TUNING digest 1384

On Tue, 14 Apr 1998 02:50:50 -0400 (EDT) tuning@eartha.mills.edu writes:

>Re: Topic No. 3


After my citation of Partch's criticism of Schoenberg's explanation of
the 12-eq chromatic scale as the first 13 partials of "I, IV and V" on C,
Paul Erlich wrote:

>> Please continue, to where Partch shows that two notes which
>> Schoenberg assigns to the same 12ET note are in fact 99
>> cents apart. I lent my copy of
>> Genesis to a friend, so I don't have the reference handy.

Continuing on p. 418 ["Genesis of a Music" 2nd ed., 1974], Partch says:

"...140.5 cents (13th [partial] of "F") and 51.3 cents (11th [partial] of
"G"),
separated by 99.2 cents, or virtually a whole "semitone", are presumed to
be the same tone, "Db"."

Partch's point is that the assignment of partials to 12-eq
representations by Schoenberg has a margin of error which nearly equals
the smallest interval in the 12-equal scale, so that the partials could
just as easily be represented by the next nearest 12-eq note as by the
one assigned by Schoenberg.
--------------------------------------------------------------------------------------
Further in the discussion, regarding over-/ undertones and dualism,
Erlich wrote:

>> ...It may have been Max Meyer who used the term "hallucination",
>> I apologize if I misattributed that word.

Meyer is indeed the one who called undertones "hallucinations". As his
book has been long out of print and is hard to find even in libraries, I
give the relevant passage in full (from Meyer, "The Musician's
Arithmetic", p.88-90 -- the idiosyncratically frequent use of italics is
Meyer's):

-----------------------------------------------

"While the pretense of Rameau that his _dualistic_ theory of harmony
(major-minor) had a _scientific_ basis had been more or less _challenged_
by his contemporary, the _physicist_ D'Alembert, a later _physicist_, A.
von Oettingen ["Harmoniesystem in dualer Entwicklung", 1866], not only
_defended_ the logic of Rameau so far as mere dualism is in question, but
even _tried to strengthen the dualism_ by a new logical derivation of the
"symmetry" upon which the notion of dualism is based.

Let us at once point out, however, that the major-minor theory of
Oettingen is worse than Rameau's in this respect: that it fits even
fewer, indeed only an exceedingly small minority of, cases of musical
experience.

Reasoning from the fact that Rameau proposed 1-3-5 as the make-up of the
major chord with 1 as the chief tone, and 3-5-15 as the make-up of the
minor chord with 5 as the chief tone, Oettingen very unreasonably invites
the reader to _form the reciprocals of the latter three numbers_, that
is, 1/3, 1/5, 1/15. Oettingen then gives these fractions a common
denominator, 15, and considers then only the numerators 5, 3 and 1 (in
5/15, 3/15 and 1/15); and he invites the reader to conclude that, because
1 thus sprang from 15 and because 1 was _the chief_ tone in the other
(major) case, therefore the chief tone of the minor chord 3-5-15 must be
the tone 15, _the father of the chief_ 1. This would mean, for example,
that among the tones C-A-E heard the tone E would have to be the chief
tone. Oettingen comes very near rewriting all the scores of all the
previously composed music in minor, for the purpose of correcting all the
"mistakes" made by the composers who were ignorant of his logical
discovery. Musicians, of course, have not been willing to sacrifice
their quite different musical experiences to Oettingen's "logic".

Hugo Riemann ["Musikalische Syntaxis", 1877] alone among the musicians
was captivated by Oettingen's attempt at imposing the duality notion of
Rameau more strongly than ever upon musical thinkers, -- captivated to
such a degree that he even had hallucinations; for at one time he
asserted the existence of "undertones", corresponding to those
reciprocals mentioned in the previous paragraph, as the acoustical
counterpart of "overtones". The major scale would then be governed by
overtones, the minor scale by undertones. What a beautiful "symmetry"!
And not only a mysterious and speculative symmetry, but a truly
acoustical symmetry, strictly scientific! Undertones versus overtones!

Unfortunately (for Riemann) acoustical science knows nothing of
undertones. These hallucinatory tones would have to be mechanically
_multiplex_ tones in the same sense in which overtones are mechanically
_partial_ tones. But this would require the imagination of a sort of
fourth dimension for space. A volume of mass under tension can indeed
vibrate in parts of itself. This is plain mechanics. But how could it
vibrate in multiples of itself? The multiples would have to lie in the
fourth, the invisible dimension.

>From all this nonsense we should learn this lesson: We ought to abstain
altogether from mixing up acoustics (which is nothing but a branch of
mechanics) with musical theory (which concerns itself with reactions of
organisms equipped with nervous tissue), notwithstanding the fact that on
occasions a knowledge of acoustics can be very valuable to certain
friends and helpers of the musician, that is, to the builder of musical
instruments and to the physician treating larynx troubles."

-----------------------------------------------

Hopefully this last bit will provide more fodder for debate among tuning
folk.

It should be observed that although Partch almost always agrees with
Meyer, and although he found room in "Genesis" to criticize Oettingen
without making particular note of the similarities of his own theories
with Oettingen [see p. 389], he does in fact utilize exactly Oettingen's
numerical method of building Utonalities.

Partch's criticism of Oettingen is only that Oettingen "held that unity
for the 'minor' triad was found in the first partial common to all three
tones. This is of course the second 2/1 above the highest tone in the
triad: 3/2 in the traid 1/1 - 6/5 - 3/2...", and that "Oettingen failed
to consider difference tones of the minor...". Otherwise, Partch placed
the same significance as Oettingen in the reciprocal proportions of the
"minor" chord, but without the acoustical allusions.

Joseph L. Monzo
monz@juno.com
4940 Rubicam St., Philadelphia, PA 19144-1809, USA
phone 215 849 6723

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