source file: mills2.txt Date: Sun, 24 Nov 1996 08:19:06 -0800 Subject: New Post from Brian McLaren From: John Chalmers 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 Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 24 Nov 1996 17:21 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05999; Sun, 24 Nov 1996 17:22:47 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA05935 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA19527; Sun, 24 Nov 1996 08:22:44 -0800 Date: Sun, 24 Nov 1996 08:22:44 -0800 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu