New Post from Brian McLaren

From: mclaren
Subject: Paul Hindemith & the harmonic series
--
Some while back Gary Morrison and others mentioned
Paul Hindemith's view of the harmonic series. This=20
issue is a xenharmonically important one because=20
Hindemith isn't the only composer with such odd
ideas...so it's worth discussing a little further.
From=20about 1880 through the 1940s it was popular
for many music theorists to add a dash of harmonic
series quasi-science to their teachings. The guy who=20
started this was Mersenne, followed by Rameau,
but the fellow who really kicked the trend=20
into high gear was Hermann Helmholtz, with help from=20
the experimental work of Michaelson, Lord Kelvin and=20
above all Dayton C. Miller.
Helmholtz proposed the notion that the human ear=20
performs a Fourier analysis on incoming sounds. This
is now known to be partly true for frequencies above
500 Hz, which leaves open the very daunting question:
what does the ear do with sounds below 500 Hz?
This question has still not been satisfactorily=20
answered, though there is overwhelming evidence
that more than one process is at work when the ear/
brain system hears a sound.
Helmholtz's theory, however, specifies *only* Fourier
analysis. This led several generations of researchers
to build mechanical Fourier analysis/resynthesis
systems: Lord Kelvin's tide predictor used 40 sine and
cosine terms added mechanically to predict tides,
while Michaelson's mechanical Fourier analyzer
reduced spectroscopic measurements to Fourier
coefficients. Far and away the most elaborate of
these systems, however, was Dayton C. Miller's
mechanical Fourier analysis/resynthesis system=20
operating on phonodeik-generated sound waveforms.
A music theorist perusing the cutting-edge scientific
literature on musical acoustics between 1900 and 1940
would have "learned" that the harmonic series is=20
fundamental to an understanding of music and=20
acoustics. The fact that the harmonic series was
used to analyze music primarily because it it had
the simplest and most straightforward mathematics=20
never seemed to occur to anyone--with a few lone
exceptions. See Llewellyn S. Lloyd's "Musical Theory
In Retrospect" in JASA, 1941, for a cogent critique
of the circular reasoning by which early 1900s musicians=20
and physicists concluded that the harmonic series=20
explained music, since all music when analyzed by=20
the Fourier transform is turned into a set of perfectly
harmonic frequencies. And since Fourier analysis
generates a set of perfectly harmonic frequencies,
therefore the harmonic series explains all music.
The problem with this way of thinking about music
and sound is that the method you're using (Fourier
analysis) is incapable of ever giving you *anything*
BUT a set of perfectly harmonic frequencies. If you
Fourier-analyze a set of inharmonic frequencies, your
output will be a set of perfectly harmonic frequencies--
albeit an infinite number of 'em. Fourier analysis,
because it's incapable of generating any output
other than a set of perfectly harmonic frequencies,
automatically *limits the types of sounds which can
be usefully analyzed*, and it also radically limits
the kinds of output you get from your analysis.
As a result, Fourier analysis drastically limits the kinds
of musical systems which can be usefully theorized
about. The problem here is that we've begun with an=20
assumption and then proceeded to use evidence derived
from it to prove that out-of-the-blue assumption, =20
with the predictable result that our logic is infallibly
mathematically true and thoroughly meaningless as
a method of probing physical reality. =20
Imagine it this way: you're a bee. You have faceted eyes.
You see the universe as facets. You develop an elaborate
theory which explains the universe in terms of facets.
Have you learned anything basic about physical reality?
Or are you merely the victim of your preconceptions,
having assumed the truth of what you set out to prove?
Appplying Fourier analysis to music produces the same
conundrum.=20
However, no music theorist realized this back at the start
of this century. (For a very telling early criticism of
the entire Fourier picture of sound, timbre and music,
see Danis Gabor's "Acoustical Quanta," 1947, Nature
(British journal), V. 4044) As a result, in the early 1900s it=20
was popular to "explain" exotic modern 12-TET music by=20
using the harmonic series. A typical example is Scriabin's
"mystic chord," which supposedly approximates members=20
11 and 13 of the harmonic series with a set of 12-tone
equal-tempered pitches. (If memory serves, the complete
"mystic chord" was 9:10:11:12:13. I may be incorrect.)
The fact that harmonics 11 and 13 were not well approximated
in 12-TET (harmonic 11 is about 50 cents away from
the nearest value available in 12) didn't seem to bother
Scriabin or his supporters. Such, apparently, was the
power of the 12-TET mindset...
