121.65^{+3.34}_{-2.90}$ years with an eccentricity of
$0.20^{+0.01}_{-0.02}$. Thanks to the inclusion of the absolute astrometry, we
find a mass ratio of $q = 1.25^{+0.19}_{-0.17}$, and a total mass of
$9.47_{-3.24}^{+5.88}$ M$_\odot$, indicating that the secondary (Ac) is the
more massive of the pair. These results strongly suggest the presence of a
fourth, unseen, member of the Albireo system. Given the current photometric
data it is likely that beta Cyg A is itself a hierarchical triple. We also
derive the systemic proper motion, line-of-sight velocity, and an orbital
parallax of the beta Cyg A system, allowing us to quantitatively assess the
hypothesis that Albireo A and B form a physically bound and genealogically
connected system. Finally, we find four potential members of a common proper
motion group with Albireo, though none anywhere as close by as the Albireo
components A to B.
