more on Schoenberg and rational implications

I was resisting saying any more today about this, but
Patrick's response to Graham really got me going.
(There will be a lot about Schoenberg in my forthcoming
"JustMusic" book, but I've written so much on him
specifically that there will ultimately be a another whole
book: "Rational Implications in Schoenberg's Harmonic
Theories").

[Graham Breed:]
>> ... I see no need to explain [Schoenberg's]
>> general style as anything but an _avoidance_ of
>> integer ratios.

[Patrick Ozzard-Low:]
> But a greater 'avoidance' (say, using 11- or 13-ET) would
> never have provided Schoenberg with the language he
> neeeded. So it surely can't be as simple as that (?)

[Monzo:]
Touch�! Great reply, Patrick!

In fact, Schoenberg intended to represent loads and
loads of ratios, and to explore the ambiguities between
them which he felt he had available by making use of
the approximations to ratios in the 12-eq scale.

[Schoenberg, from "Problems of Harmony" [1934], in
_Style and Idea_, p. 284:]
> "...interrelationship of all tones exists...because of
> their derivation from the first 13 overtones of the
> three fundamental tones [1/1, 3/2, and 4/3]..."

[Monzo:]
Schoenberg had already illustrated a partially-chromatic
scale which implied overtones thru the first 12 (i.e., an
11-limit) back in 1911 in _Harmonielehre_.

[Schoenberg, _Theory of Harmony_ [1911], p. 21:]
> "...the tempered system [meaning 12-eq], which is only
> an expedient for overcoming the difficulties of the material,
> has indeed only a limited similarity to nature. This is perhaps
> an advantage, but hardly a mark of superiority."

[Monzo:]
Throughout his _Harmonielehre_, Schoenberg postulates
an extreme fluidity of proportional implication in the 12-eq
scale, explaining all the traditional harmonic concepts in
terms of 5-limit, then progressing to higher odds/ primes
(but without further mathematical explanation) as his
interpretations of harmonic practice became more
adventurous in the chapters on "Non-Harmonic" Tones,
the Whole-Tone Scale, Chords Constructed in 4ths, and
finally Chords With Six or More Tones.

By "difficulties of the material", Schoenberg was
referring to the difficulty of accomodating the vastly
expanded pitch resources offered by an 11- or
13-limit to the practical concerns of scale size,
hand/ finger size and other instrumental limitations,
notation, etc.

The 12-eq scale is the smallest ET which gives even
an adequate approximation of 3-limit ratios, and 6-eq
and 9-eq give the same deviation from 5-limit ratios
as 12-eq, so 12-eq is the smallest ET that gives tolerably
good approximations to the ratios which are implied
in the (theoretically 5-limit) musical repertoire
Schoenberg knew.

Without using any numbers (these are my interpretation),
he stated in one of his articles that the music being
composed by he and his pupils was intended to make
use of the rational harmonic relations which were as yet
unfamiliar to listeners, that is, ratios with prime factors
higher than 5. He said that those in his circle were
focussing specifically on these new relations and
avoiding the old ones:

[Schoenberg, "Twelve-Tone Composition" [1923], in
_Style and Idea_, p. 207:]
> "In twelve-tone composition consonances (major and
> minor triads) and also the simpler dissonances (diminished
> triads and seventh chords) -- in fact almost everything
> that used to make up the ebb and flow of harmony --
> are, as far as possible, avoided."

[Monzo:]
He then immediately defended this statement:

[Schoenberg, same page:]
> "...this is not becasue of any natural law of the new art.
> It is, presumably, just one manifestation of a reaction,
> one that does not have its own special causes but derives
> from another manifestation -- which it tries to contradict,
> and whose laws are therefore the same, basically, as its
> own."

[Monzo:]
I believe what Schoenberg meant here was that the
higher-prime or higher-odd relationships follow the same
compositional procedures as the lower numbers, but
that _because they are different_, try to "contradict"
the tonal implications of the lower primes/ odds.

In other words, for example, the septimal "major 3rd"
of 9/7 embodies vastly different _harmonic_ relationships
than the 5-limit "major 3rd" of 5/4.

He then explains this "manifestation of a reaction":

[Schoenberg, same page:]
> "...At the root of all this is the unconscious urge to
> try out the new resources independently, to wrest
> from them possibilities of constructing forms, to produce
> with them alone all the effects of a clear style, of a
> compact, lucid and comprehensive presentation of
> the musical idea. To use here the old resources in
> the old sense saves trouble -- the trouble of cultivating
> the new -- but also means passing up the chance of
> enjoying whatever can *only* be attained by new
> resources when the old ones are excluded!"

[Monzo:]
Taking this literally says to me that he _should_ have
been willing to consider other ETs which represent
these higher-prime ratios better than 12-eq, even if
their approximations of 3 and 5 aren't good. 11-eq
gives a decent approximation of 7/4 and and a good
one of 11/8, with nothing resembling a 3/2 or a 5/4.
13-eq gives an even better approximation of 11/8,
and one of 13/8 which is quite good, with an OK 5/4
but lousy 3/2 and even worse 7/4. So both of these
small ETs (which Ozzard-Low probably picked pretty
arbitrarily from the possibilities available) would have
been a good choice for Schoenberg if his practice
was true to the letter of his description.

He went on to say that "a later time will perhaps"
allow composers a harmonic language which would
mix the old 3- and 5-limit relationships with the newer
ones of 7, 9, and 11. (I'm not certain if Schoenberg had as
yet included 13 in his rational implications -- there's no
evidence of this until 1934).

However, Schoenberg was a complicated man whose
writings and compositions frequently embody paradoxes.
He stipulated many times that he was "evolutionary,
not revolutionary" -- in other words, that his musical
advances were not a break with the past, but on the
contrary, the logical outgrowth of what had come
before (in the Austro-German tradition). It's been pointed
out by many writers that Schoenberg's actual music
retained far more of Brahms's influence than that of
his "more radical" pupil Webern (although I disagree with
the assessment of Webern as such a radical).

Schoenberg was clearly interested in expansion,
not replacement. This would seem to argue in favor
of his retention of 12-eq, as he certainly did not want
to eliminate completely the 3- and 5-limit rational implications
which were so important to so-called "tonal" music, and
which would have happened had he accepted 11- or
13-eq. A favorite quote from his old age is "there is still
plenty of good music to be written in C major".

Also worth examing is Schoenberg's self-appellation
"pantonal", which he coined in angry opposition to
the popularly-accepted epithet "atonal". His
argument was that since any sounds giving
recognizable pitches exhibited numerical
relationships, some kind of tonality (defined as a
mathematical centricity around a certain tone) was
automatically built in to any collection of pitches. He
preferred to think of his style as making use of all
tonalities at once, rather than as making use of no
tonality at all.

Joseph L. Monzo
monz@juno.com

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