There is one misconception here: Music practice does not follow theory. Theory follows practice, and practice follows physics. You cannot just invent a note-naming system and expect to write pleasing music in it, the note-naming system must follow what is actually pleasing to hear, and what is pleasing to hear is dictated by the physics of sound.
You see, the point about the octave is not that it is some kind of convention. The point is, that one octave equals a factor 2 in frequency. This factor 2 is the reason why all the overtones in the spectrum of the higher note fall squat on the overtones of the lower note, allowing the two notes to sound together as one.
The fourth and fifth intervals are likewise fixed by very simple fractions of frequency (the fifth is almost 3/2, the fourth 4/3). The fact that we have 12 semitones is due to the fact that
2^19 = 524288 is roughly the same as
3^12 = 531441. This allows us to close the circle of fifths. This is the first point where a power of 3 comes close enough to a power of 2 to not sound way off. You cannot just add a 13th/14th semitone to the octave without destroying this relationship.
Furthermore, the circle of fifth is the basis for choosing which of the semitones to use within a scale. I won't go into details here, as they are a bit too involved for this answer.
Bottom line is, you cannot ignore physics. Our 12 semitones to an octave "convention" follows directly from physics. So...
If you add your
H by relabeling one of the sharp/flat notes, the octave would remain unchanged, and thus sound the same.
If you instead add your
H by putting more semitones within an octave, the "octave" will sound really off.