Ok, gotta remove multiples of factors of twelve, so 2 (whole), 3 (dim), 4 (aug), 6 (tritone), 8 and 9 are out.

That leaves 1, 5, 7, 10, and 11, plus all the variants from adding n*12 (e.g. 13 or 25). So my bold assertion didn’t work out too well.

For example, there are only three distinct diminished “loops”. These each define tonal centers because each pair of tritones uniquely identifies a 5 chord. You can view a fully diminished chord as a “gateway” of sorts into 4 different keys, depending on what root note you slap on it.