An interval is the distance between two pitches. We name the major intervals based on the number of note names they contain. Since the note letter names have been assigned such that in the C-major scale the 7 notes contain no sharps or flats, it seems that intervals are inherently linked to the C-major scale, not to absolute differences in pitch.
For example, the major third (M3) going from C to E contains 4 semi-tones (C->D=2; D->E=2). In contrast, the M3 going from E to G contains only 3 semi-tones (E->F=1; F->G=2). Thus, it would seem that a M3 is not an absolute unit of distance but tied to the underlying assumption of a "default" to C-Major.
Now my question: my music theory book states an M3 contains 2 steps and a M6 contains 4.5steps. I don't understand. If "step" refers to alphabetical notes, the M3 definition makes sense, but the M6 does not. If a step refers to two semi-tones, then neither definition makes sense. As noted above, an M3 could contain either 3 or 4 semitones (1.5 or 2 steps??) depending on which note you start at. Similarly, an M6 could contain either 9 semi-tones (C-A) or 10 semi-tones (E-C) (4.5 or 5 steps??).
What does my textbook mean by "step"? Why does it imply that intervals are absolute differences in pitch when the number of semi-tones in any given interval varies? Are my example incorrect in some way?