source file: mills2.txt Date: Tue, 31 Oct 1995 10:27:18 -0800 From: "John H. Chalmers" From: mclaren Subject: Models of reality, the Fourier mindset, and negative eigenvalues in Sturm-Liouville problems --- Charles Lucy has been regularly pilloried in this forum for his doubts about the Fourier analysis model of acoustic systems. Several months ago, in Topic 1 of Digest 404, John Chalmers also stated that "I can think of no physical principle which would favor flat fifths over natural..." This is a concise expression of a general attitude among acousticians and music theorists, most of whom are not fully conversant with the physics behind the conventional textbook description of vbrating strings and air columns. In fact there are good solid physical reasons why no simple harmonic mechanical oscillator ever produces strictly harmonic oscillations, and will instead tend to generate partials which are either systematically stretched or shrunk by comparison with the expected harmonics. There are also excellent physical reasons why 2- and 3-dimensional physical oscillators will in general not exhibit either linear or harmonic oscillatory behavior at all. This point bears on Lucy's criticism of the Fourier analysis mindset that characterizes much of the discussion of this forum, and in a larger sense it also bears on the question of which tunings we use--or even contemplate using. Ultimately, the musics we choose to make are limited by our model of the world. Whenever we tune an instrument, we inevitably begin to impose a structure on physical reality: and the type of structure we choose to impose is determined by our preconceptions. In the West and Mideast, ever since the time of Pythagoras and almost certainly (before that) the Babylonians and the Egyptians, we (in the West) have conceived of the universe as ruled by number. The predictive value of number is a touchstone of classical Hellenic thought. This had profound implications for the kind of music made in Greece and the European cultures. However, other cultures bring other preconceptions to the process of tuning. There is no evidence (as Marc Perlman has pointed out) that the Javanese, the Balinese, or any of the other cultures of Southeast Asia conceive of the universe as a manifestion of number, except perhaps in the Kabalist sense implied by gematria, the geomancy of the Dogon peoples of Mali, etc. Thus their tuning systems do not arise from mathematical or physical-acoustic considerations, and as a result Javanese and Balinese music does not employ the 2:1 octave, 4:5:6 harmonies or harmonic-series timbres. The same is true of the musics found in Africa, Central Asia, most of the South Seas islands and South America. None of these cultures appear to conceive of music in the mathematical framework typical of Western thinking; indeed, as Jon Appleton has pointed out, in many other cultures music is often not analyzed at all, but regarded as belonging to same realm as magic. Thus one's preconceptions exert a potent influence on the kind of music one chooses to make, and different models of reality produce different kinds of tuning. The simple harmonic oscillator equation is one model of reality. There are many others even within the confines of Western mathematics. According to the simple harmonic motion equation, the displacement x = A*sin(sqrt(k/m)*t) + B*cos(sqrt(k/m)*t) where k is the spring constant, m is the mass, and t is the elapsed time (A and B are constants of propotionality). This is a recipe for perfectly harmonic behaviour. But the equation which describes simple harmonic motion is a drastic simplification of reality, and there are many other (more sophisticated) mathematical models which describe the same physical oscillatory system. For example, when a simple tube is excited with a laminar airflow the air in the tube will oscillate at a characteristic frequency determined by the length of the tube, the density of the air, the location and number of tone holes, etc. Increasing the airflow produces a stronger second harmonic, and so on, according to this model. But in the real world two modes of oscillation (fundamental and second harmonic) in a tube tend to "pull in" toward each other so that the fundamental rises slightly in frequency, while the second harmonic drops slightly in frequency. The simple harmonic motion model of reality cannot explain this because it views air molecules as billiard balls connected by springs, and the system cannot behave in any but a perfectly harmonic way. If instead we look at the air-filled tube as an energy exchange system from a thermodynamic viewpoint, or if we view it as a state-space system, or if we consider it from the standpoint of continuum mechanics, the reasons for the slight inharmonicity of the oscillations become obvious. Setting up a fundamental mode of oscillation in the the tube changes its acoustical admittance; a second oscillation mode will then lose energy or gain energy, depending on the phase of its compressions and rarefactions relative to that of the fundamental mode. Since the system will tend to minimize its overall potential energy the two acoustic modes will couple; and because the system is closed (negigibly energy is lost via acoustic radiation), the energy lost from the second mode of oscillation has to go somewhere and not all of it will be either transmitted out the bore of the tube or lost as friction against the walls or in the friction created by turbulent non-laminar flow around the tone holes, etc. Thus the energy lost from the second