  Frames for Hilbert spaces are interesting for mathematicians but also
important for applications e.g. in signal analysis and in physics. Both in
mathematics and physics it is natural to consider a full scale of spaces, and
not only a single one. In this paper, we study how certain frame-related
properties, as completeness or the property of being a (semi-)frame, of a
certain sequence in one of the spaces propagate to other spaces in a scale. We
link that to the properties of the respective frame-related operators, like
analysis or synthesis. We start with a detailed survey of the theory of Hilbert
chains. Using a canonical isomorphism the properties of frame sequences are
naturally preserved between different spaces. We also show that some results
can be transferred if the original sequence is considered, in particular that
the upper semi-frame property is kept in larger spaces, while the lower one to
smaller ones. This leads to a negative result: a sequence can never be a frame
for two Hilbert spaces of the scale if the scale is non-trivial, i.e. spaces
are not equal.
