First let's consider that it is Schoenberg who asks the question and uses the word "harmony". In is book THEORY OF HARMONY he defines (pag.13) harmony as
"...the study of simultaneous sounds (chords) and of how they may be
joined with respect to their architectonic, melodic, and rhythmic
values and their significance, their weight relative to one another."
It's a pretty broad definition, but the insertion of the word "chord" seems to indicate that tonal harmony is what's on his mind (and it is, indeed, mostly what the book is about).
Now this is a relatively early work from Shoenberg (written in 1910, before he developed dodecaphony and all that), so he could have broadened the definition later on to include alternative non tonal systems. However, from what little I know about the theory of dodecaphony, the word "harmony" is not used to describe that system. Words like "system", or the preferred word by Shoenberg, "method" are used. Shoenberg defined dodecaphony as a
"Method of composing with twelve tones which are related only with
one another" [Schoenberg, Arnold. 1975. As quoted in
The word "harmony" does not enter the definition, which seems to indicate that he still associated the word "harmony" with classical tonal harmony.
It's also important to remember that for Shoenberg atonalism and dodecaphony were an evolution from tonal harmony, which remained an important theoretical basis to support the composer's efforts:
"Schoenberg viewed his development as a natural progression, and he
did not deprecate his earlier works when he ventured into
Taking all this into consideration it seems to me that when asking the question in 1934 or 35, Shoenberg considered the meaning of "harmony" to be traditional tonal harmony. It seems reasonable that after two years studying with the master, there would be some common understanding of language and such critical concepts, so we can admit (even if it is Cage who is quoting Shoenberg some 20 odd years later) that in his answer Cage uses the word in the same sense.
To reinforce this view there's a very interesting paper by James Tenney, "John Cage and the Theory of Harmony" (1983). Tenney was a music theorist with a particular interest in the work of John Cage, so his saying is probably relevant in the interpretation of Cage's meaning.
In Tenney's view,
"...the very meaning of the word “harmony” has come to be so narrowly
defined that it can only be thought of as applying to the materials
and procedures of the diatonic/triadic tonal system of the last two or
And according to him, Cage thought of harmony as being a constraint or barrier (the "wall" he himself talks about in the OP's original quote) that he should escape from:
"There were many such walls, but “harmony” — in its narrowest sense (the materials and procedures of traditional, tonal, textbook harmony)
— was for Cage a particularly obstructive one:
[now quoting from Cage himself]
"Harmony, so-called, is a forced abstract vertical relation which blots out the spontaneous transmitting nature of each of the sounds
forced into it. It is artificial and unrealistic. (1954)"
It seems that Tenney's view was also that when Caged used the word "harmony" he meant mainly "traditional, tonal, textbook harmony".
It seems that by (purposely?) not using the word "harmony" to describe different systems of composition, both Shoenberg and Cage want to make explicit the freedom from the rules of "traditional" harmony. However, whereas Shoenberg still considers "harmony" as a building block for learning and to evolve to other systems, Cage (as already commented by Craig Curtis) wants to discard it altogether.
BTW, Tenney's paper is rather interesting in his attempt to broaden the concept of harmony to cover different systems to organize pitch in music.
"The word ["harmony"] has a very long and interesting history,
however, which suggests that it need not be so narrowly defined, and
that the “continued evolution of the theory of harmony” might depend
on — among other things — a broadening of our definition of