mode of oscillation will partly flow into the fundamental mode of oscillation, which in turn forces a change in its period. This example serves as a warning about the models we use to explain physical system, and consequently to justify the tuning systems we use. In the case just considered, the inharmonic behaviour is slight--as long as flow of air through the tube remains slow and laminar. Thus it can be handled by perturbation theory, as in lord Rayleigh's "Acoustics" of 1896. But as the airflow increases, the flow becomes non-laminar. Multiphonics appear; then the oscillation patterns become quasi-periodic, start to break up, and finally become completely aperiodic. The oscillation turns to noise. Viewed as a simple energy exchange system, our model cannot explain this. But if we step back yet again and realize that the energy-exchange system we've been picturing is a drastic simplification the reason for this behaviour becomes clear. If, instead, we view the partial differential equations which describe the interaction of non-linear acoustic admittance, non-laminar airflow and radiated, transmitted and absorbed mechanical energy, and boundary conditions from the viewpoint of complexity theory... Then it becomes obvious why the system beahves as it does. "Little is known concerning boundary value problems for general nonlinear differential equations..." [Courant and Hilbert, "Methods of Mathematical Physics," 1962, pg. 367. For the class of hyperbolic PDEs which describe one-dimensional physical oscillatory systems, perturbation theory is the classical method of dealing with the increased airflow described above. But as we've seen this does not work beyond a very limited regime. Nonlinear dynamics must be invoked beyond the region of non-laminar airflow, and in this regime the oscillations in the tube move from being ordinary attractors in phase space to being strange attractors, whose behaviour not only jumps back and forth between aperiodicity and quasi-peridiodicity, but also spans the complete gamut from pure noise to harmonic oscillation. Viewed in phase space, the operation of the compressions and rarefactions in the air of the tube follow orbits which can be bounded, yet not precisely predicted, and which depend crucially on initial conditions. Lest you imagine such chaotic behaviour is restricted only to airflow in tubes, note that in the case of the wave equation describing the general nonhomogeneous vibrating string "whenever an eigenvalue is negative an aperiodic motion occurs instead of the corresponding normal mode." [Courant and Hilbert, "Methods of Mathematical Physics," Vol. 1, pg 292] Thus chaotic oscillation can occur even in vibrating strings--in fact in any one-dimensional oscillator described by a Sturm-Liouville eigenvalue problem. This is not commonly known. Why? Because every textbook on acoustics which treats the wave equation and the vibrating string assumes that the egienvalues will never become negative. As a result, the true complexity of the behjvaiour of even so-called "simple" 1-D physical oscillatory systems is masked from the unquestioning students. (These simplifying assumptions are made so that the equations can be quickly and easily solved. The usual dodge is to claim that "negative eigenvalues have no physical significance." As so often in physics and engineering, this is a fudge. In fact the problem is just systematically restricted and our viewpoint successively limited until we arrive at equations which undergrad-level mathematics can dispatch with elegance in a neat closed form.) And what does all this have to do with tuning? The debate on this forum has mostly centered around a restricted set of musical tunings-- JI, meantone, a few equal temperaments which well approximate the lower members of the harmonics series. And this tiny subset of tunings is derived from simplistic physical models (as we've seen). These acoustic models are described by the usual textbook solutions of the wave equation, simple harmonic motion, and the rest of the 18th- and 19th-century baggage. But the reality can be very different from the universe predicted by these 18th-century mathematical models. "For three centuries science has successfully uncovered many of the workings of the universe, armed with the mathematics of Newton and Leibniz. It was essentially a clockwork world, one characterized by repetition and predictability. (..) Most of nature, however, is nonlinear and is not easily predicted. (..) In nonlinear systems small inputs can lead to dramatically large consequences." [Lewin, Roger, "Complexity," 1993, pg. 12] By now it should now be clear why the Fourier transform has gained such popularity, and also why Charles Lucy's doubts about it are well-founded. In "a clockwork world, one characterized by regularity and predictability," a mathematical technique which breaks all physical oscillations down into sets of perfectly periodic sinusoidal functions with no beginning and no end makes a lot of sense. In a clockwork 18th-century universe, the Fourier transform enjoys enormous descriptive power. But we now know that the 18th-century clockwork model of the world is not a complete picture. In the real world, where chaotic strange attractors characterize the action of real oscillatory systems, the Fourier transform often falls apart. And instead of describing reality, the Fourier transform can put blinders on us and prevent us from seeing the world as it really is. In some cases, this is unimportant--because the oscillations of real physical systems