"Hindemith's position may be classed with 'law of Nature'
theories, which seek to derive the procedures of musical
art from the objective properties of tone rather than from
the musical activities of people.=20
%=A0resulting natural
'laws' are presented to us as scientifically valid and
general enough to comprise the permanent and inevitable=20
aspects of the art. Musicians hesitate to question such
theories, first because it is unpopular to oppose the=20
claimed pronouncements of natural science, second
because few of them today are sufficiently prepared in
the relevant disciplines to be critical of details, and third
because they do not readily distinguish between proper=20
scientific forumlas and a species of mysticism dealing
with magic numbers.
"Hindemith's analysis begins with the familiar harmonic
series of overtones from the fundamental C, but he
discusses theoretical rather than actual overtones. (..)
"For we can 'discover' and 'prove' in the fertile harmonics
the natural origin of any musical relations whatsoever,=20
not to mention all the unmusical ones, though there is an
embarassing exception in the minor triad. (..)
"The most severe of Hindemith's troubles with Nature
occur in his account of the minor harmony, not so much=20
because Nature is notoriously coy about the matter, as=20
because Hindemith projects upon it a kind of reasoning
that should make even his scholastic predecessors wince. (..)
"Hindemith repeats some well-worn odes in praise of
just intervals, but does not investigate the facts. His
good ear notifies him of inflections, but he is not
equipped to define them properly, so that he never suspects
that Hindemith the violist plays quite normally in the=20
'abominable' Pythagorean system. As with mere facts,
so with logic, for our theorist first derides equal temp-
ermants in the manner of the least informed fiddle
teacher and elsewhere claims that we can hear only
in equal temperament. A pretty muddle is reached when
we learn that enharmonic differences in notation are
meaningless and futile and that these same enharmonic
differences in fact are also inesecapable and beautiful. (..)
"Hindemith does not study the real nature of music,=20
neither the nature of the material medium through which
the art of music comes alive, nor the nature of the human
purposes which have brought about every facet of the art.
His concern is with non-human mechanics of a non-
material world...=20
"Thus Hindemith does not give us Nature but in her stead
some more or less ingenious manipulations of cycles=20
and epicycles, of triangulations and of cubes, of alternate
multiplications and divisions...and above all the reflections
thereof in that wondrous and capacious subconscious which=20
is so convenient a repository, by definition, of everything
we do not know or cannot answer. (..)
"Hindemith prestends to be modern and scientific, and at every
turn he dazzles the unwary with thel anguage of mathematics,
of astronomy and of physics, and he even talks about=20
experiments. Yet he is not concerned with Nature but with
a revival of medieval speculation about magic numbers.
"Hindemith's 'experiments' consist of his private contemplation
of triangles and syllogisms, and he should not therefore hint to
us that his results come directly from the laboratory. (..)
"For the sake of clarity, the olden myths of musica mundana
should not be hidden in jargon taken from the natural=20
sciences." [Cazden, Norman, abstract of the paper "Hindemith
and Nature," read in Iowa City on April 18 1954; Journal
of the American Musicologycal Society, Vol. 7, Summer
1954, pp. 161-164]=20
An even weirder twist on this strange notion that=20
high members of the harmonic series could somehow
be "represented" by 12-TET pitches is the idea that
the harmonic series pitches were "out of tune"--rather
than the 12-tone equal tempered scale pitches!
Paul Hindemith uses both of these ideas in his derivation
of the 12 tone equal temperament. He then proceeds to
dismiss harmonic-series-based tunings as "out of tune"
because they deviate from the tuning of 12 tone equal
temperament (though as Cazden points out Hindemith
also praises the harmonic series effusively at other
points.)=20
In his 1931 doctoral thesis (republished as "Tuning and
Temperament in 1950) J. Murray Barbour commits the
same error--meantone and JI are "musically inferior"
because they contain large "errors"--errors measured
by comparing them with 12-tet!!!
This sounds incredible, but it was a common way of
theorizing about music between 1900 and 1940. Back
then, ethnomusicology hadn't really gotten off the
ground and there was very little understanding of the
deeply non-twelvular nature of non-western musics
throughout the world. Other cultures were presumed
to be "primitive" and their tuning systems were
assumed to be "crude approximations" either of
12 tone equal temperament, or some even simpler system
like 5-TET or 7-TET (which, most music theorists
assumed, were merely "precursors" of 12-TET). =20
It never seemed to occur to anyone between 1900 and
1940 that other cultures often performed music using
instruments which had inharmonic timbres and
therefore did not need tunings based on the harmonic
series. It never seemed to occur to anyone in
the beginning of this century that other cultures
might make music which was *more* sophisticated
than ours in certain ways, while our music was
more sophisticated than that of other cultures
in different ways. (For example, the idea that
African music was far more rhythmically complex
than western music seems to have been alien to
music theorists of the early part of this century.
