To start out: I am more-or-less familiar with all the standard musical clef notations, including F, C, and G clefs; as well as the fact that clefs can be movable (tenor clef, french violin clef...) and have octave displacements (8va/8vb). I'm also aware of the unpitched percussion clef. With the exception of the later, these are all ways to map staff positions to absolute pitches. (I should also mention that in the case of transposing instruments, like Clarinets or Horns, this absolute pitch may not be the pitch it first appears to be.)
My question is whether anyone has ever seen a clef that maps staff positions to relative pitches (i.e. scale degrees) rather than absolute pitches. I realize this would be of limited use, especially for performers, who need to know which note on their instrument to play. But I could see such a system being useful in certain harmonic or analytical contexts, where the absolute pitch is unimportant (for example, notating chord progressions, or melodic motifs, irrespective of key). I could also see it being used for initial steps of transcriptions done by-ear, where the key isn't immediately important.
The "work around" is just to pick some specific key (which may or may not be correct, and which may often end up being 'C'), and just notate in that key. That works fine, I suppose, but it makes an explicit statement about key which may not be accurate or relevant. Of course, there are also numbers and solfege symbols which can be used, but these use alphanumeric characters, rather than standard music notes, and thus lose rhythmic information. One similar thing I've seen is shape-notes, in which the shape of note heads indicate scale degree. But this is non-standard notation practice (thankfully!), and it still uses normal clefs to indicate absolute pitches. I suppose transposing instruments are probably closest to what I'm looking for, if they were notated without a specific transposition in mind.
I have not been able to find any information on Google about the existence of such a clef, so I presume it doesn't exist, but I might not be using the right keywords, or it might be very obscure, so I'm wondering if anyone else has ever come across something like this? I'd be mildly surprised if no music theoretician anywhere has ever proposed some such system.