are sometimes only a little different from the simplistic clockwork-universe description of the Fourier Theorem. In instruments like strings, winds and brasses (after the initital attack of the tone is over, and if the instruments aren't played too loudly, and if we're talking only about the notes in the middle range of these instruments) a short-time Fourier analysis tells us *something* about what's going on in *some* of the notes, during *part* of their duration. But even for this restricted class of sounds, Fourier techniques fail during the first 10 milliseconds or so of the note's attack. FFTs fail because when t < 10ms the amplitude and frequency of the component partials are both changing with great rapidity, and the Fourier transform *cannot* provide accurate information about both the period *and* the spectrum of an input function. An increase in resolution in one parameter forces a descrease in resolution in the other. This is inherent in the mathematics of the FFT, and *cannot* be sidestepped. A much greater limitation on the Fourier mindset is the fact that most of the musical instruments used by most of the cultures in the world are not violins or French horns or flutes. Most of the musical instruments used by other cultures are two- or three-dimensional oscillators which generate inharmonic partials & noise, and store energy from one vibrational cycle to the next... so that their behaviour is often extraordinarily non-linear. See Rossing's discussion of the energy storage from one cycle to the next in a tam-tam, for example. [Rossing, 1992] For such instruments, the Fourier description is a hindrance rather than an aid to understanding. And the class of tunings to which we in the west have systematically restricted ourselves--namely JI, meantone and equal temperaments with good approximations of the lower harmonics--are also inadequate for such instruments. This perhaps addresses John Chalmers' doubts about the validity of any "physical system that would favor flat fifths over natural." Simply moving from one-dimensional physical oscilaltors to two- and three-dimensional oscillators generate fifths which are either *very* flat or *very* sharp (depending on the oscillatory geometry)...indeed, the whole 18th- and 19th-century armamentarium of Western acoustic terminology is inapplicable to such physical oscillators: "fifth" and "third" and "natural harmonics" are terms without meaning for such physical systems. The tam-tam or the metallophone or the vibrating drumhead or (as we've seen) even purportedly "simple" one-dimensional systems like the vibrating string often operate in the region of complexity... a region of oscillation that lies between the complete chaos of noise and the clockwork perfection of perfect harmonicity. The Fourier view of the universe does not yield useful information when applied to such acoustic systems, or even when applied to a vibrating string characterized by negative eigenvalues in the associated Sturm-Liouville equations. One of the central revelations of complexity theory is that patterns lie hidden in chaos. Emergent order appears ex nihilio when systems reach the edge of chaotic behavior, as in woodwind multiphonics, etc. In fact the Brookhaven National Laboratory phsyicist Per Bak has developed the hypothesis that dynamical systems naturally evolve toward a critical state in which the edge of chaos spontaneously generates order. [See Bak, P. and Chen K,. in Scientific American, Jan. 1991; also see Packard, N., "Adaptation Toward the Edge of Chaos," Technical Report, Center for Complex Systems Research, University of Illinois, CCSR-88-5, 1988.] What does this have to do with tuning & music? It seems possible (if not probable) that non-just non-equal-tempered tunings represent an adaptation toward spontaneous order generated by the non-linear dynamical systems used in so many other musical cultures (i.e., two- and three-dimensional physical oscillatory systems: drums, flat or curves metal plates, non-linearly coupled oscillators like those used in parts of Africa in conjuction with resonant strings, etc). Interestingly enough, throwing away the Fourier transform does not mean a loss of predictive power. Many other analytic models for acoustic systems exist: Walsh transforms, Daubechies wavelets, Gabor's acoustic quantum, and even more recent non-linear mathematical transforms such as the slope transform [See Maragos, P., "Slope Transforms: Theory and Application to Nonlinear Signal Processing," IEEE Trans. Sig. Proc., 43(40, Paril 1995, pp. 864-877.] The fact that so astute and insightful a thinker as John Chalmers could fall into the trap of looking at all tuning systems and physical oscillators through the narrow distorting lens of the Fourier transform is an indication of the power the Fourier mindset to brainwash the unwary. While the Fourier transform is a marvellous mathematical tool, is does not describe all of acoustic reality-- only a small part of it. Thus Charles Lucy's doubts about the universal value of the FFT are well founded, and the attacks he has suffered for voicing these doubts in this forum are an indication of just how thoroughly the Fourier mindset can blind us to the wonderfully complex nature of real instruments, real tunings and real music in the real world. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 31 Oct 1995 20:59 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA08873; Tue, 31 Oct 1995 10:58:57 -0800 Date: Tue, 31 Oct 1995 10:58:57 -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