Ditto the notion that other cultures often didn't break
up a 2:1 interval into small whole number ratios,
but instead added non-just non-equal-tempered=20
intervals cumulatively, producing scales which
are best analyzed as blocks of constant numbers=20
of Herz, rather than cents. See Ellis, C., "Pre-
Instrumental Scales," J. Ethnomusiology,
1962.)
The idea that higher harmonics above 6 might
somehow be "represented" by pitches in common
12-TET chords sounds very strange to us today,
but it was an extremely common idea in the
early part of this century. =20
Two currently-cited references which date from
the period 1900-1940 still contain these weird
ideas: J. Murray Barbour's "Tuning and
Temperament" and Joseph Yasser's "Theory of
Evolving Tonality." Barbour's book was published=20
in hardback in the 1950s, but was actually a
PhD thesis from the 1930s. =20
Joseph Yasser's "Theory of Evolving Temperament"
from 1932 also propounds this weird notion of
12-TET "containing" or "representing" high members
of the harmonic series like 13. Yasser mentions=20
Scriabin's "mystic chord" approvingly, & proposes an odd-
sounding and very dissonant hexad as the fundamental
"consonant" chord of 19-tone equal temperment:
the hexad consists of the 19-tone pitches
most closely approximating harmonics 8, 9, 10,
11, 13, 14 (if memory serves). Harmonic 12
was deliberately left out because it's 3 octaves
plus a perfect fifth, and since the perfect fifth
is 1/170 of an octave off from the harmonic-series
value in 19-tone equal temperament, Yasser
proscribed the use of that 19-tone interval.
(Yasser never explained how 9 could be obtained
if the harmonic 3 was proscribed. This, as Ivor
Darreg remarked, was one of the sacred Mysteries
of the Yasser religion.)
Today, of course, this all sounds incredibly weird.
No one to my knowledge composes in 19-tone
equal temperament according to Yasser's
strange hexad system, or his "supra-diatonic"
12-out-of-19 pitches. No one to my knowledge
who has heard and performed in 19-tet thinks
that Yasser's dissonant hexads are the most
consonant chords in 19-tet. (M. Joel Mandelbaum
did once compose a serial piece using 12-of-19=20
purely as a showpiece--a serial composition which
is atonal yet which *modulates.* Try *that,*
Schoenberg!) As Ivor Darreg and Easley
Blackwood have both pointed out, 19-TET
major and minor triads sound smoother
than the equivalent structures in 12-TET...
the 19-TET minor triad particularly so.=20
Why Yasser chose to categorize the 19-tet
perfect fifth as "dissonant" is simply
inexplicable...unless you look at the=20
era he came out of.=20
It's impossible to understand the strange
reasoning behind Yasser's, Barbour's, Hindemith's
and Scriabin's theories of intonation and=20
musical harmony without realizing that to
music theorists in the early part of this
century, the harmonic series was a mistuned
approximation of 12-tone equal temperament.
Seen from that standpoint, it's obvious why
the harmonic series is "useless" above the
6th harmonic--at that point the harmonic
series begins to sound extremely "out of tune"=20
with 12-TET.=20
To us, in the 1990s, this sounds incredibly
odd--as though someone were to explain
that clean air is a badly degraded form of smog.
Yet such musical-theoretic notions were common=20
from 1900 until 1948, when serialism took=20
over academia like an epidemic of Tulipomania.
--
Incidentally, you might think that modern music
theorists have put behind them the weird delusion
that harmonic series members are "out of tune"
12-TET pitches. Alas, such strange statements
are *still* being made in contemporary music=20
textbooks. The most glaring example I have found
recently is page 27 of "Materials and Structure
of Music, Vol. 1," 3rd edition, by Christ, DeLone,
Kliewer, Lowell and Thompson., 1988. On that page, ex.
2-10 shows "harmonic series on D." But the harmonic
series members 1-10 are depicted as 12-TET notes
with no indication that that the pitches of the 12-TET
are any different from the pitches of the "Natural=20
harmonic series"(!!!)
When William Alves suggested teaching music
theory by introducing the harmonic series, this
is probably *not* what he had in mind.
--mclaren


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