Webern, parallelograms, and quarter-tones (!)

On p 12-13 of _The Path to the New Music_, Webern describes
the regular 'major' scale in the same way Schoenberg did in
his _Harmonilehre_ [1911, English trans. p 24], in terms of
the 3rd and 5th partials of the harmonic series built on the
'I', 'IV', and 'V'. It may be portrayed as the simple rational
tonal lattice:

. 9:8
. ' /
15:8 /
/ '-._ /
/ . 3:2
/ . ' /
5:4 /
/ '-._ /
/ . 1:1
/ . ' /
5:3 /
'-._ /
4:3

This description of the major scale is quite old, going back
to approximately Rameau [*]

Webern says:

[Webern, _The Path to the New Music_, p 12-13]
>
> ... the fifth [degree of the scale = the 3rd partial] is the
> first obtrusive note, that is to say it has the strongest
> affinity with the tonic. This implies that the latter note
> has the same relationship with the one a fifth lower. So here
> we have a kind of parallelogram of forces, "equilibrium" is
> produced, there is a balance between the forces pulling upwards
> and downwards.

There is nothing especially new in this, either: it is a part
of Schoenberg's _Harmonielehre_ description too [p 23], and also
has precedents, particularly in 19th-century German theory. [**]

But what struck me was the term Webern actually used to
refer to this tonal structure: a parallelogram. He's not
using it in quite the same way Fokker did in describing his
'periodicity blocks', but it think it's an interesting
premonitionary 'parallel'.

A couple of pages later:

[Webern, p 15]
>
> ...notes are natural law as related to the sense of hearing.
> ... - So, a note is...complex - a complex of fundamental and
> overtones. Now, there has been a gradual process in which music
> has gone on to exploit each successive stage of this complex
> material. This is the one path: the way in which what lay to
> hand [i.e., the paradigm of the ratios of the lowest overtones]
> was first of all drawn upon, then what lay farther off....we
> find an ever growing appropriation of nature's gifts! The
> overtone series must be regarded as, practically speaking,
> infinite. Ever subtler differentiations can be imagined, and
> from this point of view there's nothing against attempts at
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> quarter-tone music and the like; the only question is whether
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> the present time [1933] is yet ripe for them. But the path is
> wholly valid, laid down by the nature of sound. So we should
> be clear that what is attacked today is just as much a gift of
> nature as what was practised earlier.

[emphasis added by me]

So the composer who was held up by later advocates as the paradigm
of 12-tone serialism liked the idea of 'quarter-tone' [= 24-tone]
music!

I'm not very familiar with microtonality in Austria in the 1930s,
but I would think that Webern is referring mainly to Haba and
his followers.[***]

On the next few pages Webern discusses the quest for the discovery
and expression of 'unity' in the presentation of the musical idea,
and I couldn't stop imagining Partch's Tonality Diamond in my mind
as I read.

Upon re-reading this little book now, I see that there's material
here that provides quite fertile ground for an imaginative
'marriage' of serialism and JI microtonality.

-monz

[*] More specific info would be appreciated, because I know
that Rameau's major scale had 10:9 instead of 9:8, so this
lattice is not accurate for his theories.

(see Rameau, 1722, _Traite de l'Harmonie_, English trans. by
Philip Gossett, 1971, _Treatise on Harmony_, Dover, p 28.)

[**] Again, more specific info is requested. (I believe
Daniel Wolf knows more about this than anyone else posting
to this List.) I'm particularly interested in the history
of this 'equilibrium of forces' idea.

This was also an aspect of Rameau's theory, which I have
seen referred to as the 1:3:9 polarity. Rameau alludes to
it in the 1722 _Traite_ as a 4:6:9 '9th' chord [p 38 in the
English trans.], but does not seem to discuss it further in
that early book. Apparently it is explained in more detail
in one of his later treatises. (so Rameau experts are also
sought).

I would express it as 3^-1 : 3^0 : 3^1 .

[***] I'd definitely appreciate knowing more about this!
(... Daniel? Johnny Reinhard? Manuel?) I'd like to know just
how familiar Webern was with these 'attempts', since he was
congenial to them. I have Moldenhaur's very detailed Webern
biography, but not much about the microtonal goings-on of the
period and place.

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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