Hi Anthony,

>         \transposition c''

What you are looking for here is \transpose:
http://lilypond.org/doc/v2.18/Documentation/notation/changing-multiple-pitches#transpose

> Now I want also the time signature "C" to disappear... I've read the doc
> <http://lilypond.org/doc/v2.18/Documentation/notation/typesetting-mensural-music#mensural-time-signatures>,
> but don't see how.

\omit was introduced in 2.18:
http://lilypond.org/doc/v2.18/Documentation/changes/index.html

Documentation:
http://lilypond.org/doc/v2.18/Documentation/notation/visibility-of-objects

> Is there a "\omit" ?

you can basically \omit anything as soon as you know what the object it
is called:

\omit TimeSignature

If you don’t know what to look for to find the name, this page could help:
https://joramberger.de/files/lilypond_visualindex.pdf

HTH,
Joram

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