Schoenberg was the inventor of the 12-tone technique, which is generally considered the first way of writing truly atonally. Some of the rules of it were devised specifically in order to prevent any one pitch from dominating - such as having to state the tone row completely each time; avoiding octave doubling etc.
That's not quite the whole story, actually. Schönberg was only one of a number of composers who engaged in theorising about some form of pitch serialism at various times during the first quarter of the last century, Joseph Matthias Hauer, Alexander Scriabin and Nikolai Roslavets being just three others who, to one degree or another, considered something along these lines as a possibility; Schönberg just happened to go farther than any of the others in formalising it as a compositional technique. Even Liszt's as yet undiscovered (I think)
Prélude Omnitonique from the 1880s might well have been an even earlier experiment in this kind of approach.
Even those "rules of it [that] were devised specifically in order to prevent any one pitch from dominating" didn't and indeed couldn't always be guaranteed to function in practice as apparently intended because, ultimately, their results are inevitably to some degree dependent upon the subjective and experientially based responses of individual listeners - which is what I mean when I suggest that atonality and perceptions thereof are a matter of degree rather than something fixed that applies equally to all listeners.
More importantly still, especially in Schönberg's case even before his so-called free atonal" period (which was more or less from the Op. 11 piano pieces up to the Serenade op. 24), the most notable characteristic of the loosening of tonal bonds in what was nevertheless still tonal music (such as his D minor Quartet Op. 7 and E major Chamber Symphony Op. 9) was the prevention of any one
tonality from dominating at any given time and the sense of constant tonal flux arising from this kind of approach; this was a far more potent disturbance of traditional tonal progression than, say, the kind of "progressive tonality" found in Nielsen, Mahler and others' symphonic works that begin in one key and end in another not necessarily closely related one, although a sense of expanded tonality still pervaded.
I'm not sure what he meant if he described his music as not atonal. There is certainly a way of composing with a 12-tone row that incorporates a sense of tonality. Berg did that a lot, and some composers picked up on it after the war like Benjamin Frankel and Richard Rodney Bennett. There is a bit of that in some of Schoenberg's late works, but not in the ones when he'd first invented the technique.
You are right about much of this but I cannot personally agree with your final premise here because - to my ears, at least - such works as the Serenade and Suite for piano are full of tonal references and implications; they just don't have defined tonal centres. As I've suggested before, it's usually less a question of
whether a particular passage is tonal or atonal but
to what extent it might be perceived as one or the other by different listeners.
As has rightly been pointed out already, 12-note serial writing and atonality are not in any case synonymous. Furthermore, the use of a 12-note row does not of itself necessarily have to imply its serial treatment; I have myself used several 12-note rows in my writing but never treated them serially.
Best,
Alistair
Topic: Atonal music is crap